Harmonic Morphisms from Fefferman Spaces

Abstract

We study a ramification of a phenomenon discovered by Baird and Eells (in: Looijenga et al (eds) Geometry Symposium Utrecht 1980. Lecture Notes in Mathematics, Springer, Berlin, 1981) i.e. that non-constant harmonic morphisms $\Phi : {{\mathfrak {M}}}^{\textrm{N}} \rightarrow N^2$ from a $\mathrm N$-dimensional ($\textrm{N} \ge 3$) Riemannian manifold ${{\mathfrak {M}}}^{\textrm{N}}$, into a Riemann surface $N^2$, can be characterized as those horizontally weakly conformal maps having minimal fibres. We recover Baird–Eells’ result for $S^1$ invariant harmonic morphisms $\Phi : {{\mathfrak {M}}}^{2n+2} \rightarrow N^2$ from a class of Lorentzian manifolds arising as total spaces ${{\mathfrak {M}}} = C(M)$ of canonical circle bundles $S^1 \rightarrow {{\mathfrak {M}}} \rightarrow M$ over strictly pseudoconvex CR manifolds $M^{2n+1}$. The corresponding base maps $\phi : M^{2n+1} \rightarrow N^2$ are shown to satisfy $\lim _{\epsilon \rightarrow 0^+} \, \pi _{{{\mathscr {H}}}^\phi } \, \mu ^{{{\mathscr {V}}}^\phi }_\epsilon = 0$, where $\mu ^{{{\mathscr {V}}}^\phi }_\epsilon$ is the mean curvature vector of the vertical distribution ${{\mathscr {V}}}^\phi = \textrm{Ker} (d \phi )$ on the Riemannian manifold $(M, \, g_\epsilon )$, and $\{ g_\epsilon \}_{0< \epsilon < 1}$ is a family of contractions of the Levi form of the pseudohermitian manifold $(M, \, \theta )$.

1 Harmonic Morphisms in Riemannian Geometry Versus Lorentzian Geometry

A harmonic morphism is a continuous mapping $\Phi : {{\mathfrak {M}}}^{\textrm{N}} \rightarrow N^m$ of semi-Riemannian manifolds $({{\mathfrak {M}}}, \, g)$ and $(N, \, h)$ such that for every solution $v: V \rightarrow {{\mathbb {R}}}$ to $\Delta _h v = 0$, defined on an open subset $V \subset N$, the pullback $u = v \circ \Phi$ is a distribution-solution to $\Delta _g u = 0$ on $U = \Phi ^{-1} (V)$, where $\Delta _g$ and $\Delta _h$ are the Laplace–Beltrami operators of the given semi-Riemannian manifolds. Considerable attention has been given to the study of harmonic morphisms within the Riemannian category, and the main results up to 2003 were reported on in the wonderful monograph by Baird and Wood [5]. By a result of Fuglede (cf. [34]) every non-constant harmonic morphism of Riemannian manifolds is an open map. Remarkably, Fuglede’s result relies on the Harnack inequality for positive harmonic functions on a Riemannian manifold (cf. e.g. Serrin [59]) thus establishing a solid bond between the geometry of harmonic morphisms and elliptic theory (cf. e.g. [39]). However, harmonic morphisms $\Phi : {{\mathfrak {M}}}^2 \rightarrow N$, from a Lorentzian surface ${\mathfrak {M}}$ into a semi-Riemannian manifold, that aren’t open maps, do exist (cf. [5, pp. 446–448]). For morphisms from Fefferman spaces we may state (leaving definitions momentarily aside)

Theorem 1

Let M be a strictly pseudoconvex CR manifold, equipped with the positively oriented contact form $\theta \in {{\mathscr {P}}}_+ (M)$, and let N be a Riemannian manifold. Any nonconstant $S^1$ invariant harmonic morphism $\Phi : C(M) \rightarrow N$ from the total space of the canonical circle bundle $S^1 \rightarrow C(M) \rightarrow M$, endowed with the Lorentzian metric $F_\theta$ [the Fefferman metric of $(M, \, \theta )$], is an open map. Moreover, if M is compact and N is connected then N is compact and $\Phi$ is surjective.

As another contrasting feature of the semi-Riemannian case, harmonic morphisms of semi-Riemannian manifolds may be non smooth. Indeed the proof that a continuous harmonic morphism $\Phi : {{\mathfrak {M}}} \rightarrow N$ of Riemannian manifolds is actually $C^\infty$ relies (cf. [5, p. 111]) on two ingredients i.e. (i) the existence of harmonic local coordinate systems on the target manifold N, and (ii) the hypoellipticity of the Laplace–Beltrami operator $\Delta _g$ of $({{\mathfrak {M}}}, \, g)$, itself following from the ellipticity of $\Delta _g$. The known proof of the existence of harmonic local coordinates is tied (cf. DeTurck and Kazdan [22]) to the ellipticity of $\Delta _h$, although harmonic local coordinate systems on Lorentzian manifolds were used in spacetime physics as early as the work by Lanczos (cf. [47]) and Einstein himself (cf. [30]), yet without questioning their existence. Moreover, if say $({{\mathfrak {M}}}, \, g)$ is a Lorentzian manifold, then its Laplace–Beltrami operator is the geometric wave operator $\square _g$ which is not hypoelliptic. For morphisms from Fefferman spaces we may state

Theorem 2

Any continuous $S^1$ invariant harmonic morphism $\Phi : C(M) \rightarrow N$ from the Lorentzian manifold $(C(M), \, F_\theta )$ into the Riemannian manifold N is smooth.

Unique continuation (cf. [5, pp. 111–112]) doesn’t hold for harmonic maps and morphisms of semi-Riemannian manifolds (cf. [5, p. 448]). It should also be mentioned that J.H. Sampson’s unique continuation theorem for harmonic maps of Riemannian manifolds (cf. Theorem 1 in [58, p. 213]) relies on a unique continuation result for solutions to elliptic equations due to N. Aronszajn (cf. [2]) whose proof is believed to be wrong, cf. Appendix A in [25, pp. 433–434], (although the very result in [2] may hold true, at least for solutions to $\Delta _g u = 0$). Apart from a brief conjectural discussion in $\S \, 6$, unique continuation of subelliptic harmonic maps and morphisms will be addressed in further work.

By a celebrated result of Baird and Eells (cf. [3]) a smooth non-constant horizontally weakly conformal map $\Phi : {{\mathfrak {M}}}^{\textrm{N}} \rightarrow N^m$ of Riemannian manifolds is a harmonic morphism if and only if

$$\begin{aligned} (m - 2) \, {{\mathscr {H}}} \big \{ \nabla \log \lambda (\Phi ) \big \} + (N - m ) \, \Phi _*\, \mu ^{{\mathscr {V}}} = 0, \end{aligned}$$

(1)

where $\lambda (\Phi )$ and $\mu ^{{\mathscr {V}}}$ are respectively the dilation of $\Phi$ and the mean curvature vector of its fibres. Cf. also [16] for the case $m = \textrm{N} - 1$. In particular, if the target manifold is a real surface ($m = 2$) then harmonic morphisms $\Phi : {{\mathfrak {M}}} \rightarrow N^2$ have minimal fibres. The case where ${\mathfrak {M}}$ is Lorentzian has not been studied, and a Lorentzian analog to the fundamental equation

$$\begin{aligned} \tau _g (\Phi ) = - (m - 2) \, \Phi _*\, \big \{ \nabla \log \lambda (\Phi ) \big \} - (N - m) \, \Phi _*\, \mu ^{\mathscr {V}}, \end{aligned}$$

(cf. (4.5.2) in [5, p. 120]) and to the characterization (1) are not known, so far.

Given a Riemannian manifold $N^m$, the purpose of the present paper is to analyze harmonic morphisms $\Phi : {{\mathfrak {M}}}^{2n+2} \rightarrow N^m$ from the total space ${{\mathfrak {M}}} = C(M)$ of the canonical circle bundle $S^1 \rightarrow {{\mathfrak {M}}} \rightarrow M^{2n+1}$ over a strictly pseudoconvex CR manifold M, of CR dimension n. Here M is equipped with a fixed positively oriented contact form $\theta$, so that ${\mathfrak {M}}$ is a Lorentzian manifold with the corresponding Fefferman metric $g = F_\theta$. The discussion is confined to $S^1$ invariant harmonic morphisms $\Phi$ of $({{\mathfrak {M}}}, \, F_\theta )$ into $(N^m, \, h)$, whose associated base maps $\phi : M \rightarrow N$ turn out to be subelliptic harmonic morphisms, in the sense of Dragomir and Lanconelli [25]. Our result in this direction is

Theorem 3

Let M be a strictly pseudoconvex CR manifold, of CR dimension n, equipped with the positively oriented contact form $\theta \in {{\mathcal {P}}}_+ (M)$, and let $(N, \, h)$ be a m-dimensional Riemannian manifold. Let $\Phi : C(M) \rightarrow N$ be a continuous $S^1$ invariant map, and let $\phi : M \rightarrow N$ be the corresponding base map. The following statements are equivalent

(i) $\Phi$ is a harmonic morphism of the Lorentzian manifold $(C(M), \, F_\theta )$ into $(N, \, h)$, of square dilation $\Lambda (\phi ) \circ \pi$.

(ii) $\phi$ is a subelliptic harmonic morphism of the pseudohermitian manifold $(M, \, \theta )$ into $(N, \, h)$, of $\theta$-dilation $\sqrt{\Lambda (\phi )}$.

If this is the case then

(a) $\Phi$ is nondegenerate at p $\Longleftrightarrow$ $\pi (p) \in \Omega (\phi ):= M {\setminus } Z \big [ \Lambda (\phi ) \big ]$.

(b) $p \in \textrm{Crit} (\Phi ) \Longleftrightarrow \pi (p) \in \textrm{Crit} (\phi )$.

(c) $\Phi$ is degenerate at p $\Longleftrightarrow$ either $m = 1$ and $\pi (p) \in \textrm{II}_1 (\phi )$, or $m \ge 2$ and $\pi (p) \in M {\setminus } S(\phi )$.

(d) $\Phi$ is a harmonic map of the Lorentzian manifold $(C(M), \, F_\theta )$ into the Riemannian manifold $(N, \, h)$, while $\phi$ is a subelliptic harmonic map of the pseudohermitian manifold $(M, \theta )$ into $(N, \, h)$.

(e) $\Phi$ is horizontally weakly conformal, while $\phi$ is Levi conformal.

(f) If $m = 2$ i.e. $(N, \, h)$ is a real surface, then every leaf of the pullback foliation $\pi ^*\, {{\mathscr {F}}}$ of $S(\Phi )$ [the foliation of $S(\Phi )$ tangent to ${{\mathscr {V}}}^\Phi$] is a minimal submanifold of $\big ( C(M), \, F_\theta \big )$.

The equivalence (i) $\Longleftrightarrow$ (ii) in Theorem 3 was first observed by Barletta (cf. [6]) for the particular case of the Heisenberg group $M = {{\mathbb {H}}}_n$. The more general case at hand is treated in Section $\S \, 4$ of the present paper.

Let $\mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla \big ) \in C^\infty \big ( {{\mathscr {H}}}^\phi \big )$ be defined by formally replacing the Levi-Civita connection $\nabla ^{g_\theta }$ (of the Webster metric $g_\theta$) by the Tanaka–Webster connection $\nabla$ (of the pseudohermitian manifold $(M, \, \theta )$) in the ordinary mean curvature vector $\mu ^{{{\mathscr {V}}}^\phi } \equiv \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla ^{g_\theta } \big ) \in C^\infty \big ( {{\mathscr {H}}}^\phi \big )$ of the vertical distribution ${{\mathscr {V}}}^\phi = \textrm{Ker} (d \phi )$ thought of as a distribution on the Riemannian manifold $(M, \, g_\theta )$. Let $\Phi = \phi \circ \pi$ be the vertical lift of $\phi$ to the total space C(M) of the canonical circle bundle over M, equipped with the Fefferman metric $F_\theta$. To some surprise, while the tension field $\tau _{F_\theta } (\Phi )$ projects on the pseudohermitian tension field $\tau _b (\phi )$, the square dilation $\ell (\Phi )$ is the vertical lift of the square dilation $\Lambda (\phi )$, and gradients with respect to $F_\theta$ project on horizontal gradients on $(M, \, \theta )$, the term $\Phi _*\; \mu ^{{{\mathscr {V}}}^\Phi }$ (appearing in the fundamental equation (50)) doesn’t project on $\phi _*\; \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla \big )$, as one might have hoped for, to start with. In a quest for the “correct” pseudohermitian analog to the mean curvature vector of ${{\mathscr {V}}}^\phi$, we endow M with the family

$$\begin{aligned} g_\epsilon = g_\theta + \Big ( \frac{1}{\epsilon ^2} - 1 \Big ) \, \theta \otimes \theta , \;\;\;\; 0< \epsilon < 1, \end{aligned}$$

(2)

of contractions (in the sense of Strichartz [60]) of the Levi form $G_\theta$, and analyze the behavior of $\mu ^{{{\mathscr {V}}}^\phi }_\epsilon$ [the mean curvature vector of ${{\mathscr {V}}}^\phi$ as a distribution on the Riemannian manifold $(M, \, g_\epsilon )$] in the limit as $\epsilon \rightarrow 0^+$. The family of Riemannian metrics $\{ g_\epsilon \}_{0< \epsilon < 1}$ is devised such that $\big ( M, \, d_\epsilon \big ) \rightarrow \big ( M, \, d_H \big )$ as $\epsilon \rightarrow 0^+$, in the Gromov–Hausdorff distance. Here $d_\epsilon$ and $d_H$ are respectively the distance function of the Riemannian manifold $\big ( M, \, g_\epsilon \big )$, and the Carnot–Carathéodory distance function associated to the sub-Riemannian structure $\big ( H(M), \, G_\theta \big )$ (the maximally complex distribution of the CR manifold M, equipped with the Levi form, cf. [29, 60]). A comparison to the works by Barone-Adesi et al. [13], Cheng et al. [20], Malchiodi et al. [19], Danielli et al. [21], Garofalo et al. [37] and Pauls et al. [38], may reveal $\mu ^{{{\mathscr {V}}}^\phi }_{\textrm{hor}}:= \lim _{\epsilon \rightarrow 0^+} \, \mu ^{{{\mathscr {V}}}^\phi }_\epsilon$ as the appropriate candidate for the mean curvature vector [of a leaf of the foliation ${\mathscr {F}}$ tangent to ${{\mathscr {V}}}^\phi$, as a submanifold of the pseudohermitian manifold $(M, \, \theta )$]. For the time being, we establish (in the spirit of the work by Ni [53])

Theorem 4

Let $\phi : M^{2n+1} \rightarrow N^2$ be a non-constant subelliptic harmonic morphism, of the pseudohermitian manifold $(M, \, \theta )$ into the real surface $(N, \, h)$. Let $\mu ^{{{\mathscr {V}}}^\phi }_\epsilon$ be the mean curvature vector of ${{\mathscr {V}}}^\phi$, as a distribution on the Riemannian manifold $(M, \, g_\epsilon )$. Then $\pi _{{{\mathscr {H}}}^\phi } \; \mu ^{{{\mathscr {V}}}^\phi }_\epsilon \rightarrow 0$ as $\epsilon \rightarrow 0^+$, uniformly on any relatively compact domain $U \subset M$.

We revisit the notion of horizontal mean curvature of a real hypersurface in a Carnot group (cf. Capogna et al. [18], Capogna and Citti [17]) in the context of subelliptic harmonic morphisms $\phi : M^{2n+1} \rightarrow N^1$ from a pseudohermitian manifold $(M, \, \theta )$ into a $m = 1$ dimensional Riemannian manifold $N^1$. We compute the horizontal mean curvature of every leaf of the foliation ${\mathscr {F}}$ by real hypersurfaces of $S(\phi )$ [where, by Theorem 6, $S(\phi ) = M {\setminus } \textrm{Crit}(\phi )$, an open set] determined by the submersion $\phi : S(\phi ) \rightarrow N^1$. Precisely, let $\{ g_\epsilon \}_{0< \epsilon < 1}$ be the family of contractions of $G_\theta$ given by (2), and let ${{\mathscr {H}}}^\phi _\epsilon$ be the $g_\epsilon$-orthogonal complement of ${{\mathscr {V}}}^\phi$ in $(T(M), \, g_\epsilon )$. Let $\textbf{n}^\epsilon \in C^\infty \big ( {{\mathscr {H}}}^\phi _\epsilon \big )$ such that $g_\epsilon \big ( \textbf{n}^\epsilon \,, \, \textbf{n}^\epsilon \big ) = 1$. The horizontal normal $\textbf{n}^0$ is

$$\begin{aligned} \textbf{n}^0 = \frac{1}{g_\epsilon \big ( \nu ^\epsilon \,, \, \nu ^\epsilon \big )^{1/2}} \, \nu ^\epsilon \,, \;\;\;\; \nu ^\epsilon \equiv \Pi _H \, \textbf{n}^\epsilon = \textbf{n}_\epsilon - \theta \big ( \textbf{n}_\epsilon \big ) \, T, \end{aligned}$$

and the horizontal mean curvature $K_0$ of the leaves of ${\mathscr {F}}$ is

$$\begin{aligned} K_0 = \textrm{div} \big ( \textbf{n}^0 \big ) \in C^\infty ( \Omega ), \end{aligned}$$

where $\Omega = M \setminus Z(\Lambda )$ (an open set) and the divergence is computed with respect to the volume form $\Psi = \theta \wedge (d \theta )^n$. The horizontal normal and mean curvature are well defined on $\Omega$ because $\Sigma ({{\mathscr {F}}}) \subset Z(\Lambda )$ [by Theorem 6 below, and our discussion in $\S \, 7$] where $\Sigma ({{\mathscr {F}}})$ is the set of all characteristic points of the leaves of ${\mathscr {F}}$.

Theorem 5

Let $\phi : M \rightarrow N^1$ be a subelliptic harmonic morphism, of square dilation $\Lambda$. Then

(i) For every local coordinate system $(V, \, y^1 )$ on N such that $U = \phi ^{-1} (V) \subset \Omega$

$$\begin{aligned} \textbf{n}^0 = \frac{1}{\sqrt{\Lambda _0}} \, \nabla ^H \phi ^1 \,, \;\;\; \Lambda _0 = \frac{\Lambda }{h_{11} \circ \phi } \,, \;\;\; \phi ^1 = y^1 \circ \phi , \end{aligned}$$

so that

$$\begin{aligned} K_0= & {} \textrm{div} \Big ( \frac{1}{\sqrt{\Lambda _0}} \, \nabla ^H \phi ^1 \Big ) \nonumber \\= & {} - \frac{1}{\sqrt{\Lambda _0}} \, \Big \{ \Delta _b \phi ^1 + \big ( \nabla ^H \phi ^1 \big ) \, \log \sqrt{\Lambda _0} \Big \}, \end{aligned}$$

(3)

everywhere in U.

(ii) The vector field

$$\begin{aligned} \mu ^{{{\mathscr {V}}}^\phi }_{\textrm{hor}} \equiv \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla \big ) + \frac{1}{2n} \, \pi _{{{\mathscr {H}}}^\phi } \; \Big \{ \frac{2}{\theta ({{\mathscr {T}}})} \, J {{\mathscr {T}}} - \nabla _{{\mathscr {T}}} {{\mathscr {T}}} \Big \} \in C^\infty \big ( {{\mathscr {H}}}^\phi \big ), \end{aligned}$$

and the mean curvature $K_0$ are related by

$$\begin{aligned} 2 n \, g_\theta \big ( \mu ^{{{\mathscr {V}}}^\phi }_{\textrm{hor}} \,, \, \textbf{n}^0 \big )= & {} \big \{ \varphi \, T(\phi ^1 ) - 1 \big \} \; K_0 \,, \nonumber \\ \varphi ^2 \big \{ \Lambda _0 + T(\phi ^1 )^2 \big \}= & {} 1 - \theta ({{\mathscr {T}}} )^2 \,, \;\;\; {{\mathscr {T}}} = \Vert T^{{\mathscr {V}}} \Vert ^{-1} \, T^{{\mathscr {V}}}. \end{aligned}$$

(4)

Consequently

$$\begin{aligned} 2n \, \mu ^{{{\mathscr {V}}}^\phi }_{\textrm{hor}} = \alpha \, \nabla \phi ^1 \, , \;\;\; \alpha := - \frac{\Delta _b \phi ^1 + \sqrt{\Lambda _0} \, K_0}{\Lambda _0 + T(\phi ^1 )^2}. \end{aligned}$$

(5)

In particular, for any local harmonic coordinate system $(V, \, y^1 )$ on N [i.e. $\Delta _h y^1 = 0$ in V] with $U = \phi ^{-1} (V) \subset \Omega$

$$\begin{aligned} 2 n \, \Pi _H \, \mu ^{{{\mathscr {V}}}^\phi }_{\textrm{hor}} = - \frac{\Lambda _0}{\Lambda _0 + T(\phi ^1 )^2} \, K_0 \, \textbf{n}^0, \end{aligned}$$

(6)

everywhere in U.

By a result in [5, p. 448], to every non-constant harmonic morphism $\Phi : C(M) \rightarrow N$ of class $C^\infty$, there corresponds a symbol $\sigma _p (\Phi ): \big ( T_p (C(M)), \; F_{\theta , \, p} \big ) \rightarrow \big ( T_{\Phi (p)} (M), \; h_{\Phi (p)} \big )$ which is a harmonic morphism (this may fail if $\Phi \in C^{\ell }$ for some $2 \le \ell < \infty$ yet $\Phi \not \in C^{\ell + 1}$). The CR structure on M induces a natural strictly pseudoconvex CR structure on the tangent space $T_{x} (M)$ at every point $x = \pi (p)$, yet the properties of the symbol $\sigma _{x} (\phi ): T_{x} (M) \rightarrow T_{\phi (x)} (N)$ of a subelliptic harmonic morphism are not known, so far.

The paper is organized as follows. Section 2 recalls the essentials of CR and pseudohermitian geometry (by following [29]) and of subelliptic harmonic maps $\phi : M \rightarrow N$, from a pseudohermitian manifold $(M, \, \theta )$ into a Riemannian manifold $(N, \, h)$ (cf. [9, 44]). The needed material on subelliptic harmonic morphisms is reviewed in Sect. 3 (cf. [7, 25]). Theorems 1, 2 and 3 are proved in Sect. 5. The Lorentzian and pseudohermitian ramifications of the result by Baird and Eells (cf. [3]) are treated in Sect. 3, where we also prove Theorem 4. In Sect. 7 we prove Theorem 5. Sect. 8 exhibits a few examples i.e. subellitic harmonic morphisms from the Heisenberg group and from Rossi spheres. The study of the properties of the symbol of a subelliptic harmonic morphism $\phi : M \rightarrow N$ is relegated to a further paper.

Given a 3-dimensional nondegenerate CR manifold $M^3$, let $\Phi : {{\mathfrak {M}}}^4 \rightarrow N^2$ be a harmonic morphism from a 4-manifold ${{\mathfrak {M}}}^4$ equipped with the Lorentizan metric g, into the Riemann surface $N^2$, such that the vertical spaces ${{\mathscr {V}}}^\Phi _p = \textrm{Ker} \big ( d_p \Phi \big )$ are nondegenerate for every $p \in {{\mathfrak {M}}}^4$, and let ${{\mathscr {H}}}^\Phi$ be the g-orthogonal complement of ${{\mathscr {V}}}^\Phi$. By a result of J. Ventura (cf. [62]) the Ricci curvature of $\big ( {{\mathfrak {M}}}^4, \, g \big )$ may be computed in terms of i) the (square) dilation of $\Phi$, ii) the first fundamental forms of ${{\mathscr {H}}}^\Phi$ and ${{\mathscr {V}}}^\Phi$, iii) the second fundamental forms of ${{\mathscr {H}}}^\Phi$ and ${{\mathscr {V}}}^\Phi$ and their adjoints, iv) the sectional curvature of the fibers of $\Phi$, v) the mean curvature of ${{\mathscr {H}}}^\Phi$, vi) the mean curvature of the fibers of $\Phi$, and vii) the integrability 1-form of ${{\mathscr {H}}}^\Phi$. Let $g^{{\mathscr {V}}}$ and $g^{{\mathscr {H}}}$ be the bundle metrics induced on ${{\mathscr {V}}}^\Phi$ and ${{\mathscr {H}}}^\Phi$, respectively. Given $C^\infty$ functions $\sigma , \, \rho : {{\mathfrak {M}}}^4 \rightarrow (0, \, + \infty )$ the Lorentzian metric

$$\begin{aligned} \tilde{g} := \frac{1}{\sigma ^2} \, g^{{\mathscr {H}}} + \frac{1}{\rho ^2} \, g^{{\mathscr {V}}}, \end{aligned}$$

(7)

is a biconformal deformation of g. The Einstein equation

$$\begin{aligned} \textrm{Ric}_{\tilde{g}} = \Lambda \, \tilde{g}, \end{aligned}$$

(8)

recasts as a PDE system in the unknown functions $\sigma$ and $\rho$, and solving (8) for $\sigma$ and $\rho$ amounts to producing solutions (Einstein metrics) $\tilde{g}$ by biconformal deformations of an a priori given Lorentzian metric g. The approach was devised by Ventura (cf. [62]) and the method was applied to a number of space-times and morphisms $\Phi : {{\mathfrak {M}}}^4 \rightarrow N^2$ e.g. for the Schwarzschild metric g and a projection $\Phi$ from the Schwarzschild space-time onto $N^2 = S^2$. Cf. also Baird and Ventura [4], where the approach is however confined to the case of a Riemannian 4-manifold ${{\mathfrak {M}}}^4$. Given a positively oriented contact form $\theta$ on $M^3$, the Fefferman metric $g = F_\theta$ is never Einstein (cf. Lee [49]). The curvature calculations in [62] and the resulting attempt to solve (8) is then liable to produce Einstein metrics on the total space ${{\mathfrak {M}}}^4 = C(M^3 )$ of the canonical circle bundle over M, by a biconformal deformation [associated to a given harmonic morphism $\Phi$ from $C(M^3 )$ into a Riemann surface] of the Fefferman metric.

A similar problem was solved (outside harmonic morphisms theory) by Leitner (cf. [50]) who built pseudo-Einstein (cf. [29]) contact forms $\theta$ of vanishing pseudohermitian torsion, and observed that the corresponding Fefferman metric is conformally Einstein i.e. there is a $C^\infty$ function $\sigma : C(M^3 ) \rightarrow (0, \, + \infty )$ such that $\tilde{g} = \big ( 1 / \sigma ^2 \big ) \, F_\theta$ is an Einstein metric [cf. (7) with $\sigma \equiv \rho$, as $F^\theta = F_\theta ^{{\mathscr {H}}} + F_\theta ^{{\mathscr {V}}}$ for any harmonic morphism $\Phi$ with a nondegenerate vertical distribution]. The results in [50, 62] were not paralleled so far. It is an open problem, suggested by the Reviewer, to build examples of Einstein metrics on $C(M^3 )$ by a biconformal deformation of the Fefferman metric $F_\theta$, associated to a contact form $\theta$ that is neither pseudo-Einstein nor transversally symmetric (and compensating said obstructions by an appropriate choice of harmonic morphism $\Phi$).

2 Subelliptic Harmonic Maps

For notations, conventions and basic results in CR and pseudohermitian geometry, we follow the monograph by Dragomir and Tomassini [29]. Let M be a strictly pseudoconvex CR manifold, of CR dimension n, equipped with a positively oriented contact form $\theta$, and let N be a m-dimensional Riemannian manifold, with the Riemannian metric h. Let $T_{1,0}(M) \subset T(M) \otimes {{\mathbb {C}}}$ be the CR structure on M and let $H(M) = \textrm{Re} \big \{ T_{1,0} (M) \oplus T_{0,1} (M) \big \}$ be the corresponding maximally complex, or Levi, distribution. Here $T_{0,1}(M) = \overline{T_{1,0}(M)}$ and overbars denote complex conjugates. Let $J: H(M) \rightarrow H(M)$ be the natural complex structure i.e.

$$\begin{aligned} J \big ( Z + \overline{Z} \big ) = \sqrt{-1} \big ( Z - \overline{Z} \big ), \;\;\; Z \in T_{1,0}(M). \end{aligned}$$

Let $H(M)^\bot \subset T^*(M)$ be the conormal bundle i.e. the real line bundle

$$\begin{aligned} H(M)^\bot _x = \big \{ \omega \in T_x^*(M) \;: \; \textrm{Ker} (\omega ) \supseteq H(M)_x \big \}, \;\;\; x \in M. \end{aligned}$$

As is well known (cf. e.g. [29, pp. 8–9]) under the mild assumption that M is orientable, the conormal bundle is trivial i.e. $H(M)^\bot \simeq M \times {{\mathbb {R}}}$ (a vector bundle isomorphism). The set of all globally defined nowhere zero $C^\infty$ sections in $H(M)^\bot$ is denoted by ${{\mathscr {P}}}(M)$. For every $\theta \in {{\mathscr {P}}}(M)$ let $G_\theta$ be the Levi form i.e.

$$\begin{aligned} G_\theta (X, Y) = (d \theta )(X, \, J Y), \;\;\; X, Y \in H(M). \end{aligned}$$

Cf. [29, pp. 5–7]. Let ${{\mathscr {P}}}_+ (M)$ denote the set of all $\theta \in {{\mathscr {P}}}(M)$ such that $G_\theta$ is positive definite. By its very definition, strict pseudoconvexity of the given CR structure $T_{1,0}(M)$ is equivalent to ${{\mathscr {P}}}_+ (M) \ne \emptyset$. A contact form $\theta \in {{\mathscr {P}}}_+ (M)$ is termed positively oriented. Let us consider the functional $E_b: C^\infty (M, \, N) \rightarrow {{\mathbb {R}}}$ given by

$$\begin{aligned} E_b (\phi ) = \frac{1}{2} \int _\Omega \textrm{Trace}_{G_\theta } \Big ( \Pi _H \, \phi ^*\, h \Big ) \; \theta \wedge (d \theta )^n. \end{aligned}$$

Here $\Omega \subset \subset M$ is a relatively compact domain and $\Pi _H \, B$ denotes the restriction to $H(M) \otimes H(M)$ of the bilinear form B. A $C^\infty$ map $\phi : M \rightarrow N$ is subelliptic harmonic map if it is a critical point of $E_b$ i.e.

$$\begin{aligned} \frac{d}{d t} \Big \{ E_b \big ( \phi _t \big ) \Big \}_{t = 0} = 0, \end{aligned}$$

for any smooth 1-parameter variation $\{ \phi _t \}_{|t| < \epsilon } \subset C^\infty (M, N)$ of $\phi _0 = \phi$ with $\textrm{Supp} (V) \subset \Omega$, where $V = \partial \phi _t /\partial t \in C^\infty \big ( \phi ^{-1} T N \big )$ is the infinitesimal variation induced by $\{ \phi _t \}_{|t| < \epsilon }$. Subelliptic harmonic maps [from a pseudohermitian manifold $(M, \, \theta )$ into a Riemannian manifold $(N, \, h)$] were first introduced by E. Barletta et al. [9], under the name pseudoharmonic maps. Cf. also [24]. Let $\{ X_a \,: \, 1 \le a \le 2 n \}$ be a local $G_\theta$-orthonormal frame of H(M), defined on the open set U which is also the domain of a local chart $\chi : U \rightarrow {{\mathbb {R}}}^{2n+1}$. Then $X \equiv \{ \chi _*\, X_a \,: \, 1 \le a \le 2 n \}$ is a Hörmander system of vector fields on $\chi (U)$ and for any pseudoharmonic map $\phi$ the map $\phi \circ \chi ^{-1}$ is subellitic harmonic in the sense of Jost and Xu [44] i.e. as a map of $\chi (U)$ [an open set in ${{\mathbb {R}}}^{2n+1}$ equipped with the Hörmander system X] into the Riemannian manifold N (thus motivating the adopted terminology).

Let $\nabla$ and $\nabla ^h$ be respectively the Tanaka–Webster connection (cf. [29, Theorem 1.3, Definition 1.25, pp. 25–31]; see also Eq. (11) below) of $(M, \, \theta )$ and the Levi–Civita connection of $(N, \, h)$. For every $C^\infty$ map $\phi : M \rightarrow N$ let

$$\begin{aligned} B_b (\phi ) (X, Y) = D^\phi _X \phi _*\, Y - \phi _*\nabla _X Y \,, \;\;\; X, Y \in \mathfrak {X}(M), \end{aligned}$$

be the pseudohermitian second fundamental form of $\phi$. Here $\phi _*X$ is the $C^\infty$ section in the pullback bundle $\phi ^{-1} T(N) \rightarrow M$ defined by

$$\begin{aligned} \big ( \phi _*\, X \big ) (x) = (d_x \phi ) X_x \,, \;\;\; x \in M. \end{aligned}$$

Also $D^\phi = \phi ^{-1} \nabla ^h$ is the pullback of $\nabla ^h$ by $\phi$ [a connection in the vector bundle $\phi ^{-1} T(N) \rightarrow M$ parallelizing the bundle metric $h^\phi = \phi ^{-1} h$ (the pullback of h by $\phi$)]. Let us set

$$\begin{aligned} \tau _b (\phi ) = \textrm{trace}_{G_\theta } \big [ \Pi _H \, B_b (\phi ) \big ] \in C^\infty \big ( \phi ^{-1} T N \big ), \end{aligned}$$

(9)

(the pseudohermitian tension field of $\phi$) so that

$$\begin{aligned} \frac{d}{d t} \Big \{ E_b \big ( \phi _t \big ) \Big \}_{t = 0} = - \int _\Omega h^\phi \big ( V, \, \tau _b (\phi ) \big ) \; \theta \wedge (d \theta )^n, \end{aligned}$$

(the first variation formula for $E_b$) for any smooth 1-parameter variation $\{ \phi _t \}_{|t < \epsilon }$ of $\phi$ supported in $\Omega$. A $C^\infty$ map $\phi$ is subelliptic harmonic if and only if

$$\begin{aligned} \tau _b (\phi ) = 0. \end{aligned}$$

(10)

Note that $\tau _b (\phi )$ is not the full trace of $\phi ^*\, h$, but rather the trace (with respect to the Levi form $G_\theta$) of $\Pi _H \, \phi ^*\, h$ [the restriction of $\phi ^*\, h$ to $H(M) \otimes H(M)$]. Omitting a direction in the calculation of the trace [as in (9)] has far reaching consequences, as explained by Dragomir and Perrone (cf. [28]): the principal part in the subelliptic harmonic map system (10) is the sublaplacian $\Delta _b$ of $(M, \, \theta )$, a degenerate elliptic operator whose ellipticity degenerates at the cotangent directions spanned by $\theta$.

Pseudohermitian second fundamental forms were introduced by Petit (cf. [56]) who formally modified the definition of the second fundamental form (of a map of Riemannian manifolds) by replacing the Levi–Civita connection of the source manifold with the Tanaka–Webster connection. Nevertheless M does carry a natural Riemannian metric $g_\theta$, springing from the given structure $\big ( T_{1,0} (M), \, \theta \big )$, and $B_b (\phi )$ is related to the ordinary second fundamental form $B (\phi )$ of $\phi$, as a map between the Riemannian manifolds $(M, \, g_\theta )$ and $(N, \, h)$. Precisely, let $T \in \mathfrak {X}(M)$ be the Reeb vector field of $(M, \, \theta )$ i.e. the globally defined, nowhere zero tangent vector field on M, transverse to H(M), determined by $\theta (T) = 1$ and $T \, \rfloor \, d \theta = 0$. Profiting from the direct sum decomposition $T(M) = H(M) \oplus {{\mathbb {R}}} T$ one may extend the Levi form $G_\theta$ to a Riemannian metric $g_\theta$ on M [the Webster metric of $(M, \, \theta )$] by postulating that

$$\begin{aligned} g_\theta = G_\theta \;\; \text {on} \;\; H(M) \otimes H(M), \;\;\; g_\theta (X, T) = 0, \;\;\; g_\theta (T, T) = 1, \end{aligned}$$

for any $X \in H(M)$. Then $g_\theta$ is a contraction of the sub-Riemannian structure $\big ( H(M), \, G_\theta \big )$ (cf. Strichartz [60]) i.e. $d(x, y) \le \rho (x, y)$ for any $x, y \in M$, where d and $\rho$ are respectively the Riemannian distance (associated to the Webster metric) and the Carnot–Carathéodory distance (associated to the sub-Riemannian structure). Let $\nabla ^{g_\theta }$ be the Levi–Civita connection of $(M, \, g_\theta )$ and let

$$\begin{aligned} B (\phi ) (X, Y) = D^\phi _X \phi _*Y - \phi _*\nabla ^{g_\theta }_X Y, \;\;\; X,Y \in \mathfrak {X}(M), \end{aligned}$$

be the second fundamental form of $\phi$ as a map of $(M, \, g_\theta )$ into $(N, \, h )$. The tension field of $\phi$ is

$$\begin{aligned} \tau (\phi ) = \textrm{trace}_{g_\theta } \, B (\phi ) \in C^\infty \big ( \phi ^{-1} T N \big ). \end{aligned}$$

The Levi–Civita and Tanaka–Webster connections $\nabla ^{g_\theta }$ and $\nabla$ are related by (cf. [29, p. 46])

$$\begin{aligned} \nabla ^{g_\theta } = \nabla + \big ( \Omega - A \big ) \otimes T + \tau \otimes \theta + 2 \, (\theta \odot J ), \end{aligned}$$

(11)

where $\tau$ is the pseudohermitian torsion of $\nabla$ and

$$\begin{aligned} \Omega = - d \theta , \;\;\; A(X, Y) = g_\theta (X, \, \tau Y). \end{aligned}$$

A calculation relying on (11) shows that

$$\begin{aligned} \tau (\phi ) = \tau _b (\phi ) + D^\phi _T \, \phi _*\, T, \end{aligned}$$

(12)

so that the notions of a harmonic map and a subelliptic harmonic map are logically inequivalent. Let $\textrm{div}: \mathfrak {X}(M) \rightarrow C^\infty (M )$ be the divergence operator with respect to the volume form $\Psi = \theta \wedge (d \theta )^n$ i.e.

$$\begin{aligned} {{\mathcal {L}}}_X \Psi = \textrm{div} (X) \, \Psi , \end{aligned}$$

where ${{\mathcal {L}}}_X$ is the Lie derivative. The sublaplacian is the formally self-adjoint, positive, second order operator $\Delta _b$ given by

$$\begin{aligned} \Delta _b u= & {} - \textrm{div} \big ( \nabla ^H u \big ) \,, \;\;\; u \in C^2 (M),\\ \nabla ^H u= & {} \Pi _H \nabla u, \;\;\; \Pi _H = I - \theta \otimes T,\\ g_\theta (X, \, \nabla u)= & {} X(u), \;\;\; u \in C^1 (M), \;\;\; X \in \mathfrak {X} (M). \end{aligned}$$

Let $\phi : M \rightarrow N$ be a $C^\infty$ map. Let $\{ X_a \,: \, 1 \le a \le 2 n \} \subset C^\infty (U, \, H(M))$ be a $G_\theta$-orthonormal [i.e. $G_\theta (X_a \,, \, X_b ) = \delta _{ab}$] local frame and let $(V, \, y^\alpha )$ be a local coordinate system on N such that $\phi (U) \subset V$. The subelliptic harmonic map system (10) may be written locally as

$$\begin{aligned} - \Delta _b \phi ^\alpha + \sum _{a=1}^{2 n} \left\{ \begin{array}{c} \alpha \\ \beta \gamma \end{array} \right\} \, X_a \big ( \phi ^\beta \big ) \, X_a \big ( \phi ^\gamma \big ) = 0, \end{aligned}$$

(13)

where $\phi ^\alpha = y^\alpha \circ \phi$ and $\left\{ \begin{array}{c} \alpha \\ \beta \gamma , \end{array} \right\}$are the Christoffel symbols of the second kind of $h_{\alpha \beta }$. The sublaplacian is degenerate elliptic, yet subelliptic of order 1/2 and hence hypoelliptic (cf. Hörmander [42]). The study of subelliptic harmonic maps, and then the study of subelliptic harmonic morphisms (a particular sort of subelliptic harmonic maps, as introduced by Dragomir and Lanconelli [25]) fits into the larger program of Jost and Xu (cf. [44]) devoted to the study of second order quasi-linear PDE systems of variational origin whose principal part is at least hypoelliptic.

The sublaplacian $\Delta _b$ may be thought of as the linear operator of Hilbert spaces

$$\begin{aligned} \Delta _b = \big ( \nabla ^H \big )^*\circ \nabla ^H: {{\mathscr {D}}} \big ( \Delta _b \big ) \subset L^2 (M) \rightarrow L^2 (M), \end{aligned}$$

with domain

$$\begin{aligned} {{\mathscr {D}}}\big ( \Delta _b \big ) = \big \{ u \in {{\mathscr {D}}} \big ( \nabla ^H \big ) \,: \, \nabla ^H u \in {{\mathscr {D}}} \big [ (\nabla ^H )^*\big ] \big \}, \end{aligned}$$

where $\nabla ^H$ is the weak horizontal gradient and $\big ( \nabla ^H \big )^*$ is its adjoint. Then, although the subelliptic harmonic map system is but quasi-linear, weak solutions may be defined as maps $\phi \in W^{1,2}_H (M, N)$ such that for any $\varphi \in C^\infty _0 (M)$

$$\begin{aligned}{} & {} \int _U \Big \{ g_\theta \big ( \nabla ^H \phi ^\alpha \,, \, \nabla ^H \varphi \big )\\{} & {} \quad + \sum _{a=1}^{2 n} \left\{ \begin{array}{c} \alpha \\ \beta \gamma \end{array} \right\} \, X_a \big ( \phi ^\beta \big ) \, X_a \big ( \phi ^\gamma \big ) \, \varphi \Big \} \, \theta \wedge (d \theta )^n = 0. \end{aligned}$$

To make sense of the Sobolev type spaces $W^{1,2}_H (M, N)$ (the Folland–Stein spaces) the definition is either confined to target manifolds N which may be covered by a single coordinate neighborhood, or one first embeds (isometrically) N into a sufficiently high dimensional Euclidean space (by using Nash’s embedding theorem [52]). The generalized Dirichlet problem for the PDE system (13) was solved by Jost and Xu (cf. [44]) who also proved interior continuity of solutions. Finally, existence of $C^\infty$ subelliptic harmonic maps may be established by applying a result by Xu and Zuily (cf. [63]) who proved smoothness of continuous solutions to a class of PDE systems including the subelliptic harmonic map system.

3 Subelliptic Harmonic Morphisms

Definition 1

A continuous map $\phi$ of $(M, \theta )$ into (N, h) is a subelliptic harmonic morphism if for every open subset $V \subset N$, and every $C^2$ function $v: V \rightarrow {{\mathbb {R}}}$, if $\Delta _h v = 0$ in V then the pullback function $u = v \circ \phi$ is a distribution-solution to $\Delta _b u = 0$ in $U = \phi ^{-1} (V)$. $\square$

Cf. Dragomir and Lanconelli [25]. Here

$$\begin{aligned} \Delta _h v \equiv - \frac{1}{\sqrt{{\mathfrak {h}}}} \frac{\partial }{\partial y^\alpha } \Big \{ \sqrt{{\mathfrak {h}}} \, h^{\alpha \beta } \, \frac{\partial v}{\partial y^\beta } \Big \} , \;\;\; {{\mathfrak {h}}} = \det \big [ h_{\alpha \beta } \big ], \end{aligned}$$

is the Laplacian on (V, h).

Proposition 1

Every subelliptic harmonic morphism $\phi$ of the pseudohermitian manifold $(M, \, \theta )$ into the Riemannian manifold $(N, \, h)$ is smooth.

Proof

For every point $x_0 \in M$ let $(V, \, y^\alpha )$ be a harmonic local coordinate system with $\phi (x_0 ) \in V$, and let us set $\phi ^\alpha = y^\alpha \circ \phi$. Then $\Delta _b \phi ^\alpha = 0$ in $U = \phi ^{-1} (V)$ hence (as $\Delta _b$ is hypoelliptic) $\phi ^\alpha \in C^\infty (U)$. $\square$

Definition 2

A $C^\infty$ map $\phi : M \rightarrow N$ is Levi conformal if there is a continuous map $\lambda = \lambda (\phi ): M \rightarrow [0, + \infty )$ (the $\theta$-dilation of $\phi$) such that $\lambda ^2$ is $C^\infty$ and

$$\begin{aligned} G_\theta \big ( \nabla ^H \phi ^\alpha \, , \, \nabla ^H \phi ^\beta \big )_x = \lambda (x)^2 \, \delta ^{\alpha \beta }, \end{aligned}$$

(14)

for any $x \in M$ and any local normal coordinate system $(V, \, y^\alpha )$ on N with center at $\phi (x) \in V$. $\square$

We set as customary $\Lambda (\phi ) = \lambda (\phi )^2$ (the square $\theta$-dilation). For a fixed Levi conformal map $\phi : M \rightarrow N$ we abbreviate the notation to $\Lambda = \Lambda (\phi )$. Let $\phi : M \rightarrow N$ be a Levi conformal map, of $(M, \, \theta )$ into $(N, \, h)$, and let $x \in M$ be an arbitrary point. Let $\big ( V^\prime \,, \, {y^\prime }^\alpha \big )$ be an arbitrary local coordinate system on N such that $\phi (x) \in V^\prime$, and let us set ${\phi ^\prime }^\alpha = {y^\prime }^\alpha \circ \phi$. Then

$$\begin{aligned} \nabla ^H {\phi ^\prime }^\alpha = \Big ( \frac{\partial {y^\prime }^\alpha }{\partial y^\beta } \Big )^\phi \; \nabla ^H \phi ^\beta , \end{aligned}$$

(15)

on $\phi ^{-1} (V \cap V^\prime )$. Moreover [by (15) and (14)]

$$\begin{aligned}{} & {} G_\theta \big ( \nabla ^H {\phi ^\prime }^\alpha \,, \, \nabla ^H {\phi ^\prime }^\beta \big )_x \\{} & {} \quad = \Lambda (x) \; \delta ^{\mu \nu } \; \frac{\partial {y^\prime }^\beta }{\partial y^\mu } \big ( \phi (x) \big ) \; \frac{\partial {y^\prime }^\beta }{\partial y^\nu } \big ( \phi (x) \big ) = \Lambda (x) \; {h^\prime }^{\alpha \beta } \big ( \phi (x) \big ), \end{aligned}$$

i.e. if $\phi$ is Levi conformal then for any $x \in M$ and any local coordinate system $(V, \, y^\alpha )$ about $\phi (x)$

$$\begin{aligned} G_\theta \Big ( \nabla ^H \phi ^\alpha \, , \, \nabla ^H \phi ^\beta \big ) = \Lambda \; h^{\alpha \beta } \circ \phi , \end{aligned}$$

(16)

everywhere in $\phi ^{-1} (V)$.

By a result of Barletta [7] (revisited as in Appendix B of [25]) a $C^\infty$ map $\phi : M \rightarrow N$ is a subelliptic harmonic morphism of $(M, \, \theta )$ into $(N, \, h)$ if and only if $\phi$ is Levi conformal and a subelliptic harmonic map. Moreover (again by [7]) if $m > 2 n$ then every subelliptic harmonic morphism is a constant, while if $m \le 2 n$ then for every point $x \in M$ with $\lambda (x) \ne 0$ there is an open neighborhood U of x such that $\phi : U \rightarrow N$ is a $C^\infty$ submersion. Barletta’s result is a pseudohermitian analog to the Fuglede–Ishihara characterization (cf. Fuglede [34], Ishihara [43]) of harmonic morphisms between Riemannian manifolds.

Let $\phi : M \rightarrow N$ be a $C^\infty$ map and let us set

$$\begin{aligned} {{\mathscr {V}}}^\phi _x = \textrm{Ker} (d_x \phi ), \;\;\; {{\mathscr {H}}}^\phi _x = \big ( {{\mathscr {V}}}^\phi _x \big )^\bot \,, \;\;\; x \in M, \end{aligned}$$

where the orthogonal complement is meant with respect to the inner product $g_{\theta , \, x}$.

Lemma 1

Let M be a strictly pseudoconvex CR manifold, of CR dimension n, endowed with the positively oriented contact form $\theta$, and let (N, h) be a m-dimensional Riemannian manifold. Let $\phi : M \rightarrow N$ be a $C^\infty$ map. Then

(i) For every $x \in \phi ^{-1} (V)$

$$\begin{aligned} \Big \{ \big ( \nabla \phi ^\alpha \big )_x \;: \; 1 \le \alpha \le m \Big \} \subset {{\mathscr {H}}}^\phi _x. \end{aligned}$$

(ii) Let $\phi : M \rightarrow N$ be a Levi conformal map, and let $Z(\Lambda ) = \{ x \in M \;: \; \Lambda (x) = 0 \}$ be the zero set of its $\theta$-dilation. Then

$$\begin{aligned} \textrm{Crit}(\phi ) \subset Z(\Lambda ). \end{aligned}$$

(17)

Also

$$\begin{aligned} T_x \in {{\mathscr {H}}}^\phi _x \end{aligned}$$

(18)

for any $x \in Z(\Lambda ) \setminus \textrm{Crit}(\phi )$.

(iii) Let us assume that $m \le 2 n$, and let $\phi : M \rightarrow N$ be a subelliptic harmonic morphism. Then for every $x \in M {\setminus } Z(\Lambda )$

$$\begin{aligned} \Big \{ \big ( \nabla \phi ^\alpha \big )_x \; : \; 1 \le \alpha \le m \Big \}, \end{aligned}$$

(19)

is a linear basis in ${{\mathscr {H}}}^\phi _x$. In particular

$$\begin{aligned} \Big \{ x \in M \;: \; T_{x} \in {{\mathscr {H}}}_{x}^\phi \Big \} \subset Z(\Lambda ). \end{aligned}$$

Proof

(i) Let $x \in M$ and $v \in {{\mathscr {V}}}^\phi _x$. Then

$$\begin{aligned} g_{\theta \,, \, x} \big ( v, \; \big ( \nabla \phi ^\alpha \big )_x \big ) \, \Big ( \frac{\partial }{\partial y^\alpha } \Big )_{\phi (x)} = v(\phi ^\alpha ) \, \Big ( \frac{\partial }{\partial y^\alpha } \Big )_{\phi (x)} = (d_x \phi ) v = 0, \end{aligned}$$

that is

$$\begin{aligned} \big ( \nabla \phi ^\alpha \big )_x \in \big ( {{\mathscr {V}}}_x^\phi \big )^\bot = {{\mathscr {H}}}_x^\phi . \end{aligned}$$

$\square$

(ii) Let $x \in \textrm{Crit}(\phi )$. Then

$$\begin{aligned} 0 = \big ( \nabla \phi ^\alpha \big )_x = \big ( \nabla ^H \phi ^\alpha \big )_x + T \big ( \phi ^\alpha \big )_x \, T_x \Longrightarrow , \end{aligned}$$

[by the uniqueness of the direct sum decomposition $T_x (M) = H(M)_x \oplus {{\mathbb {R}}} T_x$]

$$\begin{aligned} \Longrightarrow \big ( \nabla ^H \phi ^\alpha \big )_x = 0, \end{aligned}$$

hence [by (14)] $x \in Z(\Lambda )$, accounting for (17).

Next, let $x \in Z(\Lambda ) \setminus \textrm{Crit} (\phi )$. Then [by (14)]

$$\begin{aligned} \big ( \nabla ^H \phi ^\alpha \big )_x = 0, \;\;\; 1 \le \alpha \le m, \end{aligned}$$

so that

$$\begin{aligned}{} & {} \exists \;\; \alpha \in \{ 1, \, \ldots \,, \, m \} \;: \; 0 \ne \big ( \nabla \phi ^\alpha )_x = T_x \big ( \phi ^\alpha \big ) \, T_x\nonumber \\{} & {} \quad \exists \;\; \alpha \in \{1, \, \ldots \,, \, m \} \;: \; T_x \big ( \phi ^\alpha \big ) \ne 0. \end{aligned}$$

(20)

Finally, for every $v \in {{\mathscr {V}}}_x^\phi$ [by statement (i) in Lemma 1]

$$\begin{aligned} T_x \big ( \phi ^\alpha \big ) \, g_{\theta \,, \, x} \big ( v, \; T_x \big ) = g_{\theta \,, \, x} \big ( v, \; \big ( \nabla \phi ^\alpha )_x \big ) = 0, \end{aligned}$$

yielding [by (20)] $T_x \in \big ( {{\mathscr {V}}}^\phi _x \big )^\bot = {{\mathscr {H}}}^\phi _x$. $\square$

(iii) Let $x_0 \in M \setminus Z(\Lambda )$. By a result of Barletta [7], there is an open neighborhood $U \subset M$ of $x_0$ such that $\phi : U \rightarrow N$ is a $C^\infty$ submersion. Hence, $d_x \phi : {{\mathscr {H}}}_x^\phi \rightarrow T_{\phi (x)} (N)$ is a ${{\mathbb {R}}}$-linear isomorphism, for any $x \in U$. By statement (i) in Lemma 1 it suffices to show that the system $\Big \{ \big ( \nabla \phi ^\alpha \big )_{x_0} \;: \; 1 \le \alpha \le m \Big \} \subset {{\mathscr {H}}}^\phi _x$ is free. Indeed if for some $\mu ^\alpha \in {{\mathbb {R}}}$, $1 \le \alpha \le m$,

$$\begin{aligned} 0 = \mu _\alpha \big ( \nabla \phi ^\alpha \big )_{x_0} = \mu _\alpha \Big \{ \big ( \nabla ^H \phi ^\alpha \big )_{x_0} + T_{x_0} (\phi ^\alpha ) \, T_{x_0} \Big \}, \end{aligned}$$

then $\mu _\alpha \, \big ( \nabla ^H \phi ^\alpha \big )_{x_0} = 0$ yielding $\mu ^\alpha = 0$, because [by (14)] the vectors

$$\begin{aligned} \Big \{ \big ( \nabla ^H \phi ^\alpha \big )_{x_0} \; : \; 1 \le \alpha \le m \Big \}, \end{aligned}$$

(21)

are linearly independent.

Next, let $x \in M$ be a point such that $T_x \in {{\mathscr {H}}}^\phi _x$. Either x is a critical point of $\phi$, so that [by (17)] $x \in Z(\Lambda )$, or $x \in M {\setminus } \textrm{Crit} (\phi )$. For the remainder of the proof we argue by contradiction, i.e. let us assume that $\Lambda (x ) \ne 0$. If this is the case, for any $1 \le \alpha \le m$

$$\begin{aligned} {{\mathscr {H}}}_x^\phi \ni \big ( \nabla \phi ^\alpha \big )_x - T_x (\phi ^\alpha ) \, T_x = \big ( \nabla ^H \phi ^\alpha \big )_x, \end{aligned}$$

hence (21) is a linear basis of ${{\mathscr {H}}}^\phi _x$, too, yielding ${{\mathscr {H}}}^\phi _x \subset H(M)_x$ and in particular $T_x \in H(M)_x$, a contradiction. $\square$

Let $\phi : M \rightarrow N$ be a subelliptic harmonic morphism, of the pseudohermitian manifold $(M, \theta )$, into the Riemannian manifold (N, h). For each $x \in M$ we set

$$\begin{aligned} {{\mathscr {V}}}^\phi _{H, \, x} = H(M)_x \cap {{\mathscr {V}}}^\phi _x \,, \;\;\; {{\mathscr {H}}}^\phi _{H, \, x} = H(M)_x \cap {{\mathscr {H}}}^\phi _x. \end{aligned}$$

If $x \in \textrm{Crit} (\phi )$ then

$$\begin{aligned} {{\mathscr {V}}}^\phi _{H, \, x} = H (M)_x \,, \;\;\; {{\mathscr {H}}}^\phi _{H, \, x} = \{ 0 \}. \end{aligned}$$

If $x \in M {\setminus } \textrm{Crit} (\phi )$ then the differential $d_x \phi : T_x (M) \rightarrow T_{\phi (x)} (N)$ may, or may not, be an epimorphism.

Definition 3

A regular point in the set

$$\begin{aligned} S(\phi ) = \big \{ x \in M \setminus \textrm{Crit} (\phi ) \;: \; d_x \phi \;\;\; \text {is on-to} \big \}, \end{aligned}$$

is called a submersive point of the morphism $\phi$. $\square$

At every submersive point $x \in S(\phi )$

$$\begin{aligned} \dim _{{\mathbb {R}}} {{\mathscr {H}}}^\phi _x = m, \;\;\; \dim _{{\mathbb {R}}} {{\mathscr {V}}}^\phi _x = 2 n - m + 1. \end{aligned}$$

Lemma 2

Let M and N be a strictly pseudoconvex CR manifold, of CR dimension n, and let N be an m-dimensional Riemannian manifold, such that $m \le 2 n$. Let $\phi : M \rightarrow N$ be a subelliptic harmonic morphism, of $\theta$-dilation $\sqrt{\Lambda }$. Then (i)

$$\begin{aligned} M \setminus Z(\Lambda ) \subset S(\phi ) . \end{aligned}$$

(22)

(ii) For every submersive point $x \in S(\phi )$

$$\begin{aligned}{} & {} m - 1 \le \dim _{{\mathbb {R}}} {{\mathscr {H}}}^\phi _{H, \, x} \le m, \end{aligned}$$

(23)

$$\begin{aligned}{} & {} 2 n - m \le \dim _{{\mathbb {R}}} {{\mathscr {V}}}^\phi _{H, \, x} \le 2 n - m + 1. \end{aligned}$$

(24)

Proof

(i) Let $x \in M$ with $x \not \in Z(\Lambda )$. Then, on one hand [by (ii) in Lemma 1] x is a regular point of $\phi$. On the other hand $m \le 2 n$ and $\Lambda (x) \ne 0$ so that, by a result of Barletta (cf. [7]) $\phi$ is a submersion on some neighborhood of x, and in particular x is a submersive point. $\square$

(ii) For instance, for every $x \in S(\phi )$ the relations

$$\begin{aligned}{} & {} \dim _{{\mathbb {R}}} \Big [ H(M)_x + {{\mathscr {H}}}^\phi _x \Big ] = 2 n + m - \dim _{{\mathbb {R}}} {{\mathscr {H}}}^\phi _{H, \, x} \,,\\{} & {} \quad 2 n \le \dim _{{\mathbb {R}}} \Big [ H(M)_x + {{\mathscr {H}}}^\phi _x \Big ] \le 2 n + 1, \end{aligned}$$

yield (19). $\square$

Note that [by taking complements in (22)] $M {\setminus } S(\phi ) \subset Z(\Lambda )$. Next, as a consequence of (23) and (24), the set of submersive points of $\phi$ admits the natural partition

$$\begin{aligned} S(\phi )= & {} \textrm{I}_m (\phi ) \cup \textrm{II}_m (\phi ) \cup \textrm{III}_m (\phi ), \nonumber \\ \textrm{I}_m (\phi )= & {} \big \{ x \in S(\phi ) \;: \; \dim _{{\mathbb {R}}} {{\mathscr {H}}}^\phi _{H, \, x} = m, \;\; \dim _{{\mathbb {R}}} {{\mathscr {V}}}^\phi _{H, \, x} = 2n - m \big \},\nonumber \\ \textrm{II}_m (\phi )= & {} \big \{ x \in S(\phi ) \;: \; \dim _{{\mathbb {R}}} {{\mathscr {H}}}^\phi _{H, \, x} = m - 1, \;\; \dim _{{\mathbb {R}}} {{\mathscr {V}}}^\phi _{H, \, x} = 2n - m + 1 \big \}, \nonumber \\ \textrm{III}_m (\phi )= & {} \big \{ x \in S(\phi ) \;: \; \dim _{{\mathbb {R}}} {{\mathscr {H}}}^\phi _{H, \, x} = m - 1, \;\; \dim _{{\mathbb {R}}} {{\mathscr {V}}}^\phi _{H, \, x} = 2n - m \big \}. \end{aligned}$$

(25)

Indeed, case (IV) where $\dim _{{\mathbb {R}}} {{\mathscr {H}}}^\phi _{H, \, x} = m$ and $\dim _{{\mathbb {R}}} {{\mathscr {V}}}^\phi _{H, \, x} = 2n - m + 1$ is ruled out by ${{\mathscr {H}}}^\phi _{H, \, x} \oplus {{\mathscr {V}}}^\phi _{H, \, x} \subset H(M)_x$.

The main difficulties one encounters are related to the presence of two pairs of complementary distributions on M [rather than just $\big ( H(M), \, {{\mathbb {R}}} T \big )$ as in CR geometry, or just $\big ( {{\mathscr {V}}}^\phi \,, \, {{\mathscr {H}}}^\phi \big )$ as in the theory of harmonic morphisms between Riemannian manifolds]. These distributions intersect, and the dimension of the intersections may vary from a point to another, requiring a classification of types of points, relative to a fixed subelliptic harmonic morphism $\phi$, as captured by the partition (25). Our conclusive finding is

Theorem 6

Let M be a strictly pseudoconvex CR manifold, equipped with the contact form $\theta \in {{\mathscr {P}}}_+ (M)$, and let $(N, \, h)$ be a m-dimensional Riemannian manifold. Let $\phi : M \rightarrow N$ be a subelliptic harmonic morphism, from $(M, \theta )$ into (N, h).

(i) If $m = 1$ then

$$\begin{aligned} Z(\Lambda ) = \textrm{II}_1 (\phi ) \cup \textrm{Crit} (\phi ) , \;\;\; M \setminus S(\phi ) = \textrm{Crit} (\phi ). \end{aligned}$$

(26)

(ii) If $m \ge 2$ then

$$\begin{aligned} \textrm{II}_m (\phi ) = \emptyset , \;\;\; Z(\Lambda ) = M \setminus S(\phi ). \end{aligned}$$

(27)

Consequently, for every subelliptic harmonic morphism $\phi : M \rightarrow N^m$ all points in M are submersive, except for the critical points of $\phi$ when $m = 1$, or for the zeros of the square dilation $\Lambda = \Lambda (\phi )$ when $m \ge 2$.

The proof of Theorem 6 requires a number of lemmas.

Lemma 3

Under the assumptions of Lemma  2, let $x \in S(\phi )$ be a submersive point. Then

(i) $x \in \textrm{I}_m (\phi ) \Longleftrightarrow T_x \in {{\mathscr {V}}}^\phi _x$.

(ii) $x \in \textrm{II}_m (\phi ) \Longleftrightarrow T_x \in {{\mathscr {H}}}^\phi _x$.

Proof

(i) If $x \in \textrm{I}_m (\phi )$ then $\dim _{{\mathbb {R}}} {{\mathscr {H}}}^\phi _x = m$ hence ${{\mathscr {H}}}^\phi _x = {{\mathscr {H}}}^\phi _{H, \, x} \subset H(M)_x$ yielding ${{\mathscr {V}}}^\phi _x = \big ( {{\mathscr {H}}}^\phi _x \big )^\bot \supset H(M)_x^\bot = {{\mathbb {R}}} T_x$. $\square$

Viceversa, if $T_x \in {{\mathscr {V}}}^\phi _x$ then $T_x$ is orthogonal to ${{\mathscr {H}}}^\phi _x$ i.e. ${{\mathscr {H}}}^\phi _x \subset H(M)_x$ yielding ${{\mathscr {H}}}^\phi _x = {{\mathscr {H}}}^\phi _{H, \, x}$, and consequently $\dim _{{\mathbb {R}}} {{\mathscr {H}}}^\phi _{H, \, x} = m$. The sets $\textrm{I}_m (\phi )$, $\textrm{II}_m (\phi )$ and $\textrm{III}_m (\phi )$ are mutually disjoint, so it must be that $x \in \textrm{I}_m (\phi )$. $\square$

(ii) If $x \in \textrm{II}_m (\phi )$ then $\dim _{{\mathbb {R}}} {{\mathscr {V}}}^\phi _{H, \, x} = 2 n - m + 1$, hence ${{\mathscr {V}}}^\phi _x = {{\mathscr {V}}}^\phi _{H, \, x} \subset H(M)_x$ implying that $T_x$ is orthogonal to ${{\mathscr {V}}}^\phi _x$ i.e. $T_x \in {{\mathscr {H}}}^\phi _x$. $\square$

Viceversa, if $T_x \in {{\mathscr {H}}}^\phi _x$ then $T_x$ is orthogonal to ${{\mathscr {V}}}^\phi _x$ i.e. ${{\mathscr {V}}}^\phi _x \subset H(M)_x$. It follows that the subspaces ${{\mathscr {V}}}^\phi _{H, \, x}$ and ${{\mathscr {V}}}^\phi _x$ coincide, yet the space ${{\mathscr {V}}}^\phi _x$ is $(2n - m + 1)$-dimensional, so that $x \in \textrm{II}_m (\phi )$, again because (25) is a partition.

Lemma 4

Under the assumptions of Lemma  2

$$\begin{aligned} \textrm{II}_m (\phi ) = S(\phi ) \cap Z(\Lambda ). \end{aligned}$$

(28)

Consequently

$$\begin{aligned} \textrm{I}_m (\phi ) \cap Z(\Lambda ) = \emptyset , \;\; \textrm{II}_m (\phi ) \subset Z(\Lambda ) \setminus \textrm{Crit}(\phi ), \;\; \textrm{III}_m (\phi ) \cap Z(\Lambda ) = \emptyset . \end{aligned}$$

Proof

For every $x \in \textrm{II}_m (\phi )$ [by statement (ii) in Lemma 2] $T_x \in {{\mathcal {H}}}^\phi _x$ and then [by statement (iii) in Lemma 1] $x \in Z(\Lambda )$. $\square$

To prove the opposite inclusion, let $x \in S(\phi ) \cap Z(\Lambda )$. Then $\Lambda (x) = 0$ so that [by (14)]

$$\begin{aligned} \big ( \nabla ^H \phi ^\alpha \big )_x = 0, \;\;\; 1 \le \alpha \le m. \end{aligned}$$

Consequently [by statement (i) in Lemma 1]

$$\begin{aligned} T_x (\phi ^\alpha ) \, T_x = \big ( \nabla \phi ^\alpha \big )_x \in {{\mathcal {H}}}^\phi _x \,, \;\;\; 1 \le \alpha \le m, \end{aligned}$$

and there is $\alpha \in \{ 1, \, \ldots \,, \, m \}$ such that $T_x (\phi ^\alpha ) \ne 0$, because x is a regular point. Therefore $T_x \in {{\mathcal {H}}}^\phi _x$. $\square$

Let us set $\Omega = \Omega (\phi ):= M \setminus Z(\Lambda )$ (an open subset of M). Then

$$\begin{aligned} \partial \Omega \subset Z(\Lambda ). \end{aligned}$$

(29)

Lemma 5

Under the assumptions of Lemma  2

$$\begin{aligned} S(\phi ) = \Omega (\phi ) \cup \textrm{II}_m (\phi ). \end{aligned}$$

(30)

Proof

Let $x \in S(\phi )$. We distinguish two cases, as A) $x \in Z(\Lambda )$, or B) $z \not \in Z(\Lambda )$. In case (A) [by Lemma 4]

$$\begin{aligned} x \in S(\phi ) \cap Z(\Lambda ) = \textrm{II}_m (\phi ). \end{aligned}$$

In case (B)

$$\begin{aligned} x \in M \setminus Z(\Lambda ) = \Omega (\phi ). \end{aligned}$$

To check the opposite inclusion, let $x \in \Omega (\phi ) \cup \textrm{II}_m (\phi )$. Then [by (28) and (22)] either

$$\begin{aligned} x \in \textrm{II}_m (\phi ) = S(\phi ) \cap Z(\Lambda ) \subset S(\phi ), \end{aligned}$$

or

$$\begin{aligned} x \in \Omega (\phi ) = M \setminus Z(\Lambda ) \subset S(\phi ). \end{aligned}$$

$\square$

Lemma 6

Under the assumptions of Lemma  2

(1) $\textrm{I}_m (\phi ) \subset \Omega$,

(2) $\textrm{III}_m (\phi ) = \Omega {\setminus } \textrm{I}_m (\phi )$.

Proof

(1) Given any $x \in \textrm{I}_m (\phi )$, let us show that $x \not \in Z(\Lambda )$. We argue by contradiction. If $x \in Z(\Lambda )$ then [by statement (ii) in Lemma 1, as x is a regular point] $T_x \in {{\mathscr {H}}}_x^\phi$. Yet [by statement (i) in Lemma 3] $T_x \in {{\mathscr {V}}}_x^\phi$, implying that $T_x = 0$, a contradiction.

(2) By (30) and the first statement in the current lemma

$$\begin{aligned} S(\phi ) = \Omega \cup \textrm{II}_m (\phi ) = \textrm{I}_m (\phi ) \cup \textrm{II}_m (\phi ) \cup \big [ \Omega \setminus \textrm{I}_m (\phi ) \big ], \end{aligned}$$

implying [by (25)] $\Omega {\setminus } \textrm{I}_m (\phi ) = \textrm{III}_m (\phi )$. $\square$

Lemma 7

Let $\phi : M \rightarrow N$ be a subelliptic harmonic morphism, from $(M, \theta )$ into (N, h). Then for every $x \in Z(\Lambda ) {\setminus } \textrm{Crit}(\phi )$

$$\begin{aligned} {{\mathscr {V}}}^\phi _x = H(M)_x, \end{aligned}$$

(31)

and in particular $\dim _{{\mathbb {R}}} \, (d_{x} \phi ) T_{x} (M) = 1$.

Proof

Let $x \in Z(\Lambda ) \setminus \textrm{Crit} (\phi )$. By $\Lambda (x) = 0$ and (14) [as $\phi$ is Levi conformal]

$$\begin{aligned} (\nabla ^H \phi ^\alpha )_x = 0, \;\;\; 1 \le \alpha \le m. \end{aligned}$$

Let $\{ E_a \,: \, 1 \le a \le 2 n \}$ be a local $G_\theta$-orthonormal frame of H(M), defined on an open neighborhood $U \subset M$ of x. Then

$$\begin{aligned} 0 = (\nabla ^H \phi ^\alpha )_x = \sum _{a=1}^{2n} E_{a, \, x} (\phi ^\alpha ) \, E_{a, \, x} \Longrightarrow E_{a, \, x} (\phi ^\alpha ) = 0, \end{aligned}$$

hence

$$\begin{aligned} 0 = E_{a, \, x} (\phi ^\alpha ) \, \Big ( \frac{\partial }{\partial y^\alpha } \Big )_{\phi (x)} = (d_x \phi ) E_{a, \, x} \Longrightarrow E_{a, \, x} \in {{\mathscr {V}}}^\phi _x, \end{aligned}$$

that is

$$\begin{aligned} H(M)_x \subset {{\mathscr {V}}}^\phi _x. \end{aligned}$$

(32)

As $x \not \in \textrm{Crit} (\phi )$ it must be

$$\begin{aligned} T_x \not \in {{\mathscr {V}}}_x^\phi . \end{aligned}$$

(33)

Indeed, if $T_x \in {{\mathscr {V}}}^\phi _x$ then [by (32)] $T_x (M) \subset {{\mathscr {V}}}^\phi _x$ i.e. $d_x \phi = 0$, a contradiction. Next, let $v \in {{\mathscr {V}}}^\phi _x$ so that

$$\begin{aligned} v = \lambda ^a \, E_{a, \, x} + \theta _x (v) \, T_x, \end{aligned}$$

for some $\lambda ^a \in {{\mathbb {R}}}$, $1 \le a \le 2 n$. By applying $d_x \phi$ to both members [and using (32) and (33)]

$$\begin{aligned} 0 = (d_x \phi ) v = \theta _x (v) \, (d_x \phi ) T_x \Longrightarrow \theta _x (v) = 0 \Longrightarrow v \in H(M)_x, \end{aligned}$$

that is ${{\mathscr {V}}}^\phi _x \subset H(M)_x$, yielding equality in (32). $\square$ In particular [by (31)]

$$\begin{aligned} \dim _{{\mathbb {R}}} T_x (M) = \dim _{{\mathbb {R}}} {{\mathscr {V}}}^\phi _x + \dim _{{\mathbb {R}}} (d_x \phi ) T_x (M) \Longrightarrow \dim _{{\mathbb {R}}} (d_x \phi ) T_x (M) = 1. \end{aligned}$$

$\square$

At this point we may complete the proof of Theorem 6.

(i) By Lemmas 1 and 4 the inclusion

$$\begin{aligned} Z(\Lambda ) \supset \textrm{II}_m (\phi ) \cup \textrm{Crit} (\phi ), \end{aligned}$$

holds for arbitrary $m \ge 1$. To check the opposite inclusion, let $x \in Z(\Lambda )$. Then either $x \in \textrm{Crit} (\phi )$ and we are done, or $x \not \in \textrm{Crit} (\phi )$ and then we may apply Lemma 7 to conclude that the space $(d_x \phi ) \, T_x (M)$ is 1-dimensional. Hence (as $m = 1$) $d_x \phi$ is an epimorphism, implying that $x \in S(\phi )$, and then

$$\begin{aligned} x \in S(\phi ) \cap Z(\Lambda ) = \textrm{II}_1 (\phi ), \end{aligned}$$

(according to Lemma 4). $\square$

To prove the second equality in (26) note first that

$$\begin{aligned} M \setminus S(\phi ) \supset \textrm{Crit} (\phi ), \end{aligned}$$

[by its very definition, $S(\phi )$ lies in the complementary of $\textrm{Crit} (\phi )$]. To check the opposite inclusion, let $x \in M {\setminus } S(\phi )$. We argue by contradiction i.e. we assume that $x \not \in \textrm{Crit} (\phi )$. On the other hand [by $x \not \in S(\phi )$] the differential $d_x \phi$ is not on-to, implying that $\Lambda (x) = 0$ [otherwise $\phi$ is a submersion on some neighborhood of x, a contradiction]. At this point we may apply Lemma 7 to conclude that $(d_x \phi ) T_x (M)$ is 1-dimensional, so that $d_x \phi$ is surjective i.e. $x \in S(\phi )$, a contradiction.

(ii) The proof of $\textrm{II}_m (\phi ) = \emptyset$ is by contradiction. If $\textrm{II}_m (\phi ) \ne \emptyset$, let $x \in \textrm{II}_m (\phi )$ i.e. (by Lemma 4) $x \in S(\phi ) \cap Z(\Lambda )$. Therefore $d_x \phi \ne 0$ and $\Lambda (x) = 0$ so we may apply Lemma 7 to conclude that $T_{\phi (x)} (N) = (d_x \phi ) T_x (M)$ is 1-dimensional i.e. $m = 1$, a contradiction. $\square$

To prove the second statement in (27), let $x \in Z(\Lambda )$. Then either $x \in \textrm{Crit}(\phi )$, implying that $d_x \phi$ is not on-to i.e. $x \in M {\setminus } S(\phi )$, or $x \not \in \textrm{Crit}(\phi )$ and one may apply Lemma 7 to conclude that

$$\begin{aligned} \dim _{{\mathbb {R}}} (d_x \phi ) T_x (\phi ) = 1 < m, \end{aligned}$$

hence $d_x \phi$ is not on-to i.e. $x \not \in S(\phi )$. The inclusion $Z(\Lambda ) \subset M \setminus S(\phi )$ is proved. As to the opposite inclusion, let $x \in M \setminus S(\phi )$ and let us assume that $x \not \in Z(\Lambda )$. Then $\phi$ is a submersion on some neighborhood of x, and in particular $x \in S(\phi )$, a contradiction. This yields $M {\setminus } S(\phi ) \subset Z(\Lambda )$. $\square$

4 Harmonic Morphisms from Fefferman Spaces

Let M be a strictly pseudoconvex CR manifold, of CR dimension n, and let $\theta$ be a positively oriented contact form on M. A complex p-form $\eta$ on M is of type (p, 0), or a (p, 0)-form, if $T_{0,1} (M) \, \rfloor \, \eta =0$. Let $\Lambda ^{p,0} (M) \subset \Lambda ^p T^*(M) \otimes {{\mathbb {C}}}$ be the relevant bundle. Unlike the case of complex geometry, top degree (p, 0)-forms are $(n+1, 0)$-forms [rather than (n, 0)-forms, due to the presence of the additional real cotangent direction $\theta$]. Then $\mathbb {R}_+ = \textrm{GL}^+ (1, {{\mathbb {R}}})$ acts freely on $K^0 (M) = \Lambda ^{n+1,0} (M){\setminus } \{ \mathrm{zero \ section}\}$ and

$C(M) = K^0 (M) /\mathbb {R}_+$ is a principal $S^1$ bundle over M (the canonical circle bundle). The $(2n+2)$-dimensional manifold C(M) carries the Lorentzian metric $F_\theta$ (the Fefferman metric) naturally associated to $\theta$

$$\begin{aligned} F_\theta = \pi ^* \tilde{G}_\theta + 2 (\pi ^* \theta ) \odot \sigma \, . \end{aligned}$$

(34)

Cf. Lee [49] (or [29, pp. 128–129]). Here $\tilde{G}_\theta$ is the (degenerate) bilinear form on T(M) got by requiring that $\tilde{G}_\theta = G_\theta$ on $H(M) \otimes H(M)$ and $\tilde{G}_\theta (T, V) = 0$ for any $V \in \mathfrak {X}(M)$. Also $\sigma$ is the (globally defined) real 1-form on C(M) given by

$$\begin{aligned} \sigma =\frac{1}{n+2} \left[ d\gamma + \pi ^* \left( i \, {\omega _\alpha }^\alpha - \frac{i}{2} \,g^{\alpha \overline{\beta }} \,d g_{\alpha \overline{\beta }}-\frac{\rho }{4(n+1)} \, \theta \right) \right] , \end{aligned}$$

(35)

where $\pi : C(M) \rightarrow M$ is the projection and $\gamma : \pi ^{-1} (U) \rightarrow {{\mathbb {R}}}$ is a local fiber coordinate on C(M). Also, for any local frame $\{ T_\alpha \,: \, 1 \le \alpha \le n \} \subset C^\infty \big ( U, \, T_{1,0}(M) \big )$

$$\begin{aligned} \nabla T_\beta = {\omega _\beta }^\alpha \otimes T_\alpha \,, \;\;\; g_{\alpha \overline{\beta }} = G_\theta \big ( T_\alpha \,, \, T_{\overline{\beta }} \big ), \end{aligned}$$

$$\begin{aligned} \big [ g^{\alpha \overline{\beta }} \big ] = \big [ g_{\alpha \overline{\beta }} \big ]^{-1} \,, \;\;\; R_{\alpha \overline{\beta }} = {{R_\alpha }^\gamma }_{\gamma \overline{\beta }} \,, \;\;\; \rho = g^{\alpha \overline{\beta }} \, R_{\alpha \overline{\beta }}, \end{aligned}$$

and $\rho$ is the pseudohermitian scalar curvature (cf. [29, p. 50]). By a result of Graham (cf. [40]) $\sigma$ is a connection 1-form on the canonical circle bundle [the Graham connection on C(M)]. For every tangent vector field $X\in \mathfrak {X}(M)$ let $X^\uparrow \in \mathfrak {X}(C(M))$ be the horizontal lift of X with respect to the Graham connection i.e.

$$\begin{aligned} X_p^\uparrow \in \textrm{Ker} (\sigma _p ), \;\;\; (d_p \pi ) X_p^\uparrow = X_{\pi (p)}, \end{aligned}$$

for any $p \in C(M)$. Let $S\in \mathfrak {X} (C(M))$ be the tangent to the $S^1$ action. Locally $S = [(n+2)/2] \, \big ( \partial /\partial \, \gamma \big )$. Then $T^\uparrow -S$ is a globally defined time-like vector field on C(M), hence the Lorentzian manifold $(C(M), F_\theta )$ is time oriented.

Let $\Box$ be the Laplace–Beltrami operator of the Lorentzian manifold $(C(M), F_\theta )$ (the geometric wave operator). By a result of Lee (cf. [49]) the pushforward of $\Box$ is precisely the sublaplacian $\Delta _b$ of $(M, \, \theta )$ i.e.

$$\begin{aligned} \pi _*\Box = \Delta _b. \end{aligned}$$

(36)

By a result of Barletta et al. (cf. [9]) a $C^\infty$ map $\phi : \, (M, \theta ) \rightarrow (N, h)$ is subelliptic harmonic if and only if its vertical lift $\Phi = \phi \circ \pi : (C(M), F_\theta ) \rightarrow (N, h)$ is a harmonic map.

Our main purpose in the present section is to relate ($S^1$ invariant) harmonic morphisms from $(C(M), \, F_\theta )$ to subelliptic harmonic morphisms from $(M, \theta )$, in the spirit of the geometric interpretation of subelliptic harmonic maps provided in [8], and prove Theorem 3.

The equivalence (i) $\Longleftrightarrow$ (ii) in Theorem 3 may be accounted for, as follows. Let $v: V \rightarrow {{\mathbb {R}}}$ be a $C^2$ solution to $\Delta _h v = 0$ with $V \subset N$ open, and let ${{\mathscr {U}}} = \Phi ^{-1} (V)$ and $U = \phi ^{-1} (V)$. For any $x \in U$ and $p\in \pi ^{-1} (x) \subset {{\mathscr {U}}}$ [by (36)]

$$\begin{aligned} \Box (v \circ \Phi ) \, (p) = \Box (v \circ \phi \circ \pi ) (p) = (\pi _*\Box ) (v \circ \phi ) \, (x) = \Delta _b (v \circ \phi ) \, (x), \end{aligned}$$

[hence $\Box (v \circ \Phi ) = 0$ in ${{\mathscr {U}}}$ $\Longleftrightarrow$ $\Delta _b (v\circ \phi ) = 0$ in U]. $\square$

Proof of Theorem 1

Follows from statement (i) $\Longrightarrow$ (ii) in Theorem 3, a result by S. Dragomir & E. Lanconelli (cf. Corollary 4 in [25, p. 421]), and the fact that $\pi : C(M) \rightarrow M$ is an open map. $\square$

Proof of Theorem 2

Follows from Theorem 3 and Proposition 1. $\square$

The study of harmonic morphisms in the semi-Riemannian category was started by Fuglede [35] (cf. also Parmar [54]) and the state-of-the-art up to 2003 is described in the monograph [5], where from we recall a few basic notions, confined to our needs i.e. to the case of harmonic morphisms from the Lorentzian manifold $(C(M), \, F_\theta )$ into the Riemannian manifold $(N, \, h)$.

Definition 4

A $C^\infty$ map $\Phi : C(M) \rightarrow N$ is harmonic if it is a critical point of the energy functional

$$\begin{aligned} {{\mathbb {E}}}_D (\Phi ) = \frac{1}{2} \int _{D} \textrm{Trace}_{F_\theta } \big ( \Phi ^*h \big ) \; d \, \textrm{vol}(F_\theta ), \end{aligned}$$

for any relatively compact domain $D \subset \subset C(M)$. That is

$$\begin{aligned} \frac{d}{d t} \Big \{ {{\mathbb {E}}}_D \big ( \Phi _t \big ) \Big \}_{t = 0} = 0, \end{aligned}$$

for every smooth 1-parameter variation $\{ \Phi _t \}_{|t| < \epsilon } \subset C^\infty \big ( C(M), \, N\big )$ of $\Phi _0 = \Phi$ supported in $\Omega$ i.e. $\textrm{Supp} ({{\mathbb {V}}}) \subset D$. $\square$

Here ${{\mathbb {V}}} \in C^\infty \big ( \Phi ^{-1} T N \big )$ is the infinitesimal variation induced by $\{ \Phi _t \}_{|t| < \epsilon }$ i.e.

$$\begin{aligned} {{\mathbb {V}}}_p = (d_{(p, 0 )} \Phi \big ) \Big ( \frac{\partial }{\partial t} \Big )_{(p, 0)} \,, \;\;\; p \in C(M). \end{aligned}$$

Also

$$\begin{aligned} d \, \textrm{vol}(F_\theta ) = \sqrt{- G} \, d \gamma \wedge d u^1 \wedge \cdots \wedge d u^{2n+1}, \end{aligned}$$

is the canonical volume form on C(M), associated to the Lorentzian metric $F_\theta$, where we have set

$$\begin{aligned} G= & {} \det \big [ g_{rs} \big ] \,, \;\;\; g_{rs} = F_\theta \big ( \partial _r \,, \, \partial _s \big ), \\ \partial _s\equiv & {} \frac{\partial }{\partial u^s} \,, \;\;\; 0 \le s \le 2n+1, \;\;\; u^0 = \gamma \,,\\ u^A= & {} x^A \circ \pi \,, \;\;\; 1 \le A \le 2 n + 1, \end{aligned}$$

and $(U, \, x^A )$ is an arbitrary local coordinate system on M. The Euler–Lagrange equations of the variational principle $\delta \, {{\mathbb {E}}}_D (\Phi ) = 0$ are

$$\begin{aligned} \tau _{F_\theta } (\Phi ) = 0, \end{aligned}$$

(37)

where $\tau _{F_\theta } (\Phi ) \in C^\infty (\Phi ^{-1} T N)$ is the tension field of $\Phi$ i.e.

$$\begin{aligned} \tau _{F_\theta } (\Phi )= & {} \textrm{trace}_{F_\theta } \big [ \beta _{F_\theta } (\Phi ) \big ], \\ \beta _{F_\theta } (\Phi ) ( A, B)= & {} D^\Phi _A \Phi _*B - \Phi _*\nabla ^{F_\theta }_A B, \\ D^\Phi= & {} \Phi ^{-1} \nabla ^h \,, \;\;\; A, \, B \in \mathfrak {X}\big ( C(M) \big ). \end{aligned}$$

Let $\Phi : C(M) \rightarrow N$ be a $C^\infty$ map. For each point $p \in C(M)$ we set

$$\begin{aligned} {{\mathscr {V}}}_p^\Phi = \textrm{Ker } \big ( d_p \Phi \big ), \;\;\; {{\mathscr {H}}}_p^\Phi = \left( {{\mathscr {V}}}_p^\Phi \right) ^\perp \,, \end{aligned}$$

(the perp space is meant with respect to $F_\theta$).

Definition 5

$\Phi : C(M) \rightarrow N$ is nondegenerate at $p \in C(M)$ if ${{\mathscr {V}}}_p^\Phi$ is a nondegenerate subspace of the inner product space $\big ( T_p (C(M)), \, F_{\theta , \, p} \big )$. Otherwise $\Phi$ is degenerate at $p\in C(M)$. $\square$

We also recall (cf. [5, p. 444], or Fuglede [35]).

Definition 6

Let $\Phi : C(M) \rightarrow N$ be a $C^1$ map, and let $p \in C(M)$ be a point. $\Phi$ is horizontally weakly conformal at p provided that

(i) If $p \in C(M) \setminus \textrm{Crit}(\Phi )$ and ${{\mathscr {V}}}_p^\Phi$ is nondegenerate, then the differential $d_p \Phi : \mathcal {H}_p^\Phi \rightarrow T_{\Phi (p)} (N)$ is on-to, and there is a unique nonzero number $L (p) \in \mathbb {R} {\setminus } \{ 0 \}$ such that

$$\begin{aligned} h_{\Phi (p)} \big ( (d_p \Phi ) X, \, (d_p \Phi ) Y \big ) = L (p) \, F_{\theta , \, p} (X, Y), \end{aligned}$$

(38)

for any $X, Y \in \mathcal {H}_p^\Phi$.

(ii) If $p \in C(M)$ and ${{\mathscr {V}}}_p^\Phi$ is degenerate, then

$$\begin{aligned} {{\mathscr {H}}}_p^\Phi \subset {{\mathscr {V}}}_p^\Phi , \end{aligned}$$

(39)

[i.e. $F_{\theta , \, p} (X, Y) = 0$ for any $X, Y \in {{\mathscr {H}}}_p^\Phi$]. The number L(p) is the (square) dilation at p. $\square$

It is customary to set $L (p) = 0$ when $p \in \textrm{Crit}(\Phi )$ or ${{\mathscr {V}}}_p^\Phi$ is degenerate. The resulting function $L: C(M) \rightarrow \mathbb {R}$ [the (square)¹dilation of $\Phi$] is continuous. Also $\Phi \in C^\infty \Longrightarrow L \in C^\infty$. Occasionally we refine the notation to $L = L(\Phi )$. We shall need the following characterization of horizontal weak conformality (cf. [5, pp. 444–445]).

Lemma 8

Let $\Phi : C(M) \rightarrow N$ be a $C^1$ map, and let $p \in C(M)$. The following statements are equivalent

(i) $\Phi$ is horizontally weakly conformal at $p \in C(M)$, with dilation L(p).

(ii) There is $L(p) \in {{\mathbb {R}}}$ such that

$$\begin{aligned} (\Phi ^*h )_p = L (p) \, F_{\theta , \, p} \;\; {\textrm{on}} \;\; {{\mathscr {H}}}_p^\Phi \times {{\mathscr {H}}}_p^\Phi , \end{aligned}$$

and $L(p) \ne 0 \Longrightarrow d_p \Phi$ is on-to.

(iii) There is $L(p) \in {{\mathbb {R}}}$ such that, for every local coordinate system $\big ( V, \, y^i )$ on N about $\Phi (p)$

$$\begin{aligned} F_{\theta } (\nabla \Phi ^i, \nabla \Phi ^j )_p = L(p) \, h^{ij} \big ( \Phi (p) \big ) \,, \;\;\; 1\le i, \, j \le m, \end{aligned}$$

where $\Phi ^i = y^i \circ \Phi$.

Here $\nabla u = \nabla ^{F_\theta } u$ is the gradient of $u \in C^1 (C(M))$ with respect to the Fefferman metric i.e. $F_\theta (\nabla u, \, X ) = X(u)$ for any $X \in \mathfrak {X}(C(M))$.

We shall need the following result (the semi-Riemannian version of the Fuglede–Ishihara theorem, cf. Theorem 14.6.2 in [5, p. 447], or Fuglede [35])

Theorem 7

A $C^2$ map $\Phi : C(M) \rightarrow N$ is a harmonic morphism of the Lorentzian manifold $(C(M), \, F_\theta )$ into the Riemannian manifold $(N, \, h)$ if and only if $\Phi$ is both a harmonic map, and a horizontally weakly conformal map.

We now attack the remaining part of the proof of Theorem 3. We start by observing that

Lemma 9

The dilation $L(\Phi )$ is $S^1$-invariant.

Proof

Indeed the distributions ${{\mathscr {V}}}^\Phi$ and ${{\mathscr {H}}}^\Phi$ are invariant by right translations with respect to the natural action of $S^1$ on C(M) i.e. for every $p \in C(M)$ and $a \in S^1$

$$\begin{aligned} (d_p\, R_a ) \, {{\mathscr {V}}}^\Phi _p = {{\mathscr {V}}}^\Phi _{p\cdot a} \quad \textrm{and}\quad \,(d_p \, R_a ) \, {{\mathscr {H}}}^\Phi _p = {{\mathscr {H}}}^\Phi _{p\cdot a}. \end{aligned}$$

Here $R_a: C(M) \rightarrow C(M)$ denotes the right translation with $a \in S^1$. Next [as $\Phi$ is horizontally weakly conformal]

$$\begin{aligned} (\Phi ^* h)_p = L(\Phi )_p \, F_{\theta , p} \;\; \text {on} \;\; {{\mathscr {H}}}^\Phi _p \times {{\mathscr {H}}}^\Phi _p. \end{aligned}$$

(40)

Let $a\in S^1$ and $u, \, v \in {{\mathscr {H}}}^\Phi _{p}$, and let us set

$$\begin{aligned} u^\prime = (d_p R_a) u, \;\;\; v^\prime = (d_p R_a) v. \end{aligned}$$

Then [by (40)]

$$\begin{aligned}{} & {} L(\Phi )_{p\cdot a} \, F_{\theta , \, p\cdot a} (u^\prime , \, v^\prime ) = h_{\Phi (p)} \big ( d_p (\Phi \circ R_a) u, \, d_p (\Phi \circ R_a) v \big ) \\{} & {} \quad = h_{\phi (p)} \big ( (d_p \Phi ) u, \, (d_p \Phi ) v \big ) = L(\Phi )_p \, F_{\theta , p } (u, v), \end{aligned}$$

and [by $S^1 \subset \textrm{Isom} (C(M), F_\theta )$]

$$\begin{aligned} F_{\theta , \, p\cdot a} (u^\prime , \, v^\prime ) = \big ( R_a^*F_\theta \big )_p \big ( u, \, v \big ) = F_{\theta , \, p } (u, v), \end{aligned}$$

yielding

$$\begin{aligned} \big [ L(\Phi )_{p\cdot a} - L(\Phi )_p \big ] \, F_{\theta , \, p} (u, \, v) = 0, \end{aligned}$$

so that $L(\Phi )_{p\cdot a} = L(\Phi )_p$ when ${{\mathscr {H}}}^\Phi _p$ is nondegenerate, and $L(\Phi )_p = 0$ when ${{\mathscr {H}}}^\Phi _p$ is degenerate. Once again, as the right translation $R_a$ is an isometry, the degeneracy of ${{\mathscr {H}}}^\Phi _p$ implies that of ${{\mathscr {H}}}^\Phi _{p \cdot a}$, and hence $L(\Phi )_{p\cdot a} = 0$.

$\square$

Next, we relate the horizontal weak conformality condition on $\Phi = \phi \circ \pi$ to the Levi conformality condition on $\phi$. Let us set

$$\begin{aligned} \Phi ^j = y^j \circ \Phi , \;\;\; \phi ^j = y^j \circ \phi , \;\;\; 1 \le j \le m. \end{aligned}$$

Let $\{E_a \,: \, 1 \le a \le 2n \}$ be a local $G_\theta$-orthonormal frame of H(M)

$$\begin{aligned} G_\theta \big ( E_a \,, \, E_b \big ) = \delta _{ab} \,, \;\;\; 1 \le a, \, b \le 2n, \end{aligned}$$

defined on the open set $U \subset M$. Then

$$\begin{aligned}{} & {} \{ \mathcal {E}_\alpha \,: \, 0 \le \alpha \le 2n+1\},\\{} & {} \mathcal {E}_0 = T^\uparrow - S, \;\;\; \mathcal {E}_a = E^\uparrow _a \,, \;\;\; 1 \le a \le 2n, \;\;\; \mathcal {E}_{2n+1} = T^\uparrow + S, \end{aligned}$$

is a local $F_\theta$-orthonormal frame of T(C(M)) i.e.

$$\begin{aligned} F_\theta \big ( {{\mathcal {E}}}_\alpha \,, \, {{\mathcal {E}}}_\beta \big ) = \epsilon _\alpha \, \delta _{\alpha \beta } \,, \;\;\; \epsilon _0 = - 1, \;\; \epsilon _j = 1, \;\; 1 \le j \le 2n + 1, \end{aligned}$$

on $\pi ^{-1} (U)$. Then

$$\begin{aligned} \nabla \Phi ^j= \lambda ^\alpha \, \mathcal {E}_\alpha , \end{aligned}$$

for some $\lambda ^\alpha \in C^\infty \big ( \pi ^{-1} (U) \big )$. Contracting with $F_\theta$ one gets

$$\begin{aligned} \lambda ^0 = - T^\uparrow (\Phi ^j), \quad \lambda ^a = E_a^\uparrow (\Phi ^j), \quad \lambda ^{2n+1} = T^\uparrow (\Phi ^j), \end{aligned}$$

namely

$$\begin{aligned} \nabla \Phi ^ j = \sum _{a=1}^{2n} E_a^\uparrow (\Phi ^j) E_a^\uparrow + 2 T^\uparrow (\Phi ^j) S. \end{aligned}$$

(41)

Consequently

$$\begin{aligned} \pi _*\, \nabla \Phi ^j = \big ( \nabla ^ H \phi ^j \big )^\pi \, . \end{aligned}$$

(42)

We recall the following (cf. Proposition 14.5.4 in [5, p. 445])

Proposition 1

A $C^\infty$ map $\Phi : C(M) \rightarrow N$ is horizontally weakly conformal at $p\in C(M)$ with (square) dilation L(p) if and only if one of the following statements holds

(i) $L(p)\ne 0$, $d_p \Phi$ is on-to, and $(\Phi ^*h )_p = L (p) \, F_{\theta , \, p}$ on ${{\mathscr {H}}}_p^\Phi \times {{\mathscr {H}}}_p^\Phi$.

(ii) $p\in \textrm{Crit} (\Phi )$ [so that $L(p)=0$ and $d_p\Phi =0$].

(iii) ${{\mathscr {V}}}^\Phi _p$ is degenerate and ${{\mathscr {H}}}^\Phi _p \subset {{\mathscr {V}}}^\Phi _p$ [so that $L(p) = 0$ yet $d_p \Phi \ne 0$].

Statement (a) in Theorem 3 is proved in two steps i.e. we show that

Lemma 10

$\Phi$ is nondegenerate at p $\Longleftrightarrow$ $L(\Phi )_p > 0$ $\Longleftrightarrow$ $\pi (p) \in \Omega (\phi )$.

We proceed by distinguishing between the cases contemplated by Proposition 1. To start with, let us assume that $L (p) \ne 0$. Then $d_p \Phi \ne 0$, the restriction of $d_p \Phi$ to ${{\mathscr {H}}}^\Phi _p$ is surjective, and (38) holds. Moreover $\Phi$ is nondegenerate at p and [by (iii) in Lemma 8] $\{ (\nabla \Phi ^j)_p \, \ 1 \le j \le m\}$ is a linear basis in ${{\mathscr {H}}}_p^\Phi$. Once again by (iii) in Lemma 8

$$\begin{aligned} L (\Phi )_p \; h^{jj} (\Phi (p)) = F_{\theta , \, p} (\nabla \Phi ^j, \nabla \Phi ^j) =, \end{aligned}$$

[by (41) together with the fact that S is lightlike and $F_\theta$-orthogonal to each $E_a^\uparrow$]

$$\begin{aligned} = \sum _{a=1}^{2n} \Big [ E_a^\uparrow (\Phi ^j) (p) \Big ]^2 \ge 0. \end{aligned}$$

Hence [as h is Riemannian] $0 \ne L (\Phi )_p \ge 0$ i.e. $L (\Phi )_p > 0$. Thus $L (\Phi )_p > 0$ is a necessary condition for the nondegeneracy of $\Phi$ at p. Clearly, it also suffices [if $L(\Phi )_p > 0$ then $d_p \Phi$ is onto and ${{\mathscr {H}}}^\Phi _p$ is space-like by (ii)of Lemma 8]. So $\Phi$ is nondegenerate at p $\Longleftrightarrow$ $L (\Phi )_p >0$. Next [by Lemma 9] there is a $C^\infty$ function $\ell (\Phi ): M \rightarrow (0, \, + \infty )$ such that $L (\Phi )= \ell (\Phi ) \circ \pi$. The horizontal weak conformality condition on $\Phi$ is then

$$\begin{aligned} F_\theta \big ( \nabla \Phi ^i \, , \, \nabla \Phi ^j \big )_p = \ell (\Phi )_{\pi (p)} \, h^{ij} (\Phi (p)), \end{aligned}$$

(43)

or [by (34)]

$$\begin{aligned}{} & {} \big ( \pi ^*\tilde{G}_\theta \big ) (\nabla \Phi ^i, \nabla \Phi ^j )_p + (\pi ^*\theta ) (\nabla \Phi ^i )_p \; \sigma ( \nabla \Phi ^j )_p + (\pi ^*\theta ) (\nabla \Phi ^j )_p \; \sigma ( \nabla \Phi ^i )_p \\{} & {} \quad = \ell (\Phi )_{\pi (p)} \; h^{ij} (\phi (\pi (p))), \end{aligned}$$

hence [by (42), and then by $\big ( \pi ^*\theta \big ) ( \nabla \Phi ^j ) = 0$]

$$\begin{aligned} G_\theta (\nabla ^H \phi ^i \, , \, \nabla ^H \phi ^j )_x = \ell (\Phi )_x \; h^{ij} (\phi (x)) , \;\;\; x = \pi (p ). \end{aligned}$$

(44)

Next [by the Levi-conformality condition on $\phi$]

$$\begin{aligned} G_\theta ( \nabla ^H \phi ^i \,, \, \nabla ^H \phi ^j )_x = \Lambda (x) \; h^{ij} (\phi (x)), \end{aligned}$$

hence [as h is positive definite and $\Lambda (\phi )_x = \ell (\Phi )_x$]

$$\begin{aligned} L (\Phi )_p = \Lambda (\phi )_{\pi (p)}, \end{aligned}$$

so that $L (\Phi ) (p) > 0 \Longleftrightarrow x \in M {\setminus } [Z(\Lambda (\phi ))]$. $\square$

Let us now examine the case $L(p) = 0$, when either $p\in \textrm{Crit } (\Phi )$ or $d_p \Phi \ne 0$, ${{\mathscr {V}}}_p^\Phi$ is degenerate, and ${{\mathscr {H}}}_p^\Phi \subset {{\mathscr {V}}}_p^\Phi$.

(b) If $p \in \textrm{Crit } (\Phi )$ then $x = \pi (p) \in \textrm{Crit } (\phi )$ and conversely. Indeed let $d_p \Phi = 0$. Then $\big ( \nabla \Phi ^j \big )_p = 0$ hence [by (41)]

$$\begin{aligned} 0 = (d_p \pi ) (\nabla \Phi ^ j)_p = (\nabla ^H \phi ^j)_x, \end{aligned}$$

so that $(\nabla \phi ^j)_x = T(\phi ^j)_x \; T_x$. Yet

$$\begin{aligned} 0 = (d_p \Phi ^j ) T^\uparrow _p = (d_x \phi ^j ) T_x = T (\phi ^j )_x, \end{aligned}$$

so that $(\nabla \phi ^j )_x = 0$ i.e. $d_x \phi = 0$. $\square$

(c) If $d_p \Phi \ne 0$, ${{\mathscr {V}}}_p^\Phi$ is degenerate, and ${{\mathscr {H}}}_p^\Phi \subset {{\mathscr {V}}}_p^\Phi$, then (equivalently) $x \in Z(\Lambda ) \setminus \textrm{Crit} (\phi )$. Indeed, if $x \not \in Z(\Lambda )$ then [by our discussion of the case $L(p) \ne 0$] $\Phi$ is nondegenerate at p, while if $x \in \textrm{Crit} (\phi )$ then $p \in \textrm{Crit} (\Phi )$ [by statement (b) in Theorem 3]. The proof of statement (c) in Theorem 3 is now completed by applying Theorem 6 to $x = \pi (p) \in Z(\Lambda ) {\setminus } \textrm{Crit} (\phi )$.

5 Harmonic Morphisms Within Foliation Theory

Let $\phi : M \rightarrow N$ be a subelliptic harmonic morphism of $(M, \, \theta )$ into $(N, \, h)$, of $\theta$-dilation $\lambda (\phi )$, and let $\Phi = \phi \circ \pi$ be its vertical lift [a harmonic morphism of square dilation $L(\Phi ) = \lambda ^2 (\phi ) \circ \pi$]. Let $S(\phi )$ be the set of all submersive points of the morphism $\phi$ (cf. Definition 3 above). The connected components of the fibres of $\phi : S(\phi ) \rightarrow N$ are the leaves of a foliation ${{\mathscr {F}}}$ of $S(\phi )$. Let us set

$$\begin{aligned} S(\Phi ):= \pi ^{-1} \big [ S(\phi ) \big ] \subset C(M). \end{aligned}$$

Then $\Phi : S(\Phi ) \rightarrow N$ is a submersion and the corresponding foliation of $S(\Phi )$ is the pullback of ${\mathscr {F}}$ by $\pi$ i.e. the foliation $\pi ^*{{\mathscr {F}}}$ of C(M) whose tangent bundle is

$$\begin{aligned} T(\pi ^*{{\mathscr {F}}}) = T ({{\mathscr {F}}})^\uparrow \oplus \textrm{Ker} (d \pi ). \end{aligned}$$

The horizontal lift is meant with respect to the Graham connection $\sigma$. Cf. Molino [51, p. 54], and Dragomir and Nishikawa [26]. Cf. also [10].

Let $Q = \nu ({{\mathscr {F}}}) = T(M)/{{\mathscr {V}}}^\phi$ be the transverse bundle, and let $\Pi : T(M) \rightarrow Q$ be the projection. Let $\sigma _Q: Q \rightarrow {{\mathscr {H}}}^\phi$ be the vector bundle isomorphism

$$\begin{aligned} \sigma _Q (s) = {{\mathcal {H}}} \, Y = Y^{{\mathcal {H}}}, \;\;\; s \in Q, \;\; Y \in T(M), \;\; \Pi \, (Y) = s, \end{aligned}$$

and let $g_Q$ be the Riemannian bundle metric

$$\begin{aligned} g_Q \big ( s, \, r \big ) = g_\theta \big ( \sigma _Q (s), \, \sigma _Q (r) \big ), \;\;\; s, \, r \in Q. \end{aligned}$$

Let us consider the Q-valued symmetric 2-form $\alpha$ on ${{\mathscr {V}}}^\phi \otimes {{\mathscr {V}}}^\phi$, the bundle endomorphism $W(Z): {{\mathscr {V}}}^\phi \rightarrow {{\mathscr {V}}}^\phi$, and the basic 1-form $\kappa \in \Omega ^1_B ({{\mathscr {F}}})$, given by

$$\begin{aligned} g_{{\mathscr {F}}} \big ( X, \, X^\prime \big )= & {} g_\theta \big ( X, \, X^\prime \big ), \;\;\; \alpha (X, \, X^\prime ) = \Pi \, \nabla ^{g_\theta }_X X^\prime \,,\\ g_{{\mathscr {F}}} \big ( W(Z) X, \; X^\prime \big )= & {} g_Q \big ( \alpha (X, \, X^\prime ), \, Z \big ), \\ \kappa (Z )= & {} \textrm{Trace} \, W(Z), \end{aligned}$$

for any $X, \, X^\prime \in {{\mathscr {V}}}^\phi$ and $Z \in {{\mathscr {H}}}^\phi$. We follow the notations and conventions in Tondeur [61]. Let $\chi _{{\mathscr {F}}} \in \Omega ^p (M)$ [with $p = 2 n - m + 1$] be the tangential volume form i.e.

$$\begin{aligned}{} & {} \chi _{{\mathscr {F}}} \big ( Y_1 \,, \, \ldots \,, \, Y_p \big ) = \det \Big [ g_\theta \big ( Y_i \,, \, E_j \big ) \Big ]_{1 \le i, \, j \le p} \,,\\{} & {} \quad \big \{ E_i \;: \; 1 \le i \le p \big \} \equiv \big \{ V_j \,, \; {{\mathscr {T}}} \;: \; 1 \le j \le 2 n - m \big \}, \\{} & {} \quad Y_1 \,, \, \ldots \,, \, Y_p \in T(M). \end{aligned}$$

Note that ${{\mathscr {H}}}^\phi \, \rfloor \, \chi _{{\mathscr {F}}} = 0$. Rummler’s formula is (cf. Eq. (6.17) in [61, p. 66])

$$\begin{aligned} \big ( {{\mathscr {L}}}_Z \; \chi _{{\mathscr {F}}} \big ) \big |_{{{\mathscr {V}}}^\phi } = - \kappa (Z) \; \chi _{{\mathscr {F}}} \, \big |_{{{\mathscr {V}}}^\phi }, \end{aligned}$$

(45)

where ${{\mathscr {L}}}_Z$ is the Lie derivative. Next, let

$$\begin{aligned} \textrm{div}_{{\mathscr {F}}} (X) = \sum _{i=1}^p g_\theta \big ( \nabla ^{g_\theta }_{E_i} X, \; E_i \big ), \;\; X \in {{\mathscr {V}}}^\phi , \end{aligned}$$

[globally defined, as the trace of $X^\prime \in {{\mathscr {V}}}^\phi \mapsto \pi ^{{{\mathscr {V}}}^\phi } \, \nabla ^{g_\theta }_{X^\prime } X$] be the divergence operator along the leaves. Similar to (45)

$$\begin{aligned}{} & {} \big ( {{\mathscr {L}}}_X \; \chi _{{\mathscr {F}}} \big ) \big ( X_1 \,, \, \ldots \,, \, X_p \big ) = \textrm{div}_{{\mathscr {F}}} (X) \; \chi _{{\mathscr {F}}} \big ( X_1 \,, \, \ldots \,, \, X_p \big ),\\{} & {} \quad X_1 \,, \, \ldots \,, \, X_p \in {{\mathscr {V}}}^\phi . \end{aligned}$$

Indeed

$$\begin{aligned}{} & {} \big ( {{\mathscr {L}}}_X \; \chi _{{\mathscr {F}}} \big ) \big ( E_1 \,, \, \ldots \,, \, E_p \big ) = X \big ( \chi _{{\mathscr {F}}} \big ( E_1 \,, \, \ldots \,, \, E_p \big ) \big )\\{} & {} \quad + - \sum _{j=1}^p \chi _{{\mathscr {F}}} \big ( E_1 \,, \, \ldots \,, \, E_{j - 1} \,, \, \big [ X, \, E_j \big ] \,, \, E_{j+1} \,, \, \ldots \,, \, E_p \big ) =, \end{aligned}$$

[by $\chi _{{\mathscr {F}}} \big ( E_1 \,, \, \ldots \,, \, E_p \big ) = 1$ and $\displaystyle {\pi _{{{\mathscr {V}}}^\phi } \, \big [ X, \, E_j \big ] = \sum _{i=1}^p g_\theta \big ( \big [ X, \, E_j \big ], \, E_i \big ) \, E_i}$]

$$\begin{aligned} = - \sum _{i=1}^p g_\theta \big ( \big [ X, \, E_i \big ], \; E_i \big ) = - \sum _{i=1}^p g_\theta \big ( \nabla ^{g_\theta }_X E_i - \nabla ^{g_\theta }_{E_i} X, \; E_i \big ) =, \end{aligned}$$

[by $\nabla ^{g_\theta } g_\theta = 0$ and $\Vert E_i \Vert = 1$]

$$\begin{aligned} =\sum _{i=1}^p g_\theta \big ( \nabla ^{g_\theta }_{E_i} X, \; E_i \big ) = \textrm{div}_{{\mathscr {F}}} \, (X). \end{aligned}$$

$\square$

5.1 Mean Curvature of Fibres

Let $({{\mathfrak {M}}}^{\textrm{N}}, \, g)$ be a $\mathrm N$-dimensional semi-Riemannian manifold, equipped with the semi-Riemannian metric g, and let D be a linear connection on ${\mathfrak {M}}$. Let ${{\mathscr {D}}}$ be a $C^\infty$ distribution on ${\mathfrak {M}}$, of rank $1 \le r \le \textrm{N} - 1$, and such that ${{\mathscr {D}}}_x$ is a nondegenerate subspace of $(T_x ({{\mathfrak {M}}}), \, g_x )$, for any $x\in {{\mathfrak {M}}}$. Let ${{\mathscr {D}}}^\bot$ be the orthogonal complement of ${\mathscr {D}}$, and let $\pi ^\bot : T({{\mathfrak {M}}}) \rightarrow {{\mathscr {D}}}^\bot$ be the projection associated to the direct sum decomposition $T(M) = {{\mathscr {D}}} \oplus {{\mathscr {D}}}^\bot$. Let us consider the bilinear form $B_{{\mathscr {D}}} = B_{{\mathscr {D}}} (g, \, D )$ given by

$$\begin{aligned} B_{{\mathscr {D}}} (X, \, Y) = \pi ^\bot \; \nabla _X Y, \;\;\; X, \, Y \in {{\mathscr {D}}}. \end{aligned}$$

Next, let $\mu ^{{\mathscr {D}}} = \mu ^{{\mathscr {D}}}(g, \, D )$ be given by

$$\begin{aligned} \mu ^{{\mathscr {D}}} = \frac{1}{r} \, \textrm{Trace}_g \; B_{{\mathscr {D}}} \in C^\infty \big ( {{\mathscr {D}}}^\bot \big ) . \end{aligned}$$

(46)

When $D = \nabla ^g$ [the Levi-Civita connection of $(\mathfrak {M}, \, g)$] $\mu ^{{\mathscr {D}}} = \mu ^{{\mathscr {D}}} (g, \, \nabla ^g )$ is the mean curvature vector of ${{\mathcal {D}}}$ (cf. e.g. Definition 1.26 in [29, p. 37]). Given a subelliptic harmonic morphism $\phi : M \rightarrow N$ under the assumptions of Theorem 6, we consider both the mean curvature vector of ${{\mathscr {V}}}^\phi$ in $(M, \, g_\theta )$

$$\begin{aligned} \mu ^{{{\mathscr {V}}}^\phi } \equiv \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \, , \, \nabla ^{g_\theta } \big ) \in C^\infty \big ( {{\mathscr {H}}}^\phi \big ), \end{aligned}$$

(47)

and its pseudohermitian analog [got by replacing the Levi–Civita connection of $(M, \, g_\theta )$ by the Tanaka–Webster connection $\nabla$ of $(M, \, \theta )$]

$$\begin{aligned} \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla \big ) \in C^\infty \big ( {{\mathscr {H}}}^\phi \big ). \end{aligned}$$

From now on, let us assume that $m\ge 2$ so that [by Theorem 6]

$$\begin{aligned} S (\phi ) = \Omega (\phi ) = M \setminus Z \big ( \Lambda (\phi ) \big ).\end{aligned}$$

By arguing as in the proof of Theorem 3 [case $L(p ) \ne 0$] for every $p \in S (\Phi ) = \pi ^{-1} \big [ S(\phi ) \big ]$ the horizontal space ${{\mathscr {H}}}^\Phi _p$ is space-like i.e. $F_{\theta , \, p}$ is positive definite on ${{\mathscr {H}}}^\Phi _p$. Consequently, for every $p \in S (\Phi )$ the vertical space ${{\mathscr {V}}}^\Phi _p$ has index $\textrm{ind} \, {{\mathscr {V}}}^\Phi _p = 1$, i.e. $F_{\theta , \, p}$ has signature $(1, \, 2 n - m + 1)$ on ${{\mathscr {V}}}^\Phi _p$. Therefore

Lemma 11

C(M) is foliated by $(2n - m + 2)$-dimensional Lorentzian manifolds, whose normal bundles are spacelike.

Let $\beta _p$ be the inverse of $d_p \Phi : {{\mathscr {H}}}^\Phi _p \simeq T_{\Phi (p)} (N)$ [$\beta : \Phi ^{-1} T(N) \rightarrow {{\mathscr {H}}}^\Phi$ is the horizontal lift, a vector bundle isomorphism].

Lemma 12

Let $\phi : M \rightarrow N$ be a subelliptic harmonic morphism of $(M, \, \theta )$ into $(N, \, h)$, of $\theta$-dilation $\lambda (\phi )$, and let $\Phi = \phi \circ \pi : C(M) \rightarrow N$ be its vertical lift $\big ($a harmonic morphism of square dilation $\ell (\Phi )^2 = \big [ \lambda (\phi ) \circ \pi \big ]^2\big )$. The second fundamental form $B_{F_\theta } (\Phi )$ of $\Phi$ satisfies

$$\begin{aligned}{} & {} B_{F_\theta } (\Phi ) (H, H) \nonumber \\{} & {} \quad = 2 \, H \left( \log \, \ell (\Phi ) \right) \; \Phi _*H - F_\theta (H, H) \; \Phi _*\nabla \left( \log \, \ell (\Phi )\right) , \end{aligned}$$

(48)

$$\begin{aligned}{} & {} B_{F_\theta } (\Phi ) (V, V) = - \Phi _*\nabla ^{F_\theta }_{V} V \,,\nonumber \\{} & {} \quad H \in C^\infty \big ( S(\Phi ), \; {{\mathscr {H}}}^\Phi \big ), \;\;\; V \in C^\infty \big ( S(\Phi ), \; {{\mathscr {V}}}^\Phi \big ), \end{aligned}$$

(49)

everywhere in $S(\Phi )$.

The proof of Lemma 12 is a verbatim repetition of the arguments in [5, pp. 119–120]. Let us check for instance (48). To this end, let $\{ Z_\alpha \,: \, 1 \le \alpha \le m \}$ be a local h-orthonormal frame of T(N), defined on the open set $V \subset N$. Then [by (ii) in Lemma 8]

$$\begin{aligned} \big \{ \ell (\Phi ) \, \beta \, Z_\alpha \,: \, \alpha \le j \le m \big \}, \end{aligned}$$

is a local $F_\theta$-orthonormal frame for ${{\mathscr {H}}}^\Phi$, defined on ${{\mathscr {U}}} = \Phi ^{-1} (V)$. Let $Y \in \mathfrak {X}(V)$ and let $H = \beta \, Y \in C^\infty ({{\mathscr {U}}}, \, {{\mathscr {H}}}^\Phi )$. Then

$$\begin{aligned} 2 \, \pi _{{{\mathscr {H}}}^\Phi } \left\{ \nabla ^{F_\theta }_{H} H \right\} = 2 \, \ell (\Phi )^2 \, \sum _{\alpha =1}^m \, F_\theta \big ( \nabla ^{F_\theta }_{H} H, \, \beta \, Z_\alpha \big ) \, \, \beta \, Z_\alpha =, \end{aligned}$$

(by the explicit expression of $\nabla ^{F_\theta }$ as a Levi–Civita connection, cf. e.g. Proposition 2.3 in [45, 46, Vol. I, p. 160])

$$\begin{aligned}{} & {} = \ell (\Phi )^2 \, \sum _{\alpha =1}^m \Big \{ 2 \, H \big ( F_\theta ( H, \, \beta \, Z_\alpha ) \big ) - (\beta \, Z_\alpha ) \big ( F_\theta (H, H) \big ) +\\{} & {} \quad - 2 \, F_\theta (H, \, [H, \, \beta \, Z_\alpha ]) \Big \} \, \beta \, Z_\alpha =, \end{aligned}$$

[by (ii) in Lemma 8 i.e.

$$\begin{aligned} F_\theta ( H, \, \beta \, Z_\alpha ) = \ell (\Phi )^{-2} \, h(H, \, Z_\alpha ) \circ \Phi , \end{aligned}$$

and by

$$\begin{aligned} \Phi _*\big [ H, \hat{Z}_\alpha \big ] = \big [ Y, \, Z_\alpha \big ] \circ \Phi , \end{aligned}$$

cf. e.g. Proposition B.1 in² [27, pp. 303–304]]

$$\begin{aligned}{} & {} = \ell (\Phi )^2 \, \sum _{\alpha =1}^m \Big \{ 2 \, H \big ( \ell (\Phi )^{-2} \big ) \, h(Y, \, Z_\alpha ) \circ \Phi \\{} & {} \quad + \, 2 \, \ell (\Phi )^{-2} \, H \big ( h(Y, \, Z_\alpha ) \circ \Phi \big ) - (\beta \, Z_\alpha ) \big ( \ell (\Phi )^{-2} \big ) \, h(Y, \, Y) \circ \Phi \, +\\{} & {} \quad - \ell (\Phi )^{-2} \, (\beta \, Z_\alpha ) \big ( h(Y, \, Y)\circ \Phi \big ) - 2 \, \ell (\Phi )^{-2} h(Y, \, [Y, \, Z_\alpha ]) \circ \Phi \Big \} \, \beta \, Z_\alpha . \end{aligned}$$

Next [again by Proposition 2.3 in [45, 46, Vol. I, p. 160], applied to the Levi–Civita connection $\nabla ^h$]

$$\begin{aligned} \pi _{{{\mathscr {H}}}^\Phi } \Big \{ \nabla ^{F_\theta }_{H} H \Big \}= & {} \left( \nabla ^h_{Y} Y \right) \circ \Phi \, + \sum _{\alpha = 1}^m \Big \{ - 2 \, H (\log \, \ell (\Phi ) ) \, h(Y, \, Z_\alpha ) \\{} & {} + ( \beta \, Z_\alpha )(\log \, \ell (\Phi ) ) \, h(Y, \, Y)^\Phi \Big \} \; \beta \, Z_\alpha , \end{aligned}$$

and

$$\begin{aligned} \sum _{\alpha =1}^m (\beta \, Z_\alpha )(\log \, \ell (\Phi )) \; h(Y, \, Y)^\Phi \hat{Z}_\alpha = F_\theta (H, \, H) \; \pi _{{{\mathscr {H}}}^\Phi } \, \nabla \, \log \, \ell (\Phi ), \end{aligned}$$

so that

$$\begin{aligned} \beta _{F_\theta } (\Phi ) (H, H)= & {} D^\Phi _{H}\Phi _*H - \Phi _*\, \pi _{{{\mathscr {H}}}^\Phi } \, \Big \{ \nabla ^{F_\theta }_{H} H \Big \} \\= & {} 2 \, H \big ( \log \, \ell (\Phi ) \big ) \Phi _*H - F_\theta (H, \, H) \, \Phi _*\, \nabla \, \log \, \ell (\Phi ). \end{aligned}$$

$\square$

Lemma 13

Under the assumptions of Lemma  12, the tension field of $\Phi$ is given by

$$\begin{aligned} \tau _{F_\theta } (\Phi ) = - (m-2) \, \Phi _*\, \nabla \, \log \, \ell (\Phi ) - (2n - m + 2) \, \Phi _*\, \mu ^{{{\mathscr {V}}}^\Phi }, \end{aligned}$$

(50)

everywhere in $S(\Phi )$ [the set of submersive points of $\Phi$].

Proof

Let

$$\begin{aligned}{} & {} \big \{ H_\alpha \,: \, 1 \le \alpha \le m \big \} \subset {{\mathscr {H}}}^\Phi \,, \\{} & {} \quad \big \{ V_k \,: \, 0 \le k \le 2n - m + 1 \big \} \subset {{\mathscr {V}}}^\Phi \,, \;\;\; F_\theta ( V_0 \,, \, V_0 ) = - 1, \end{aligned}$$

be local $F_\theta$-orthonormal frames. Then

$$\begin{aligned} \tau _{F_\theta } (\Phi )= & {} \textrm{Trace}_{F_\theta } \, \big \{ \beta _{F_\theta } (\Phi ) \big \} \\= & {} \sum _{\alpha =1}^m \beta _{F_\theta } (\Phi ) (H_\alpha \,, \, H_\alpha ) + \sum _{k=1}^{2n - m + 1} \beta _{F_\theta } (\Phi ) (V_k \,, \, V_k) -\beta _{F_\theta } (\Phi ) (V_0 \,, \, V_0), \end{aligned}$$

and [by Lemma 12]

$$\begin{aligned} \beta _{F_\theta } (\Phi ) \big ( H_\alpha \,, \, H_\alpha \big ) = 2 \, H_\alpha \big ( \log \, \ell (\Phi ) \big ) \, \Phi _*\, h_\alpha - \Phi _*\, \nabla \, \log \, \ell (\Phi ), \end{aligned}$$

so that

$$\begin{aligned} \sum _{\alpha =1}^m \beta _{F_\theta } (\Phi ) (H_\alpha \,, \, H_\alpha ) = 2 \, \Phi _*\, \nabla \, \log \, \ell (\Phi ) - m \, \Phi _*\, \nabla \, \log \, \ell (\Phi ). \end{aligned}$$

Also

$$\begin{aligned}{} & {} \beta _{F_\theta } (\Phi ) (V_k \,, \, V_k) = - \Phi _*\nabla ^{F_\theta }_{V_k} V_k \,,\\{} & {} \quad \sum _{k=1}^{2n-m+1} \beta _{F_\theta } (\Phi ) (V_k \,, \, V_k) - \beta _{F_\theta } (\Phi ) (V_0 \,, \, V_0) = - (2n - m + 2) \; \Phi _*\, \mu ^{{{\mathscr {V}}}^\Phi }. \end{aligned}$$

$\square$

Next, we project (50) on the base manifold M, so that to get a subelliptic version of the fundamental equation for a harmonic morphism (cf. e.g. Eq. (4.5.2) in [5, p. 129]), applying to the base map $\phi$. We start by recalling the following result³ (relating the Levi–Civita connection $\nabla ^{F_\theta }$ of $(C(M), \, F_\theta )$ to the Tanaka–Webster connection $\nabla$ of $(M, \, \theta )$, cf. [9, p. 26] or [1])

Lemma 14

For any $X, Y\in H(M)$

$$\begin{aligned}{} & {} \nabla ^{F_\theta }_{X^\uparrow }\, Y^\uparrow = \left( \nabla _X \, Y\right) ^\uparrow + \Omega (X, Y) \, T^\uparrow - \left[ A(X, Y) + (d\sigma ) (X^\uparrow , Y^\uparrow )\right] S \,, \end{aligned}$$

(51)

$$\begin{aligned}{} & {} \nabla ^{F_\theta }_{X^\uparrow }\, T^\uparrow = \left( \tau X + \varphi X \right) ^\uparrow , \end{aligned}$$

(52)

$$\begin{aligned}{} & {} \nabla ^{F_\theta }_{T^\uparrow }\, X^\uparrow = \left( \nabla _T \, X + \varphi X\right) ^\uparrow + 2 (d\sigma )(X^\uparrow , T^\uparrow ) S, \end{aligned}$$

(53)

$$\begin{aligned}{} & {} \nabla ^{F_\theta }_{X^\uparrow }\, S = \nabla ^{F_\theta }_{S}\, X^\uparrow = \frac{1}{2} \, (JX)^\uparrow , \end{aligned}$$

(54)

$$\begin{aligned}{} & {} \nabla ^{F_\theta }_{T^\uparrow }\, T^\uparrow = V^\uparrow \,, \end{aligned}$$

(55)

$$\begin{aligned}{} & {} \nabla ^{C(M)}_{S}\, S = \nabla ^{C(M)}_{S}\, T^\uparrow = \nabla ^{C(M)}_{T^\uparrow }\, S = 0 , \end{aligned}$$

(56)

where $\varphi : H(M) \rightarrow H(M)$ and $V\in H(M)$ are given by

$$\begin{aligned} G_\theta (\varphi X, \, Y ) =(d \sigma ) (X^\uparrow \,, \, Y^\uparrow ), \;\;\; G_\theta (V, \, Y) = 2 \, (d \sigma ) (T^\uparrow , Y^\uparrow ). \end{aligned}$$

The tension field $\tau _{F_\theta } (\Phi )$ may be shown to project on $\tau _b (\phi )$ i.e.

$$\begin{aligned} \tau _{F_\theta } (\Phi ) = \tau _b (\phi ) \circ \pi . \end{aligned}$$

(57)

To prove (57) let $\{ E_a\,: \, 1 \le a \le 2n \}$ be a local $G_\theta$-orthonormal frame of H(M), defined on the open set $U \subset M$. Then

$$\begin{aligned} \big \{ E_a^\uparrow \,, \; T^\uparrow \pm S \;: \; 1 \le a \le 2n \big \}, \end{aligned}$$

is a local $F_\theta$-orthonormal frame for $T \big ( C(M) \big )$, defined on ${{\mathscr {U}}} = \pi ^{-1} (U)$, so that

$$\begin{aligned} \tau _{F_\theta } (\Phi )= & {} \sum _{a=1}^{2n} \Big \{ D^\Phi _{E_a^\uparrow } \Phi _*\, E_a^\uparrow - \Phi _*\, \nabla ^{F_\theta }_{E_a^\uparrow } E_a^\uparrow \Big \} + D^\Phi _{T^\uparrow +S} \Phi _* (T^\uparrow + S) \\{} & {} - \Phi _*\, \nabla ^{F_\theta }_{T^\uparrow + S} (T^\uparrow +S) -D^\Phi _{T^\uparrow -S} \Phi _* (T^\uparrow - S) + \Phi _* \nabla ^{C(M)}_{T^\uparrow -S} (T^\uparrow -S). \end{aligned}$$

Also, for every $X \in H(M)$ [by Lemma 14 and $\Phi _*S = 0$]

$$\begin{aligned}{} & {} D^\Phi _{X^\uparrow } \Phi _*\, X^\uparrow = \Big \{ D^\phi _{X} \phi _*\, X \Big \} \circ \pi \,,\\{} & {} \quad \Phi _*\, \nabla ^{F_\theta }_{X^\uparrow } X^\uparrow = \left( \phi _*\, \nabla _{X} \, X \right) \circ \pi \,,\\{} & {} \quad D^\Phi _{T^\uparrow +S} \Phi _*\, (T^\uparrow + S) - D^\Phi _{T^\uparrow -S} \Phi _*\, (T^\uparrow - S) = 2 \, D^\Phi _S \, \Phi _*\, T^\uparrow = 0,\\{} & {} \quad - \nabla ^{F_\theta }_{T^\uparrow +S} (T^\uparrow +S) + \nabla ^{F_\theta }_{T^\uparrow -S} (T^\uparrow -S) = - 2 \, \Big \{ \nabla ^{F_\theta }_{T^\uparrow }\, S + \nabla ^{F_\theta }_S\, T^\uparrow \Big \} = 0. \end{aligned}$$

Finally

$$\begin{aligned} \tau _{F_\theta } (\Phi ) = \sum _{a=1}^{2n} \Big \{ D^\phi _{E_a} \phi _*\, E_a - \phi _*\, \nabla _{E_a} \, E_a \Big \} \circ \pi = \tau _b (\phi ) \circ \pi .\end{aligned}$$

$\square$

The gradient $\nabla \, \log \, \ell (\Phi )$ may be shown to project on the horizontal gradient $\nabla ^H \, \log \, \lambda (\phi )$. Indeed [by arguing as in the proof of (41)]

$$\begin{aligned} \nabla \, \log \, \ell (\Phi ) = \sum _{a=1}^{2n} E_a^\uparrow \big ( \log \, \ell (\Phi ) \big ) \; E_a^\uparrow + 2 \, T^\uparrow \big ( \log \, \ell (\Phi ) \big ) \, S. \end{aligned}$$

Also [by $\ell (\Phi ) = \lambda (\phi ) \circ \pi$] $X^\uparrow (\log \, \ell (\Phi )) = X ( \log \, \lambda (\phi )) \circ \pi$ hence

$$\begin{aligned} \Phi _*\, \nabla \, \log \, \ell (\Phi )= \big \{ \phi _*\, \nabla ^H \, \log \, \lambda (\phi ) \big \} \circ \pi . \end{aligned}$$

$\square$

Next, we seek to project the mean curvature vector

$$\begin{aligned} \mu ^{{{\mathscr {V}}}^\Phi } = \mu ^{{{\mathscr {V}}}^\Phi } \big ( F_\theta \,, \, \nabla ^{F_\theta } \big ) \in C^\infty \big ( {{\mathscr {H}}}^\Phi \big ). \end{aligned}$$

We need to produce a local $F_\theta$-orthonormal frame of ${{\mathscr {V}}}^\Phi$, adapted to the decomposition

$$\begin{aligned} T \big ( C(M) \big ) = H(M)^\uparrow \oplus {{\mathbb {R}}} T^\uparrow \oplus {{\mathbb {R}}} S, \end{aligned}$$

[and allowing for the use of Lemma 14]. We start from building a local $g_\theta$-orthonormal frame of ${{\mathscr {V}}}^\phi$, adapted to the decomposition $T(M) = H(M) \oplus {{\mathbb {R}}}$. Once again, difficulties arise from the fact that the pairs of complementary distributions $(H(M), \, {{\mathbb {R}}} T )$ and $\big ( {{\mathscr {V}}}^\phi \,, \, {{\mathscr {H}}}^\phi )$ intersect, and then the use of Theorem 6 is crucial in ascertaining that the intersections have constant ranks on certain open sets. Indeed

$$\begin{aligned} \dim _{{\mathbb {R}}} \big ( {{\mathscr {V}}}^\phi _H \big )_x = 2n - m, \end{aligned}$$

at every point $x \in S (\phi )= \Omega (\phi ) = \textrm{I}_m (\phi ) \cup \textrm{III}_m (\phi )$, provided that $m\ge 2$. Let then

$$\begin{aligned} \big \{ V_j \,: \, 1 \le j \le 2n - m \big \} \subset C^\infty (U, \, {{\mathscr {V}}}^\phi _H ), \end{aligned}$$

be a $G_\theta$-orthonormal frame, defined on the open set $U \subset \Omega (\phi )$. Let us set

$$\begin{aligned} T^{{\mathscr {V}}}:= \pi _{{{\mathscr {V}}}^\phi } \, T \in {{\mathscr {V}}}^\phi \,, \;\;\; T^{{\mathscr {H}}}:= T - T^{{\mathscr {V}}} \in {{\mathscr {H}}}^\phi . \end{aligned}$$

Our discussion in Sect. 3 shows that $T^{{\mathscr {V}}}_x \ne 0$ for every $x \in S(\phi )$, hence one may set

$$\begin{aligned} {{\mathscr {T}}}:= \frac{1}{\Vert T^{{\mathscr {V}}} \Vert } \; T^{{\mathscr {V}}} \in C^\infty \big ( \Omega (\phi ), \, {{\mathscr {V}}}^\phi \big ). \end{aligned}$$

Lemma 15

$\big \{ V_j \,, \; {{\mathscr {T}}} \,: \, 1\le j \le 2n - m \big \}$ is a $g_\theta$-orthonormal frame for ${{\mathscr {V}}}^\phi$, defined on $U \subset \Omega (\phi )$.

Proof

Note that ${{\mathscr {T}}}_x \ne 0$ for every $x \in \Omega (\phi )$ [otherwise $T_{x_0} \in {{\mathscr {H}}}^\phi _{x_0}$ for some $x_0 \in \Omega (\phi )$ i.e. $\textrm{II}_m (\phi ) \ne \emptyset$, a contradiction] and $\Vert {{\mathscr {T}}} \Vert = 1$. Also

$$\begin{aligned} g_\theta (V_j \,, \, {{\mathscr {T}}}) = \frac{1}{\Vert T^{{\mathscr {V}}} \Vert } \, g_\theta \big ( V_j \,, \, T^{{\mathscr {V}}}+ T^{{\mathscr {H}}} \big ) = \frac{1}{\Vert T^{{\mathscr {V}}} \Vert } \, g_\theta ( V_j \,, \, T) = 0.\end{aligned}$$

Also $\theta ({{\mathscr {T}}}) \ne 0$ everywhere in $\Omega (\phi )$. Indeed, if $\theta ({{\mathscr {T}}})_{x_0} = 0$ for some $x_0 \in \Omega (\phi )$ then

$$\begin{aligned} {{\mathscr {T}}}_{x_0} \in H(M)_{x_0} \cap {{\mathscr {V}}}^\phi _{x_0} = \big ( {{\mathscr {V}}}^\phi _H \big )_{x_0} \Longrightarrow \dim _{{\mathbb {R}}} \big ( {{\mathscr {V}}}^\phi _H \big )_{x_0} = 2 n - m + 1, \end{aligned}$$

a contradiction. $\square$

Using the local frame provided by Lemma 15 one may relate the mean curvature vector $\mu ^{{{\mathscr {V}}}^\phi } \equiv \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla ^{g_\theta } \big )$ to its pseudohermitian analog $\mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla \big )$ i.e.

$$\begin{aligned}{} & {} (2 n - m + 1) \, \mu ^{{{\mathscr {V}}}^\phi } = (2 n - m + 1) \, \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla \big ) \nonumber \\{} & {} \quad + \pi _{{{\mathscr {H}}}^\phi } \Big \{ \theta ({{\mathscr {T}}}) \, \big ( \tau + 2 \, J \big ) \, {{\mathscr {T}}} - \big ( \textrm{Trace}_{g_\theta } \; \Pi _{{{\mathscr {V}}}^\phi } \, A \Big ) \, T \Big \}, \end{aligned}$$

(58)

where $\Pi _{{{\mathscr {V}}}^\phi } \, A$ denotes the restriction of A to ${{\mathscr {V}}}^\phi \otimes {{\mathscr {V}}}^\phi$. Indeed [by (11)]

$$\begin{aligned} \nabla ^{g_\theta }_V V= & {} \nabla _V V - A(V, \, V) \, T + \theta (V) \, \tau \, V + 2 \, \theta (V) \, J \, V, \\ \nabla ^{g_\theta }_{{\mathscr {T}}} {{\mathscr {T}}}= & {} \nabla _{{\mathscr {T}}} {{\mathscr {T}}} - A ({{\mathscr {T}}}, \, {{\mathscr {T}}}) \, T + u \big ( \tau + 2 \, J \big ) \, {{\mathscr {T}}}, \end{aligned}$$

for any $V \in {{\mathscr {V}}}^\phi _H$. Hence

$$\begin{aligned}{} & {} \quad = \pi _{{{\mathscr {H}}}^\phi } \Big \{ \sum _{j=1}^{2n - m} \nabla ^{g_\theta }_{V_j} V_j + \nabla ^{g_\theta }_{{\mathscr {T}}} {{\mathscr {T}}} \Big \}\\{} & {} \quad = \pi _{{{\mathscr {H}}}^\phi } \Big \{ \sum _{j = 1}^{2n - m} \nabla _{V_j} V_j + \nabla _{{\mathscr {T}}} {{\mathscr {T}}} - \big ( \textrm{Trace}_{g_\theta } \, \Pi _{{{\mathscr {V}}}^\phi } \, A \big ) \, T + u \, ( \tau + 2 \, J \big ) \, {{\mathscr {T}}} \Big \}. \end{aligned}$$

$\square$

For further use, note that [by (11)]

$$\begin{aligned} \sum _{j=1}^{2n - m} g_\theta \big ( \nabla _{V_j} \, {{\mathscr {T}}}, \; V_j \big ) = \textrm{div}_{{\mathscr {F}}} \, ({{\mathscr {T}}}) + \theta ({{\mathscr {T}}} ) \, \big [ A({{\mathscr {T}}}, \, {{\mathscr {T}}}) - \textrm{Trace}_{g_\theta } \; \Pi _{{{\mathscr {V}}}^\phi } \, A \big ].\nonumber \\ \end{aligned}$$

(59)

Using the Graham connection $\sigma$ to lift $\{ V_j \,, \, {{\mathscr {T}}} \}$, one produces the local frame $\big \{ V_j^\uparrow \,, \; {{\mathscr {T}}}^\uparrow \,, \; S \;: \; 1 \le j \le 2n - m \big \}$ for ${{\mathscr {V}}}^\Phi$, defined on ${{\mathscr {U}}} = \pi ^{-1} (U) \subset S(\Phi )$.

Lemma 16

Let us set

$$\begin{aligned} u:= \theta ({{\mathscr {T}}} ), \;\;\;\; v:= \frac{2 - u^2}{u} \,, \;\;\;\; u, \, v \in C^\infty \big ( \Omega (\phi ) \big ). \end{aligned}$$

Then

$$\begin{aligned} \Big \{ V_j^\uparrow \,, \;\; {{\mathscr {T}}}^\uparrow + u \, S, \;\; {{\mathscr {T}}}^\uparrow - v \, S \;: \; 1 \le j \le 2n - m \Big \}, \end{aligned}$$

is a $F_\theta$-orthonormal frame for ${{\mathscr {V}}}^\Phi$ with ${{\mathscr {T}}}^\uparrow - v \, S$ timelike.

The proof is straightforward. Note that

$$\begin{aligned} |u| = \big | g_\theta \big ( {{\mathscr {T}}}, \; T \big ) \big | \le \Vert {{\mathscr {T}}} \Vert \; \Vert T \Vert = 1, \end{aligned}$$

and in particular $|v| \ge 1$.

Lemma 17

$$\begin{aligned} (2n - m + 2) \; \Phi _*\; \mu ^{{{\mathscr {V}}}^\Phi }= & {} (2n - m + 1) \, \phi _*\, \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla \big ) \nonumber \\{} & {} + \phi _*\left\{ \frac{2}{\theta ({{\mathscr {T}}})}\, J \, {{\mathscr {T}}} - \nabla _{{\mathscr {T}}} {{\mathscr {T}}} \right\} \circ \pi . \end{aligned}$$

(60)

Proof

We start by computing the needed components of $B_{{{\mathscr {V}}}^\Phi }$. By Lemma 14 and $S \in {{\mathscr {V}}}^\Phi$

$$\begin{aligned}{} & {} B_{{{\mathscr {V}}}^\Phi } \big ( X^\uparrow \,, \, X^\uparrow \big ) = \pi _{{{\mathscr {H}}}^\Phi } \, \nabla ^{F_\theta }_{X^\uparrow } X^\uparrow \\{} & {} \quad = \pi _{{{\mathscr {H}}}^\Phi } \, \Big \{ \big ( \nabla _X X \big )^\uparrow - A (X, \, X) \, S \Big \} = \pi _{{{\mathscr {H}}}^\Phi } \, \big ( \nabla _X X \big )^\uparrow \,,\\{} & {} \qquad B_{{{\mathscr {V}}}^\Phi } (S, \, S) = \pi _{{{\mathscr {H}}}^\Phi } \, \nabla ^{F_\theta }_S S = 0. \end{aligned}$$

The calculation of $B_{{{\mathscr {V}}}^\Phi } \big ( {{\mathscr {T}}}^\uparrow \,, \, S \big )$ is a bit trickier. One first decomposes ${{\mathscr {T}}} = \Pi _H \, {{\mathscr {T}}} + u \, T$ and then [again by Lemma 14 and $S (u \circ \pi ) = 0$]

$$\begin{aligned} \nabla ^{F_\theta }_S {{\mathscr {T}}}^\uparrow = \frac{1}{2} \, \big ( J \, \Pi _H \, {{\mathscr {T}}} \big )^\uparrow , \end{aligned}$$

i.e. (as $J \, T = 0$)

$$\begin{aligned} B_{{{\mathscr {V}}}^\Phi } \big ( {{\mathscr {T}}}^\uparrow \,, \, S \big ) = \frac{1}{2} \, \pi _{{{\mathscr {H}}}^\Phi } \, \big ( J \, {{\mathscr {T}}} \big )^\uparrow . \end{aligned}$$

Next (by Lemma 16 and $u + v = 2/u$)

$$\begin{aligned}{} & {} (2 n - m + 2 ) \, \mu ^{{{\mathscr {V}}}^\Phi } \\{} & {} \quad = \textrm{Trace}_{F_\theta } \, B_{{{\mathscr {V}}}^\Phi } = \sum _{j=1}^{2n - m} B_{{{\mathscr {V}}}^\Phi } \big ( V_j^\uparrow \,, \, V_j^\uparrow \big ) + B_{{{\mathscr {V}}}^\Phi } \big ( {{\mathscr {T}}}^\uparrow + u \, S, \, {{\mathscr {T}}}^\uparrow + u \, S \big ) \\{} & {} \qquad - B_{{{\mathscr {V}}}^\Phi } \big ( {{\mathscr {T}}}^\uparrow - v \, S, \, {{\mathscr {T}}}^\uparrow - v \, S \big ) \\{} & {} \quad = \sum _{j=1}^{2n - m} B_{{{\mathscr {V}}}^\Phi } \big ( V_j^\uparrow \,, \, V_j^\uparrow \big ) + \big ( u^2 - v^2 \big ) \, B_{{{\mathscr {V}}}^\Phi } (S, \, S) + 2 (u + v) \, B_{{{\mathscr {V}}}^\Phi } \big ( {{\mathscr {T}}}^\uparrow \,, \, S \big ) \\{} & {} \quad = \pi _{{{\mathscr {H}}}^\Phi } \, \Big [ \sum _{j=1}^{2n - m} \nabla _{V_j} V_j + \frac{2}{u} \, J \, {{\mathscr {T}}} \Big ]^\uparrow . \end{aligned}$$

Substitution from

$$\begin{aligned} \pi _{{{\mathscr {H}}}^\phi } \sum _{j=1}^{2n - m} \nabla _{V_j} V_j = (2 n - m + 1) \, \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla \big ) - \pi _{{{\mathscr {H}}}^\phi } \, \nabla _{{\mathscr {T}}} {{\mathscr {T}}}, \end{aligned}$$

yields (60). $\square$

Summing up [by Lemmas 13 to 17] the fundamental equation (50) projects on

$$\begin{aligned} \tau _b (\phi )= & {} - \frac{m - 2}{2} \, \phi _*\, \nabla ^H \log \, \Lambda (\phi ) - (2 n - m + 1) \, \phi _*\, \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla \big ) \nonumber \\{} & {} + - \phi _*\, \Big \{ \frac{2}{\theta ({{\mathscr {T}}})} \, J \, {{\mathscr {T}}} - \nabla _{{\mathscr {T}}} {{\mathscr {T}}} \Big \}. \end{aligned}$$

(61)

Besides from the foliation ${\mathscr {F}}$ tangent to ${{\mathscr {V}}}^\phi \big |_{S(\phi )}$ [the portion of the vertical bundle ${{\mathscr {V}}}^\phi$ over the (open) set $\Omega (\phi ) = S(\phi ) = \textrm{I}_m (\phi ) \cup \textrm{III}_m (\phi )$ of all submersive points], the manifold M comes equipped with the Reeb foliation i.e. the codimension 2n foliation ${\mathscr {R}}$ of M tangent to T. The case where ${\mathscr {R}}$ is a subfoliation of ${\mathscr {F}}$ is closest to the Riemannian case i.e. (61) becomes

$$\begin{aligned} \tau (\phi ) = - \frac{m - 2}{2} \, \phi _*\, \nabla \log \Lambda (\phi ) - (2 n - m + 1 ) \, \phi _*\, \mu ^{{{\mathscr {V}}}^\phi }, \end{aligned}$$

(62)

which is the fundamental Eq. (4.5.2) in [5, p. 120], for $\phi : M \rightarrow N$ as a map of the Riemannian manifolds $(M, \, g_\theta )$ and $(N, \, h)$. Indeed, at each point $x \in \textrm{I}_m (\phi )$ [equivalently $(d_x \phi ) \, T_x = 0$] one has $\tau _b (\phi )_x = \tau (\phi )_x$ [by (12)] and $\mu ^{{{\mathscr {V}}}^\phi }_x = \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla \big )_x$ [by (60)] hence (61) becomes

$$\begin{aligned} \tau (\phi ) = - \frac{m - 2}{2} \, \phi _*\, \nabla \log \Lambda (\phi ) - (2 n - m + 1 ) \, \phi _*\, \mu ^{{{\mathscr {V}}}^\phi } + \phi _*\, \nabla _{{\mathscr {T}}} {{\mathscr {T}}}, \end{aligned}$$

along $\textrm{I}_m (\phi )$. If ${{\mathscr {R}}} \subset {{\mathscr {F}}}$, i.e. $\phi _*\, T = 0$ everywhere in $\Omega (\phi )$, then $T^{{\mathscr {V}}} = T$ and $u = 1$ on the whole open set $\Omega (\phi )$, hence $\nabla T = 0$ yields $\nabla _{{\mathscr {T}}} {{\mathscr {T}}} = 0$ on $\Omega (\phi )$. $\square$

Corollary 1

Let $\phi : M \rightarrow N$ be a subelliptic harmonic morphism of the pseudohermitian manifold $(M, \, \theta )$ into the real surface $(N, \, h)$.

(i) If the Reeb foliation is a subfoliation of ${\mathscr {F}}$ [the foliation of $\Omega (\phi )$ tangent to ${{\mathscr {V}}}^\phi$] then every leaf of ${\mathscr {F}}$ is a minimal submanifold of the Riemannian manifold $(M, \, g_\theta )$.

(ii) If $(d_x \phi ) T_x \ne 0$ for some $x \in M {\setminus } \textrm{Crit} (\phi )$, then

$$\begin{aligned} (2 n - 1) \, \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla \big ) = \pi _{{{\mathscr {H}}}^\phi } \, \Big \{ \nabla _{{\mathscr {T}}} {{\mathscr {T}}} - \frac{2}{\theta ({{\mathscr {T}}})} \, J \, {{\mathscr {T}}} \Big \}. \end{aligned}$$

Proof

(i) If $m = 2$ then (62) yields $\mu ^{{{\mathscr {V}}}^\phi } = 0$. $\square$

(ii) Follows from (61).

5.2 $\epsilon$-Contractions

Let $p = 2 n - m + 1$. To some surprise, the term $p \; \phi _*\, \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla \big ) + \phi _*\, \big \{ 2 \, \theta ({{\mathscr {T}}})^{-1} \, J \, {{\mathscr {T}}} - \nabla _{{\mathscr {T}}} {{\mathscr {T}}} \big \}$ replaces the term $p \; \phi _*\, \mu ^{{{\mathscr {V}}}^\phi }$ [occurring in the fundamental equation (62), in the Riemannian setting]. Besides the term $\phi _*\, \mu ^{{{\mathscr {V}}}^\phi } (g_\theta \,, \, \nabla )$ that one might have hoped for to start with, the fundamental equation for a subelliptic harmonic morphism contains the additional term $\phi _*\, \big \{ 2 \, \theta ({{\mathscr {T}}})^{-1} \, J \, {{\mathscr {T}}} - \nabla _{{\mathscr {T}}} {{\mathscr {T}}} \big \}$ whose geometric meaning is so far unknown. So, given an immersion $f: L \rightarrow M$ of a p-dimensional manifold L into the pseudohermitian manifold $(M, \, \theta )$, what is the “correct” pseudohermitian analog to the mean curvature vector (of an isometric immersion)?

Pseudohermitian geometry (on a strictly pseudoconvex CR manifold) embeds into sub-Riemannian geometry. One may construct families of contractions $\{ g_\epsilon \}_{0< \epsilon < 1}$ of the Levi form $G_\theta$ [so that the norm of the Reeb vector T is $O(\epsilon ^{-1} )$] and examine Riemannian geometric objects in the limit as $\epsilon \rightarrow 0^+$, in an attempt to discover new pseudohermitian invariants. Cf. e.g. the approaches by Barletta et al. [12] and Capogna and Citti [17].

Let $0< \epsilon < 1$ and let $g_\epsilon$ be the Riemannian metric

$$\begin{aligned} g_\epsilon (X, Y) = G_\theta (X, Y), \;\;\; g_\epsilon (X, T) = 0, \;\;\; g_\theta (T, T) = \epsilon ^{-2}, \end{aligned}$$

for any $X, \, Y \in H(M)$. Equivalently

$$\begin{aligned} g_\epsilon = g_\theta + \left( \frac{1}{\epsilon ^2}-1 \right) \, \theta \otimes \theta , \end{aligned}$$

(63)

(the $\epsilon$-contraction of $G_\theta$, cf. Strichartz [60], Barletta et al. [12]). To illustrate our strategy, let us assume that, for every $0< \epsilon < 1$, the map $\phi : (M, \, g_\epsilon ) \rightarrow (N, \, h)$ is horizontally weakly conformal, with square dilation $\Lambda _\epsilon$ i.e. for any $x_0 \in M \setminus \textrm{Crit} (\phi )$ and any local coordinate system $\big ( V, \; y^\alpha \big )$ on N with $\phi (x_0 ) \in V$

$$\begin{aligned} m \, \Lambda _\epsilon = \big ( h_{\alpha \beta } \circ \phi \big ) \, g_{\epsilon } \big ( \nabla ^\epsilon \, \phi ^\alpha \, , \, \nabla ^\epsilon \, \phi ^\beta \big ). \end{aligned}$$

(64)

Here $\nabla ^\epsilon$ is the gradient with respect to $g_\epsilon$. Choose $V \subset N$ such that $U = \phi ^{-1} (V) \subset M$ is a relatively compact domain. A straightforward calculation (relying on (63), cf. also [12]) leads to

$$\begin{aligned} \nabla ^\epsilon \, \phi ^\alpha = \nabla ^H \, \phi ^\alpha + \epsilon ^2 \; \theta (\nabla \, \phi ^\alpha ) \; T, \end{aligned}$$

yielding

$$\begin{aligned} g_{\epsilon } \big ( \nabla ^\epsilon \, \phi ^\alpha \,, \; \nabla ^\epsilon \, \phi ^\beta \big ) = G_{\theta } \big ( \nabla ^H \, \phi ^\alpha \,, \; \nabla ^H \, \phi ^\beta \big ) + \epsilon ^2 \; \theta \big ( \nabla \phi ^\alpha \big ) \; \theta \big ( \nabla \phi ^\beta \big ), \end{aligned}$$

and in particular

$$\begin{aligned}{} & {} \sup _{x \in U} \, \big | m \, \Lambda _\epsilon (x) - G_{\theta } \big ( \nabla ^H \, \phi ^\alpha \,, \; \nabla ^H \, \phi ^\beta \big )_x \; h_{\alpha \beta } \big ( \phi (x) \big ) \big | \le C(\phi ) \, \epsilon ^2 \,,\\{} & {} \quad C(\phi ) = \sup _{x \in U} \, \theta \big ( \nabla \phi ^\alpha \big )_x \; \theta \big ( \nabla \phi ^\beta \big )_x \; h_{\alpha \beta } \big ( \phi (x) \big ), \end{aligned}$$

hence

$$\begin{aligned} \Lambda _\epsilon \rightarrow \, \frac{1}{m} \, G_{\theta } \big ( \nabla ^H \, \phi ^\alpha \,, \; \nabla ^H \, \phi ^\beta \big ) \; h_{\alpha \beta } \circ \phi \,, \;\;\; \epsilon \rightarrow 0^+, \end{aligned}$$

uniformly on U, and the Levi conformality condition (14) is got, in the limit as $\epsilon \rightarrow 0^+$, from the horizontal weak conformality condition on $\phi : (M, \, g_\epsilon ) \rightarrow (N, \, h)$.

Let $\phi : M \rightarrow N$ be a subelliptic harmonic morphism of the pseudohermitian manifold $(M, \, \theta )$ into the Riemannian manifold $(N, \, h)$, of $\theta$-dilation $\lambda ( \phi )$. We shall compute the mean curvature vector $\mu ^{{{\mathscr {V}}}^\phi }_\epsilon$ of the vertical distribution ${{\mathscr {V}}}^\phi$ on the Riemannian manifold $(M, \, g_\epsilon )$, and examine the behavior of $\mu ^{{{\mathscr {V}}}^\phi }_\epsilon$ as $\epsilon \rightarrow 0^+$, in an attempt to discover the “correct” pseudohermitian analog to the ordinary mean curvature vector. To this end, let ${{\mathscr {H}}}_\epsilon ^\phi$ be the $g_\epsilon$-orthogonal complement of ${{\mathscr {V}}}^\phi$. Let us set

$$\begin{aligned} B_\epsilon (X, Y)= & {} \pi _{{{\mathscr {H}}}_\epsilon ^\phi } \, \nabla ^\epsilon _X Y, \;\;\; X, Y \in {{\mathscr {V}}}^\phi \,, \\ \mu _\epsilon ^{{{\mathscr {V}}}^\phi }= & {} \frac{1}{2n-m+1} \; { \mathrm Trace}_{g_\epsilon } \, B_\epsilon , \end{aligned}$$

i.e. $\mu _\epsilon ^{{{\mathscr {V}}}^\phi }$ is the mean curvature vector of ${{\mathscr {V}}}^\phi$. Here $\nabla ^\epsilon$ is the Levi-Civita connection of $(M, g_\epsilon )$. Let us set

$$\begin{aligned} {{\mathscr {T}}}_\epsilon = \frac{1}{g_\epsilon ({{\mathscr {T}}}, \, {{\mathscr {T}}})^{1/2}} \; {{\mathscr {T}}} = \frac{\epsilon }{\sqrt{\epsilon ^2 + (1 - \epsilon ^2 ) \, u}} \; {{\mathscr {T}}}. \end{aligned}$$

Then $\big \{ V_j \;, \; {{\mathscr {T}}}_\epsilon \;: \; 1 \le j \le 2 n - m \big \}$ is a local $g_\epsilon$-orthonormal frame of ${{\mathscr {V}}}^\phi$, adapted to the decomposition $T(M) = H(M) \oplus {{\mathbb {R}}}T$ [which is both $g_\theta$ and $g_\epsilon$ orthogonal]. Let

$$\begin{aligned} B_\epsilon (X, \, Y) = \pi _{{{\mathscr {H}}}^\phi _\epsilon } \; \nabla ^\epsilon _X Y, \;\;\; X, \, Y \in {{\mathscr {V}}}^\phi , \end{aligned}$$

[the second fundamental form of $L \hookrightarrow (M, \, g_\epsilon )$, for every leaf $L \in \big [ M {\setminus } \textrm{Crit} (\phi ) \big ] \big / {{\mathscr {F}}}$].

Lemma 18

For every $X \in \mathfrak {X}(M)$

$$\begin{aligned} \pi _{{{\mathscr {H}}}^\phi _\epsilon } \, X = \pi _{{{\mathscr {H}}}^\phi } \, X + \frac{(1 - \epsilon ^2 ) u}{\epsilon ^2 + (1 - \epsilon ^2 ) \, u^2} \, \Big \{ u \, g_\theta ({{\mathscr {T}}}, \, X ) - \theta (X) \Big \} \; {{\mathscr {T}}} . \end{aligned}$$

(65)

Proof

Let $\pi ^\epsilon _{{{\mathscr {V}}}^\phi }: T(M) \rightarrow {{\mathscr {V}}}^\phi$ be the projection associated with the direct sum decomposition $T(M) = {{\mathscr {V}}}^\phi \oplus {{\mathscr {H}}}^\phi _\epsilon$. For every $X \in T(M)$

$$\begin{aligned} \pi _{{{\mathscr {H}}}^\phi _\epsilon } \, X{} & {} = X - \pi ^\epsilon _{{{\mathscr {V}}}^\phi } \, X \\{} & {} = X - \sum _{j=1}^{2n - m} g_\epsilon \big ( V_j \,, \, X \big ) \; V_j - g_\epsilon \big ( {{\mathscr {T}}}_\epsilon \,, \, X \big ) \; {{\mathscr {T}}}_\epsilon =, \end{aligned}$$

[by (63)]

$$\begin{aligned}{} & {} = X - \sum _{j=1}^{2n - m} g_\theta \big ( V_j \,, \, X \big ) \; V_j \, \\{} & {} \quad + - \frac{\epsilon ^2}{\epsilon ^2 + (1 - \epsilon ^2 ) \, u^2} \; \Big \{ g_\theta ({{\mathscr {T}}}, \, X) + \Big ( \frac{1}{\epsilon ^2} - 1 \Big ) \, u \, \theta (X) \Big \} \; {{\mathscr {T}}} \\{} & {} = X - \sum _{j=1}^{2n - m} g_\theta \big ( V_j \,, \, X \big ) \; V_j - g_\theta ({{\mathscr {T}}}, \, X ) \; {{\mathscr {T}}} \\{} & {} \quad + \frac{(1 - \epsilon ^2 ) u}{\epsilon ^2 + (1 - \epsilon ^2 ) \, u^2} \, \Big \{ u \, g_\theta ({{\mathscr {T}}}, \, X ) - \theta (X) \Big \} \; {{\mathscr {T}}}. \end{aligned}$$

$\square$

We shall need (cf. Lemma 2 in [12, pp. 11–12])

Lemma 19

The Levi–Civita connection $\nabla ^\epsilon$ of $(M, \, g_\epsilon )$ and the Tanaka–Webster connection $\nabla$ of $(M, \, \theta )$ are related by

$$\begin{aligned} \nabla ^\epsilon _X\, Y= & {} \nabla _X\, Y + \left\{ \Omega (X, Y) - \epsilon ^2 A(X, Y)\right\} T \,, \\ \nabla ^\epsilon _X\, T= & {} \tau X + \frac{1}{\epsilon ^2} \; JX, \;\;\; \nabla ^\epsilon _T\, X = \nabla _T\, X + \frac{1}{\epsilon ^2} \; JX, \;\;\; \nabla ^\epsilon _T \, T = 0, \end{aligned}$$

for any $X, Y\in H(M)$.

Lemma 20

$$\begin{aligned} (2 n - m + 1) \; \mu ^{{{\mathscr {V}}}^\phi }_\epsilon{} & {} = (2 n - m + 1 ) \; \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla \big ) \nonumber \\{} & {} \quad + - \frac{(1 - \epsilon ^2 ) \, u^2}{\epsilon ^2 + (1 - \epsilon ^2 ) \, u^2} \; \Big \{ \pi _{{{\mathscr {H}}}^\phi } \; \nabla _{{\mathscr {T}}} \, {{\mathscr {T}}} + \textrm{div}_{{\mathscr {F}}} \, ({{\mathscr {T}}} ) \; {{\mathscr {T}}} \Big \} \nonumber \\{} & {} \quad + - \frac{(1 - \epsilon ^2 ) \, u^3}{\epsilon ^2 + (1 - \epsilon ^2 ) \, u^2} \; \Big \{ A ({{\mathscr {T}}}, \, {{\mathscr {T}}}) - \textrm{Trace}_{g_\theta } \, \Pi _{{{\mathscr {V}}}^\phi } \, A \Big \} \; {{\mathscr {T}}} \nonumber \\{} & {} \quad + \frac{2 \, u}{\epsilon ^2 + (1 - \epsilon ^2 ) \, u^2} \; \pi _{{{\mathscr {H}}}^\phi } \, J \, {{\mathscr {T}}} \nonumber \\{} & {} \quad + \epsilon ^2 \, \big ( \textrm{Trace}_{g_\theta } \, \Pi _{{{\mathscr {V}}}^\phi } \, A \big ) \; \Big \{ - T + \frac{u}{\epsilon ^2 + (1 - \epsilon ^2 ) \, u^2} \; {{\mathscr {T}}} \Big \} \nonumber \\{} & {} \quad + \frac{(1 - \epsilon ^2 ) \, \epsilon ^2 \, u^2}{\epsilon ^2 + (1 - \epsilon ^2 ) \, u^2} \; A({{\mathscr {T}}}, \, {{\mathscr {T}}}) \; {{\mathscr {T}}} \nonumber \\{} & {} \quad + \frac{\epsilon ^2 \, u}{\epsilon ^2 + (1 - \epsilon ^2 ) u^2} \, \pi _{{{\mathscr {H}}}^\phi } \, \tau \, {{\mathscr {T}}} - \frac{\epsilon ^2 \, (1 - \epsilon ^2 ) \, u}{\big [ \epsilon ^2 + (1 - \epsilon ^2 ) u^2 \big ]^2} \, {{\mathscr {T}}} (u) \, {{\mathscr {T}}}.\nonumber \\ \end{aligned}$$

(66)

Proof

$$\begin{aligned}{} & {} (2 n - m + 1) \, \mu ^{{{\mathscr {V}}}^\phi }_\epsilon = \sum _{j=1}^{2n - m} B_\epsilon \big ( V_j \,, \, V_j \big ) + B_\epsilon \big ( {{\mathscr {T}}}_\epsilon \,, \, {{\mathscr {T}}}_\epsilon \big ), \nonumber \\{} & {} \quad B_\epsilon \big ( V_j \,, \, V_j \big ) = \pi _{{{\mathscr {H}}}^\phi _\epsilon } \; \nabla ^\epsilon _{V_j} V_j =, \end{aligned}$$

(67)

[by Lemma 19, as $V_j \in H(M)$]

$$\begin{aligned} = \pi _{{{\mathscr {H}}}^\phi _\epsilon } \; \Big \{ \nabla _{V_j} V_j - \epsilon ^2 \, A \big ( V_j \,, \, V_j \big ) \, T \Big \} =, \end{aligned}$$

[by (65) in Lemma 18 with $X = \nabla _{V_j} V_j - \epsilon ^2 \, A \big ( V_j \,, \, V_j \big ) \, T$]

$$\begin{aligned}{} & {} = \pi ^{{{\mathscr {H}}}^\phi } \; \nabla _{V_j} V_j - \epsilon ^2 \, A \big ( V_j \,, \, V_j \big ) \, T^{{\mathscr {H}}} + \frac{(1 - \epsilon ^2 ) \, u}{\epsilon ^2 + (1 - \epsilon ^2 ) \, u^2} \\{} & {} \quad \times \Big \{ u \, g_\theta \big ( {{\mathscr {T}}}, \; \nabla _{V_j} V_j \big ) + \epsilon ^2 \, \big ( 1 - u^2 \big ) \, A \big ( V_j \,, \, V_j \big ) \Big \} \; {{\mathscr {T}}}, \end{aligned}$$

or

$$\begin{aligned} B_\epsilon \big ( V_j \,, \, V_j \big )= & {} \pi ^{{{\mathscr {H}}}^\phi } \, \nabla _{V_j} V_j + \frac{(1 - \epsilon ^2 ) \, u^2}{\epsilon ^2 + (1 - \epsilon ^2 ) \, u^2} \; g_\theta \big ( {{\mathscr {T}}}, \; \nabla _{V_j} V_j \big ) \, {{\mathscr {T}}} \nonumber \\{} & {} + \epsilon ^2 \, A \big ( V_j \,, \, V_j \big ) \; \Big \{ - T + \frac{u}{\epsilon ^2 + (1 - \epsilon ^2 ) \, u^2} \; {{\mathscr {T}}} \Big \}. \end{aligned}$$

(68)

Next

$$\begin{aligned}{} & {} B_\epsilon \big ( {{\mathscr {T}}}_\epsilon \,, \, {{\mathscr {T}}}_\epsilon \big ) = \frac{\epsilon ^2}{\epsilon ^2 + (1 - \epsilon ^2 ) u^2} \; B_\epsilon \big ( {{\mathscr {T}}}, \, {{\mathscr {T}}} \big ),\\{} & {} B_\epsilon \big ( {{\mathscr {T}}}, \, {{\mathscr {T}}} \big ) = \pi _{{{\mathscr {H}}}^\phi _\epsilon } \; \nabla ^\epsilon _{{\mathscr {T}}} {{\mathscr {T}}}, \end{aligned}$$

and [by Lemma 19, and ${{\mathscr {T}}} = \Pi _H \, {{\mathscr {T}}} + u \, T$, and $T \, \rfloor \, A = 0$, $\tau \, T = 0$, $J \, T = 0$]

$$\begin{aligned}{} & {} \nabla ^\epsilon _{{\mathscr {T}}} {{\mathscr {T}}} = \nabla _{\Pi _H \, {{\mathscr {T}}}} \, \Pi _H \, {{\mathscr {T}}} + u \, \nabla _T \, \Pi _H \, {{\mathscr {T}}} + u \, \Big ( \tau + \frac{2}{\epsilon ^2} \, J \Big ) \, {{\mathscr {T}}} \\{} & {} \qquad \qquad \quad + \big \{ {{\mathscr {T}}} (u) - \epsilon ^2 \, A ({{\mathscr {T}}}, \, {{\mathscr {T}}} ) \big \} \; T, \\{} & {} \nabla _T \, \Pi _H \, {{\mathscr {T}}} = \nabla _T {{\mathscr {T}}} - T(u) \, T, \\{} & {} \nabla _{\Pi _H \, {{\mathscr {T}}}} \, \Pi _H \, {{\mathscr {T}}} = \nabla _{{\mathscr {T}}} {{\mathscr {T}}} - u \, \nabla _T \, {{\mathscr {T}}} + \big \{ u \, T(u) - {{\mathscr {T}}} (u) \big \} \, T, \end{aligned}$$

or

$$\begin{aligned} \nabla ^\epsilon _{{\mathscr {T}}} \, {{\mathscr {T}}} = \nabla _{{\mathscr {T}}} \, {{\mathscr {T}}} + u \, \Big ( \tau + \frac{2}{\epsilon ^2} \, J \Big ) \, {{\mathscr {T}}} - \epsilon ^2 \, A( {{\mathscr {T}}}, \, {{\mathscr {T}}}) \; T. \end{aligned}$$

(69)

Note that [by (69)]

$$\begin{aligned}{} & {} \pi _{{{\mathscr {H}}}^\phi } \, \nabla ^\epsilon _{{\mathscr {T}}} {{\mathscr {T}}} = \pi _{{{\mathscr {H}}}^\phi } \, \nabla _{{\mathscr {T}}} \, {{\mathscr {T}}} + u \, \pi _{{{\mathscr {H}}}^\phi } \, \Big ( \tau + \frac{2}{\epsilon ^2} \, J \Big ) \, {{\mathscr {T}}} - \epsilon ^2 \, A( {{\mathscr {T}}}, \, {{\mathscr {T}}}) \; T^{{\mathscr {H}}} \,,\\{} & {} g_\theta \big ( {{\mathscr {T}}}, \; \nabla ^\epsilon _{{\mathscr {T}}} \, {{\mathscr {T}}} \big ) = \big ( 1 - \epsilon ^2 \big ) \, u \, A ({{\mathscr {T}}}, \, {{\mathscr {T}}} ),\\{} & {} \theta \big ( \nabla ^\epsilon _{{\mathscr {T}}} \, {{\mathscr {T}}} \big ) = \theta \big ( \nabla _{{\mathscr {T}}} \, {{\mathscr {T}}} \big ) - \epsilon ^2 \, A ({{\mathscr {T}}}, \, {{\mathscr {T}}}), \\{} & {} \theta \big ( \nabla _{{\mathscr {T}}} \, {{\mathscr {T}}} \big ) = {{\mathscr {T}}} (u). \end{aligned}$$

Let us apply $\pi ^{{{\mathscr {H}}}^\phi _\epsilon }$ to both sides of (69) so that [by (65) in Lemma 18]

$$\begin{aligned} B_\epsilon ({{\mathscr {T}}}, \, {{\mathscr {T}}}){} & {} = \Pi _{{{\mathscr {H}}}^\phi } \, \nabla _{{\mathscr {T}}} \, {{\mathscr {T}}} + u \; \pi _{{{\mathscr {H}}}^\phi } \circ \Big ( \tau + \frac{2}{\epsilon ^2} \, J \Big ) \, {{\mathscr {T}}} \nonumber \\{} & {} \quad + A ({{\mathscr {T}}}, \, {{\mathscr {T}}}) \, \big ( u \, {{\mathscr {T}}} - \epsilon ^2 \, T \big ) - \frac{(1 - \epsilon ^2 ) \, u}{\epsilon ^2 + (1 - \epsilon ^2 ) \, u^2} \, {{\mathscr {T}}} (u) \, {{\mathscr {T}}}.\nonumber \\ \end{aligned}$$

(70)

Then [by (68) and (70)]

$$\begin{aligned} (2 n - m + 1) \, \mu ^{{{\mathscr {V}}}^\phi }_\epsilon{} & {} = \sum _{j=1}^{2n - m} \Big \{ \pi _{{{\mathscr {H}}}^\phi } \, \nabla _{V_j} V_j + \frac{(1 - \epsilon ^2 ) \, u^2}{\epsilon ^2 + (1 - \epsilon ^2 ) \, u^2} \, g_\theta \big ( {{\mathscr {T}}}, \, \nabla _{V_j} V_j \big ) \, {{\mathscr {T}}} \nonumber \\{} & {} \quad + \epsilon ^2 \, A (V_j \,, \, V_j ) \, \Big [ - T + \frac{u}{\epsilon ^2 + (1 - \epsilon ^2 ) u^2} \; {{\mathscr {T}}} \Big ] \Big \} \nonumber \\{} & {} \quad + \frac{\epsilon ^2}{\epsilon ^2 + (1 - \epsilon ^2 ) \, u^2} \, \Big \{ \pi _{{{\mathscr {H}}}^\phi } \, \nabla _{{\mathscr {T}}} {{\mathscr {T}}} + u \; \pi _{{{\mathscr {H}}}^\phi } \circ \Big ( \tau + \frac{2}{\epsilon ^2} \, J \Big ) \, {{\mathscr {T}}} \nonumber \\{} & {} \quad + + A({{\mathscr {T}}}, \, {{\mathscr {T}}}) \, \big ( u \, {{\mathscr {T}}} - \epsilon ^2 \, T \big ) - \frac{(1 - \epsilon ^2 ) \, u}{\epsilon ^2 + (1 - \epsilon ^2 ) \, u^2} \; {{\mathscr {T}}} (u) \; {{\mathscr {T}}} \Big \}.\nonumber \\ \end{aligned}$$

(71)

Using

$$\begin{aligned}{} & {} (2 n - m + 1) \; \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla \big ) = \pi _{{{\mathscr {H}}}^\phi } \, \Big \{ \sum _{j=1}^{2n - m} \nabla _{V_j} V_j + \nabla _{{\mathscr {T}}} {{\mathscr {T}}} \Big \},\\{} & {} \qquad \textrm{Trace}_{g_\theta } \; \Pi _{{{\mathscr {V}}}^\phi } \, A = \sum _{j=1}^{2n - m} A\big ( V_j \,, \, V_j \big ) + A({{\mathscr {T}}}, \, {{\mathscr {T}}} ),\\{} & {} \sum _{j=1}^{2n - m} g_\theta \big ( {{\mathscr {T}}}, \, \nabla _{V_j} V_j \big ) \, {{\mathscr {T}}} \\{} & {} \quad = - \Big \{ \textrm{div}_{{\mathscr {F}}} \, ({{\mathscr {T}}} ) + u \, \Big [ A ({{\mathscr {T}}}, \, {{\mathscr {T}}}) - \textrm{Trace}_{g_\theta } \, \Pi _{{{\mathscr {V}}}^\phi } \, A \Big ] \Big \} \; {{\mathscr {T}}}, \end{aligned}$$

equation (71) simplifies to (66). $\square$

Proof of Theorem 4

Next [by (66) with $p = 2 n - m + 1$]

$$\begin{aligned}{} & {} \Big \Vert p \; \Big [ \pi _{{{\mathscr {H}}}^\phi } \; \mu ^{{{\mathscr {V}}}^\phi }_\epsilon - \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla \big ) \Big ] + \pi _{{{\mathscr {H}}}^\phi } \, \Big [ \nabla _{{\mathscr {T}}} \, {{\mathscr {T}}} - \frac{2}{u} \, J \, {{\mathscr {T}}} \Big ] \Big \Vert \\{} & {} \quad \le \epsilon ^2 \, \Big | \textrm{Trace}_{g_\theta } \; \Pi _{{{\mathscr {V}}}^\phi } \; A \Big | \; \big \Vert \pi _{{{\mathscr {V}}}^\phi } \, T \big \Vert + \frac{\epsilon ^2}{\epsilon ^2 + (1 - \epsilon ^2 ) u^2} \; \Big \{ \Big \Vert \pi _{{{\mathscr {H}}}^\phi } \; \nabla _{{\mathscr {T}}} {{\mathscr {T}}} \Big \Vert \\{} & {} \qquad + |u| \; \Big [ \Big \Vert \pi _{{{\mathscr {H}}}^\phi } \; \tau \, {{\mathscr {T}}} \Big \Vert + \frac{2 (1 - u^2 )}{u^2} \; \Big \Vert \pi _{{{\mathscr {H}}}^\phi } \; J \, {{\mathscr {T}}} \Big \Vert \Big ] \Big \}. \end{aligned}$$

Let $U \subset M$ be a relatively compact domain and $a = \inf _U u^2$, so that $0 < a \le 1$. Indeed if $a = 0$ then for every $\nu \in {{\mathbb {N}}}$

$$\begin{aligned} \frac{1}{\nu } > 0 = \inf _U u^2, \end{aligned}$$

hence there is $x_\nu \in U$ such that $u(x_\nu )^2 < 1/\nu$. There is a subsequence, denoted by the same symbol $x_\nu$, such that $x_\nu \rightarrow x_0$ as $\nu \rightarrow \infty$, for some $x_0 \in \overline{U}$. Thus

$$\begin{aligned} 0 = u (x_0 ) = \big \Vert T^{{\mathscr {V}}} \big \Vert _{x_0} \Longrightarrow T_{x_0} \in {{\mathscr {H}}}^\phi _{x_0}, \end{aligned}$$

a contradiction. Let $\varphi (t) = 2 (1 - t) / t$, $0 < t \le 1$, so that $\sup _U \varphi (u) = \varphi (a)$. Note that

$$\begin{aligned} \big \Vert \pi _{{{\mathscr {H}}}^\phi } \, T \big \Vert= & {} \big \Vert T - u \, {{\mathscr {T}}} \big \Vert \le 1 + |u| \, \Vert {{\mathscr {T}}} \Vert \le 2,\\ \Big \Vert \pi _{{{\mathscr {H}}}^\phi } \; \nabla _{{\mathscr {T}}} {{\mathscr {T}}} \Big \Vert\le & {} \big \Vert \pi _{{{\mathscr {H}}}^\phi } \big \Vert \; \big \Vert \nabla _{{\mathscr {T}}} {{\mathscr {T}}} \Vert ,\\ \Big \Vert \pi _{{{\mathscr {H}}}^\phi } \; \tau \, {{\mathscr {T}}} \Big \Vert\le & {} \big \Vert \pi _{{{\mathscr {H}}}^\phi } \big \Vert \; \big \Vert \tau \big \Vert , \;\;\; \Big \Vert \pi _{{{\mathscr {H}}}^\phi } \; J \, {{\mathscr {T}}} \Big \Vert \le \big \Vert \pi _{{{\mathscr {H}}}^\phi } \big \Vert , \end{aligned}$$

where $\big \Vert \pi _{{{\mathscr {H}}}^\phi } \big \Vert$ is the operator norm. Finally

$$\begin{aligned}{} & {} \Big \Vert p \; \Big [ \pi _{{{\mathscr {H}}}^\phi } \; \mu ^{{{\mathscr {V}}}^\phi }_\epsilon - \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla \big ) \Big ] + \pi _{{{\mathscr {H}}}^\phi } \, \Big [ \nabla _{{\mathscr {T}}} \, {{\mathscr {T}}} - \frac{2}{u} \, J \, {{\mathscr {T}}} \Big ] \Big \Vert \\{} & {} \quad \le 2 \, \epsilon ^2 \, \Big | \textrm{Trace}_{g_\theta } \; \Pi _{{{\mathscr {V}}}^\phi } \; A \Big | + \frac{\epsilon ^2}{a + \epsilon ^2 (1 - a)} \; \big \Vert \pi _{{{\mathscr {H}}}^\phi } \big \Vert \; \Big \{ \big \Vert \nabla _{{\mathscr {T}}} {{\mathscr {T}}} \Vert + \Vert \tau \Vert + \varphi (a) \Big \}. \end{aligned}$$

Consequently

$$\begin{aligned}{} & {} (2 n - m + 1) \; \pi _{{{\mathscr {H}}}^\phi } \; \mu ^{{{\mathscr {V}}}^\phi }_\epsilon \rightarrow (2 n - m + 1 ) \; \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla \big ) \\{} & {} \quad + \pi _{{{\mathscr {H}}}^\phi } \, \Big \{ \frac{2}{u} \, J \, {{\mathscr {T}}} - \nabla _{{\mathscr {T}}} \, {{\mathscr {T}}} \Big \}, \;\;\;\; \epsilon \rightarrow 0^+ , \end{aligned}$$

uniformly on $U \subset M$. $\square$

Let us set by definition

$$\begin{aligned} \mu ^{{{\mathscr {V}}}^\phi }_{\textrm{hor}}:= & {} \pi _{{{\mathscr {H}}}^\phi } \; H \big ( {{\mathscr {V}} }^\phi \big ), (2 n - m + 1) \; H \big ( {{\mathscr {V}}}^\phi \big )\nonumber \\:= & {} (2 n - m + 1 ) \; \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla \big ) +\frac{2}{u} \, \pi _{{{\mathscr {H}}}^\phi } \, J \, {{\mathscr {T}}} - \pi _{{{\mathscr {H}}}^\phi } \, \nabla _{{\mathscr {T}}} \, {{\mathscr {T}}} \nonumber \\{} & {} \quad + - \Big \{ \textrm{div}_{{\mathscr {F}}} ({{\mathscr {T}}}) + u \; \Big [ A ({{\mathscr {T}}}, \, {{\mathscr {T}}}) - \textrm{Trace}_{g_\theta } \, \Pi _{{{\mathscr {V}}}^\phi } \, A \Big ] \Big \} \; {{\mathscr {T}}} . \end{aligned}$$

(72)

When T is tangent to the leaves of ${\mathscr {F}}$

$$\begin{aligned} \mu ^{{{\mathscr {V}}}^\phi }_{\textrm{hor}} = H \big ( {{\mathscr {V}}}^\phi \big ) = \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla \big ). \end{aligned}$$

By Lemma 20

$$\begin{aligned} \lim _{\epsilon \rightarrow 0^+} \, \mu ^{{{\mathscr {V}}}^\phi }_\epsilon = \mu ^{{{\mathscr {V}}}^\phi }_{\textrm{hor}}. \end{aligned}$$

On the other hand, by definition (72) Eq. (61) becomes

$$\begin{aligned} \tau _b (\phi ) = - \frac{m - 2}{2} \, \phi _*\, \log \Lambda (\phi ) - (2 n - m + 1) \, \phi _*\, \mu ^{{{\mathscr {V}}}^\phi }_{\textrm{hor}}, \end{aligned}$$

so that $\tau _b (\phi ) = 0$ and $m = 2$ yield $\mu ^{{{\mathscr {V}}}^\phi }_{\textrm{hor}} = 0$. Theorem 4 is proved.

6 Unique Continuation

Let $\Omega \subset {{\mathbb {R}}}^N$ be a domain, $\alpha > 0$, $x_0 \in \Omega$, and $1 \le p \le \infty$. A measurable function $u: \Omega \rightarrow {{\mathbb {C}}}$ has a zero of order $\alpha$ at $x_0$ in the p-mean if

$$\begin{aligned} \int _{B_r (x_0 )} \big | u(x) \big |^p \; d \, x = O \big ( r^{p \alpha + N} \big ). \end{aligned}$$

An ordinary zero of order $\alpha$ [i.e. except on a set of measure zero $\big | u(x) \big | = O \big ( \big | x - x_0 \big |^\alpha \big )$ as $x \rightarrow x_0$] corresponds to a zero in the p-mean for $p = \infty$. Let L be a second order linear elliptic operator. By a result of N. Aronszajn (cf. [2]) if u is a solution to

$$\begin{aligned} \big | L u (x) \big |^2 \le M \, \Big \{ \sum _{i=1}^N \Big | \frac{\partial u}{\partial x^i} \Big |^2 + |u(x)|^2 \Big \}, \end{aligned}$$

and u has a zero of infinite order in the 1-mean at some $x_0 \in \Omega$, then⁴$u \equiv 0$ in $\Omega$. Aronszajn’s proof to his result was criticized in [25, pp. 433–434], because of Aronszajn’s claim that a pair of conformally related Riemannian metrics (associated to the symbol of L) have the same geodesics. We conjecture that the arguments in [2] may be reconsidered within conformal Riemannian geometry i.e. by understanding the conformal properties of geodesic spheres, based on the use of conformal geodesics (cf. e.g. [33]). The result itself in [2] may nevertheless be true, and if that is the case it yields Sampson’s unique continuation theorem for harmonic maps of Riemannian manifolds (cf. Theorem 1 in [58, p. 213]). Let Let $\{ X_a \,: \, 1 \le a \le 2 n \}$ be a $G_\theta$-orthonormal frame of H(M), defined on the open set $U \subset M$, and let $\Omega \subset U$ be a domain. Let $u = \big ( u^1 \,, \, \ldots \,, \, u^m \big ): \Omega \rightarrow {{\mathbb {R}}}^m$ be a solution to

$$\begin{aligned} \big | \Delta _b u^\alpha \big | \le C \, \Big \{ \sum _{a, \, \beta } \big | X_a \big ( u^\beta \big ) \big | + \sum _\beta \big | u^\beta \big | \Big \} . \end{aligned}$$

(73)

We conjecture that, if u has a zero of infinite order at some point of $\Omega$ then $u \equiv 0$ in $\Omega$. Should the conjecture be true, one has

Corollary 2

Let $\phi , \, \psi : M \rightarrow N$ be two subelliptic harmonic maps, from the connected pseudohermitian manifold $(M, \, \theta )$ into the Riemannian manifold $(N, \, h)$. If $\phi$ and $\psi$ agree on some open set, then they are identical.

Proof

Let $\chi = (y^1 \,, \, \ldots \,, \, y^m )$ be a local coordinate system on N, whose domain is a ball $V = \chi ^{-1} \big [ B_r (\xi _0 ) \big ]$, and let $\{ X_a \;: \; 1 \le a \le 2 n \}$ be a local $G_\theta$-orthonormal frame of H(M), defined on the open set $U \subset M$ such that $\phi (U) \cup \psi (U) \subset V$. Let us set $u^\alpha := \phi ^\alpha - \psi ^\alpha$ so that [as both $\phi$ and $\psi$ are harmonic maps]

$$\begin{aligned} \Delta _b u^\alpha= & {} \sum _{a=1}^{2n} \Big [ \Big ( \Big \{ \begin{array}{c} \alpha \\ \beta \gamma \end{array} \Big \} \circ \phi \Big ) \, X_a (\phi ^\beta ) \, X_a (\phi ^\gamma ) \\{} & {} +- \Big ( \Big \{ \begin{array}{c} \alpha \\ \beta \gamma \end{array} \Big \} \circ \psi \Big ) \, X_a (\psi ^\beta ) \, X_a (\psi ^\gamma ) \Big ] \\= & {} \sum _{a=1}^{2n} \Big [ \Big ( \Big \{ \begin{array}{c} \alpha \\ \beta \gamma \end{array} \Big \} \circ \phi \Big ) \, X_a ( u^\beta ) \, \big [ X_a (\phi ^\gamma ) + X_a (\psi ^\gamma ) \big ] \\{} & {} + - \Big [ \Big \{ \begin{array}{c} \alpha \\ \beta \gamma \end{array} \Big \} \circ \psi - \Big \{ \begin{array}{c} \alpha \\ \beta \gamma \end{array} \Big \} \circ \phi \Big ] \, X_a (\psi ^\beta ) \, X_a (\psi ^\gamma ) \Big ]. \end{aligned}$$

Let $x \in U$ and let $\xi = \phi (x)$ and $\eta = \psi (x)$. By the mean value theorem, there is $0< \tau < 1$ such that

$$\begin{aligned}{} & {} \Big \{ \begin{array}{c} \alpha \\ \beta \gamma \end{array} \Big \} \big ( \chi ^{-1} (\eta ) \big ) - \Big \{ \begin{array}{c} \alpha \\ \beta \gamma \end{array} \Big \} \big ( \chi ^{-1} (\xi ) \big ) \\{} & {} \quad = \frac{\partial \Big \{ \begin{array}{c} \alpha \\ \beta \gamma \end{array} \Big \}}{\partial y^\mu } \big [ \chi ^{-1} \big ( ( 1 - \tau ) \, \xi + \tau \, \eta \big ) \big ] \, \big ( \eta ^\mu - \xi ^\mu \big ). \end{aligned}$$

By eventually shrinking U the derivatives $X_a (\phi ^\beta )$ and $X_a (\psi ^\beta )$ are bounded, so that $u^\alpha$ satisfy (73). $\square$

We conjecture that the uniqueness continuation result by Garofalo and Lanconelli (cf. Theorem 1.2 in [36, p. 319]) on the solutions to $\Delta _b u + V(x) \, u = 0$ carries over to equations of the form $\Delta _b u + f(u) = 0$ [with the nonlinear term f(u) as considered by Birindelli and Prajapat [15] (cf. also Birindelli and Lanconelli [14]) for different purposes] with applications to the unique continuation of subelliptic harmonic maps.

7 Horizontal Mean Curvature

Let $\phi : M \rightarrow N$ be a subelliptic harmonic morphism, of the pseudohermitian manifold $(M, \, \theta )$ into the Riemannian manifold $(N, \, h)$. Let ${\mathscr {F}}$ be the foliation of $S(\phi )$ by maximal integral manifolds of ${{\mathscr {V}}}^\phi$. A point $x \in S(\phi )$ is a characteristic point of ${{\mathscr {F}}}$ if

$$\begin{aligned} H(M)_x \subset {{\mathscr {V}}}^\phi _x. \end{aligned}$$

(74)

Let $\Sigma ({{\mathscr {F}}})$ be the set of all characteristic points of ${\mathscr {F}}$. If $x \in \Sigma ({{\mathscr {F}}})$ and $L \in S(\phi ) /{{\mathscr {F}}}$ is the leaf of ${\mathscr {F}}$ passing through x, then x is a characteristic point of L, e.g. in the sense of L. Capogna & G. Citti (cf. [17, p. 7]). The inclusion (74) yields

$$\begin{aligned} 2 n \le \textrm{dim}_{{\mathbb {R}}} \; {{\mathscr {V}}}^\phi _x \le 2 n + 1, \end{aligned}$$

(75)

hence one has equality in (74) unless $x \in \textrm{Crit} (\phi )$. Yet $S(\phi )$, and hence $\Sigma ({{\mathscr {F}}})$, contains no critical points. Also [by (75)]

$$\begin{aligned} \Sigma ({{\mathscr {F}}}) \ne \emptyset \Longrightarrow m = 1. \end{aligned}$$

This limitation doesn’t occur in [17] (where the ambient space M is a Carnot group, the rank of whose first stratus, or horizontal plane, is in general smaller than the dimension of ${\mathscr {F}}$). For the remainder of the present section we confine ourselves to subelliptic harmonic morphisms $\phi : M^{2n+1} \rightarrow N^1$ i.e. $m = 1$ (so that every leaf of ${\mathscr {F}}$ is a real hypersurface in $S(\phi )$). By Theorem 6 (with $m = 1$)

$$\begin{aligned} Z(\Lambda )= & {} \textrm{II}_1 (\phi ) \cup \textrm{Crit} (\phi ), \;\;\; S(\phi ) = M \setminus \textrm{Crit}(\phi ), \\ \Omega= & {} S(\phi ) \setminus \textrm{II}_1 (\phi ), \;\;\; \textrm{I}_1 (\phi ) \subset \Omega \,, \;\;\; \textrm{III}_1 (\phi ) = \Omega \setminus \textrm{I}_1 (\phi ), \end{aligned}$$

where [by Lemmas 5, 6]

$$\begin{aligned} \textrm{I}_1 (\phi )= & {} \Big \{ x \in S(\phi ) \;: \; {{\mathscr {H}}}^\phi _x \subset H(M)_x \,, \;\; \textrm{dim}_{{\mathbb {R}}} \, {{\mathscr {V}}}^\phi _{H, \, x} = 2 n - 1 \Big \},\\ \textrm{II}_1 (\phi )= & {} \big \{ x \in S(\phi ) \;: \; {{\mathscr {V}}}^\phi _x = H(M)_x \big \}, \\ \textrm{III}_1 (\phi )= & {} \big \{ x \in S(\phi ) \;: \; {{\mathscr {H}}}^\phi _{H, \, x} = (0), \;\; \textrm{dim}_{{\mathbb {R}}} \, {{\mathscr {V}}}^\phi _{H, \, x} = 2 n - 1 \big \},\\ \textrm{I}_1 (\phi ) \cup \textrm{III}_1 (\phi )= & {} \Omega , \;\;\; \textrm{II}_1 (\phi ) = \Sigma ({{\mathscr {F}}}). \end{aligned}$$

Let $\{ g_\epsilon \}_{0< \epsilon < 1}$ be the family of contractions of the Levi form $G_\theta$ given by (2), and let $\textbf{n}^\epsilon \in C^\infty \big ( S(\phi ), \, {{\mathscr {H}}}^\phi _\epsilon \big )$ such that $g_\epsilon \big ( \textbf{n}^\epsilon \,, \, \textbf{n}^\epsilon \big ) = 1$. Next, let

$$\begin{aligned} \nu ^\epsilon := \Pi _H \, \textbf{n}^\epsilon = \textbf{n}^\epsilon - \theta \big ( \textbf{n}^\epsilon \big ) \, T, \end{aligned}$$

(76)

be the projection of $\textbf{n}^\epsilon$ on H(M).

Lemma 21

For every $x \in S(\phi )$, the following statements are equivalent

(i) $x \in \Sigma ({{\mathscr {F}}})$.

(ii) $\nu ^\epsilon _x = 0$ for any $0< \epsilon < 1$.

Proof

(i) $\Longrightarrow$ (ii). Let $x \in \Sigma ({{\mathscr {F}}})$, so that

$$\begin{aligned} H(M)_x = {{\mathscr {V}}}^\phi _x \; \bot _{g_{\epsilon , x}} \; {{\mathscr {H}}}^\phi _{\epsilon \,, \, x} \ni \textbf{n}^\epsilon , \end{aligned}$$

i.e. for every $X \in H(M)_x$

$$\begin{aligned} 0 = g_{\epsilon \,, \, x} \big ( X, \, \textbf{n}^\epsilon _x \big ) = g_{\theta \,, \, x} \big ( X, \, \textbf{n}^\epsilon _x \big ) + \Big ( \frac{1}{\epsilon ^2} - 1 \Big ) \; \theta _x (X) \, \theta \big ( \textbf{n}^\epsilon \big )_x, \end{aligned}$$

yielding [by $\theta _x (X) = 0$]

$$\begin{aligned} \textbf{n}^\epsilon _x \; \bot _{g_{\theta , x}} \; H(M)_x, \end{aligned}$$

or $\textbf{n}^\epsilon _x = \lambda \, T_x$ for some $\lambda \in {{\mathbb {R}}}$. Then [by (76)] $\nu ^\epsilon _x = 0$. $\square$

(ii) $\Longrightarrow$ (i). Let $\nu ^\epsilon _x = 0$. Then [by (76)] $\textbf{n}^\epsilon _x = \lambda \, T_x$ with $\lambda := \theta \big ( \textbf{n}^\epsilon \big )_x$. As $\textrm{Sing} (\textbf{n}^\epsilon ) = \emptyset$ and $\textrm{Sing} (T) = \emptyset$, it must be $\lambda \ne 0$. Hence $\textbf{n}^\epsilon _x \; \bot _{g_{\theta , x}} \; H(M)_x$ implying [by (2)] that $\textbf{n}^\epsilon _x \; \bot _{g_{\epsilon , x}} \; H(M)_x$. On the other hand $\textbf{n}^\epsilon _x \; \bot _{g_{\epsilon , x}} \; {{\mathscr {V}}}^\phi _x$ so that (by the uniqueness of the $g_{\epsilon \,, \, x}$-orthogonal complement of $\textbf{n}^\epsilon _x$) it must be $H(M)_x = {{\mathscr {V}}}^\phi _x$ i.e. $x \in \Sigma ({{\mathscr {F}}})$. $\square$

Let us set

$$\begin{aligned} \textbf{n}^0 (x):= & {} \frac{1}{\sqrt{f_\epsilon (x)}} \; \nu ^\epsilon _x \,, \;\;\; x \in \Omega \setminus \Sigma ({{\mathscr {F}}}), \nonumber \\ f_\epsilon:= & {} g_\epsilon \big ( \nu ^\epsilon \,, \, \nu ^\epsilon \big ) \in C^\infty \big ( \Omega , \, {{\mathbb {R}}}_+ \big ), \end{aligned}$$

(77)

with ${{\mathbb {R}}}_+ = [0, \, + \infty )$. According to the terminology by L. Capogna et al. (cf. [17, p. 7]) $\textbf{n}^0$ is the horizontal normal (on the leaves of ${\mathscr {F}}$).

Lemma 22

For each $x \in \Omega$ the function

$$\begin{aligned} \epsilon \in (0, \, 1) \longmapsto f_{\epsilon } (x)^{-1 /2} \; \nu _x^\epsilon \in H(M)_x, \end{aligned}$$

is constant i.e. $\textbf{n}^0 (x)$ in (77) doesn’t depend on $0< \epsilon < 1$.

Proof

Note that [by (76)]

$$\begin{aligned} f_\epsilon = 1 - \frac{1}{\epsilon ^2} \; \theta \big ( \textbf{n}^\epsilon \big )^2. \end{aligned}$$

(78)

Also, for any $\epsilon , \, \epsilon ^\prime \in (0, \, 1)$

$$\begin{aligned} g_{\epsilon ^\prime } = g_\epsilon + \Big ( \frac{1}{{\epsilon ^\prime }^2} - \frac{1}{\epsilon ^2} \Big ) \; \theta \otimes \theta . \end{aligned}$$

(79)

Also [by $\textrm{dim}_{{\mathbb {R}}} \, {{\mathscr {H}}}^\phi _x = 1$ for any $x \in \Omega$] $\textbf{n}^{\epsilon ^\prime } = \lambda \, \textbf{n}^\epsilon$ for some $C^\infty$ function $\lambda : \Omega \rightarrow {{\mathbb {R}}} {\setminus } \{ 0 \}$. Thus [by (79)]

$$\begin{aligned} 1 = g_{\epsilon ^\prime } \big ( \textbf{n}^{\epsilon ^\prime } \,, \, \textbf{n}^{\epsilon ^\prime } \big ) = \lambda ^2 \, \Big \{ 1 + \Big ( \frac{1}{{\epsilon ^\prime }^2} - \frac{1}{\epsilon ^2} \Big ) \; \theta \big ( \textbf{n}^\epsilon \big )^2 \Big \}, \end{aligned}$$

yielding $f_{\epsilon ^\prime } (x)^{-1 /2} \; \nu _x^{\epsilon ^\prime } = f_{\epsilon } (x)^{-1 /2} \; \nu _x^\epsilon$ for every $x \in \Omega$. $\square$

For every $C^1$ vector field X on M, its divergence with respect to the volume form $\Psi = \theta \wedge (d \theta )^n$ is given by

$$\begin{aligned} {{\mathscr {L}}}_X \, \Psi = \textrm{div} (X) \; \Psi , \end{aligned}$$

where ${{\mathscr {L}}}_X$ is the Lie derivative at the direction X.

Definition 7

The horizontal mean curvature of ${\mathcal {F}}$ is

$$\begin{aligned} K_0 = \textrm{div} \, \big ( \textbf{n}^0 \big ) \in C^\infty ( \Omega ) . \end{aligned}$$

(80)

$\square$

Let

$$\begin{aligned} d \, \textrm{vol} (g_\theta )= & {} \sqrt{{\mathfrak {g}}} \, d x^1 \wedge \cdots \wedge d x^{2n+1} \,,\\ {{\mathfrak {g}}}= & {} \det \big [ g_{jk} \big ] \,, \;\;\; g_{jk} = g_\theta \big ( \partial _j \,, \, \partial _k \big ), \;\;\; \partial _j \equiv \frac{\partial }{\partial x^j}, \end{aligned}$$

be the canonical volume form of the oriented Riemannian manifold $(M, \, g_\theta )$. Then (cf. e.g. [9])

$$\begin{aligned} d \, \textrm{vol} (g_\theta ) = C_n \, \Psi , \end{aligned}$$

for some constant $C_n > 0$ depending only on the CR dimension n. Hence the divergence operator in (80) is the ordinary Riemannian divergence on $(M, \, g_\theta )$. The volume form $\Psi$ is parallel with respect to the Tanaka–Webster connection $\nabla$ of $(M, \, \theta )$, hence $\textrm{div} (X)$ can be computed as the trace of the covariant derivative $\nabla X$. Therefore, if $x \in \Omega$ and $\{ X_a \;: \; 1 \le a \le 2 n \}$ is a local $G_\theta$-orthonormal frame of H(M), defined on a neighborhood $U \subset \Omega$ of x, then [as $\{ X_a \,, \, T \,: \, 1 \le a \le 2 n \}$ is a local orthonormal frame of T(M) on U]

$$\begin{aligned} K_0 (x) = \sum _{a=1}^{2n} g_\theta \big ( \nabla _{X_a} \textbf{n}^0, \, X_a \big )_x + g_\theta \big ( \nabla _T \textbf{n}^0 \,, \, T \big )_x, \end{aligned}$$

hence [by $\nabla _T \textbf{n}^0 \in H(M)$, as H(M) is parallel with respect to $\nabla$, and by $\nabla g_\theta = 0$]

$$\begin{aligned} K_0 (x) = \sum _{a=1}^{2n} \Big \{ X_a \big ( g_\theta (\textbf{n}^0 \, , \, X_a ) \big ) _x - g_\theta \big ( \textbf{n}^0 \, , \, \nabla _{X_a} X_a \big )_x \Big \}. \end{aligned}$$

(81)

To draw a parallel between the considerations in the present paper and those in the work by Capogna et al. (cf. [17]) let $M = {{\mathbb {H}}}_n$ be the Heisenberg group i.e. the noncommutative Lie group ${{\mathbb {H}}}_n = {{\mathbb {C}}}^n \times {{\mathbb {R}}}$ with the group law

$$\begin{aligned} (z, \, t) \cdot (w, \, s)= & {} \big ( z + w, \; t + s + 2 \, \textrm{Im} (z \cdot \overline{w}) \big ),\\{} & {} t, \, s \in {{\mathbb {R}}}, \;\;\; z, \, w \in {{\mathbb {C}}}^n \,, \;\;\; z \cdot \overline{w} = \delta ^{\alpha \beta } \, z_\alpha \, \overline{w}_\beta , \end{aligned}$$

equipped with the strictly pseudoconvex, left invariant, CR structure $T_{1,0}({{\mathbb {H}}}_n )$ spanned by

$$\begin{aligned} L_\alpha \equiv \frac{\partial }{\partial z_\alpha } + i \, \overline{z}_\alpha \, \frac{\partial }{\partial t} \,, \;\;\; 1 \le \alpha \le n, \end{aligned}$$

[so that $\overline{L}_\alpha$ are the Lewy operators] and with the contact form

$$\begin{aligned} \theta _0 = d t + i \sum _{\alpha = 1}^n \big ( z_\alpha \, d \overline{z}_\alpha - \overline{z}_\alpha \, d z_\alpha \big ) \in {{\mathscr {P}}}_+ ({{\mathbb {H}}}_n ). \end{aligned}$$

The work [17] deals with an arbitrary Carnot group G, yet in general the horizontal plane H of G may lack a complex structure. Also, if the horizontal plane admits a complex structure $J: H \rightarrow H$, and the corresponding almost CR structure $\textrm{Eigen} \big ( J^{{\mathbb {C}}}, \, + i \big ) \subset H \otimes {{\mathbb {C}}}$ is formally integrable, then in general the CR codimension of the resulting CR structure is $> 1$. So for comparison reasons, between the theory developed here and the geometric foundations on which [17] relies, we confine ourselves to the Heisenberg group $G = {{\mathbb {H}}}_n$, which is both a Carnot group and a strictly pseudoconvex CR manifold (isomorphic to the boundary of the Siegel domain in ${{\mathbb {C}}}^{n+1}$). If this is the case

$$\begin{aligned} X_\alpha \equiv \frac{1}{\sqrt{2}} \Big ( L_\alpha + \overline{L}_\alpha \Big ) = \frac{1}{\sqrt{2}} \Big ( \frac{\partial }{\partial x_\alpha } - 2 \, y_\alpha \, \frac{\partial }{\partial t} \Big ), \end{aligned}$$

$$\begin{aligned} X_{n + \alpha } \equiv \frac{i}{\sqrt{2}} \Big ( L_\alpha - \overline{L}_\alpha \Big ) = \frac{1}{\sqrt{2}} \Big ( \frac{\partial }{\partial y_\alpha } + 2 \, x_\alpha \, \frac{\partial }{\partial t} \Big ), \end{aligned}$$

is a (globally defined) $G_{\theta _0}$-orthonormal frame of the Levi distribution $H({{\mathbb {H}}}_n )$. The Reeb vector field and the Tanaka–Webster connection of $\big ( {{\mathbb {H}}}_n \,, \, \theta _0 \big )$ are $T \equiv \partial /\partial t$ and

$$\begin{aligned} \nabla _{L_A} L_B= & {} 0, \;\;\; A, \, B \in \big \{ 0, \, 1, \, \cdots \,, \, n, \, \overline{1}, \, \cdots \,, \, \overline{n} \big \},\\ L_0\equiv & {} T, \;\;\; L_{\overline{\alpha }} \equiv \overline{L}_\alpha \,, \;\;\; 1 \le \alpha \le n. \end{aligned}$$

Consequently $\nabla _{X_a} X_a = 0$, so that for every subelliptic harmonic morphism $\phi : {{\mathbb {H}}}_n \rightarrow N^1$ our formula (81) becomes

$$\begin{aligned} K_0 = \sum _{a=1}^{2n} X_a \big ( g_{\theta _0} (\textbf{n}^0 \,, \, X_a ) \big ) \end{aligned}$$

which is formula (2.2) in [17, p. 7]. Going back to the general case, let us observe that $T^{{\mathscr {V}}}_x \ne 0$ for every $x \in \Omega$. Otherwise $T_x \in {{\mathscr {H}}}^\phi _x$ for some $x \in \Omega$, hence [by Lemma 3] $x \in \textrm{II}_1 (\phi ) = \Sigma ({{\mathscr {F}}})$, a contradiction. Therefore the vector field

$$\begin{aligned} {{\mathscr {T}}} = \big \Vert T^{{\mathscr {V}}} \big \Vert ^{-1} \, T^{{\mathscr {V}}} \in C^\infty \big ( \Omega \setminus \Sigma ({{\mathscr {F}}}), \; {{\mathscr {V}}}^\phi \big ), \end{aligned}$$

(considered by us in Sect. 5.1, though confined to the case $m \ge 2$) is well defined for $m = 1$, as well. In particular, Lemma 15 applies to the case $m = 1$, producing a local $G_\theta$-orthonormal frame

$$\begin{aligned}{} & {} \big \{ V_j \;, \; {{\mathscr {T}}} \;: \; 1 \le j \le 2 n - 1 \big \} \subset C^\infty \big ( U, \; {{\mathscr {V}}}^\phi \big ),\\{} & {} V_j \in C^\infty \big (U, \, {{\mathscr {V}}}^\phi _H \big ), \;\;\; U \subset \Omega . \end{aligned}$$

Let us complete $\{ V_j \;: \; 1 \le j \le 2 n - 1 \}$ to a local $G_\theta$-orthonormal frame

$$\begin{aligned} \{ V_a \,: \, 1 \le a \le 2 n \} \subset C^\infty (U, \, H(M)). \end{aligned}$$

If $(V, \, y^1 )$ is a local coordinate system on $N^1$ and $U = \phi ^{-1} (V)$ then

$$\begin{aligned} V_{2 n} = \frac{1}{\sqrt{\Lambda _0}} \; \nabla ^H \, \phi ^1 \, , \;\;\;\; \phi ^1 \equiv y^1 \circ \phi , \;\;\; \Lambda _0 \equiv \frac{\Lambda }{h_{11} \circ \phi }. \end{aligned}$$

(82)

Indeed $\nabla ^H \phi ^1 = \lambda ^a \, V_a$ and

$$\begin{aligned} 0= & {} \phi _*\, V_j = V_j \big ( \phi ^1 \big ) \, \Big ( \frac{\partial }{\partial y^1} \Big )^\phi = g_\theta \big ( V_j \,, \, \nabla ^H \phi ^1 \big ) \, \Big ( \frac{\partial }{\partial y^1} \Big )^\phi \\= & {} \lambda ^j \; \Big ( \frac{\partial }{\partial y^1} \Big )^\phi \Longrightarrow \lambda ^j = 0, \end{aligned}$$

i.e. $\nabla ^H \phi ^1 = \lambda ^{2n} \, V_{2 n }$. By the Levi conformality condition

$$\begin{aligned} \Lambda (x) \; h^{1 1} \big ( \phi (x) \big ) = G_\theta \big ( \nabla ^H \phi ^1 \,, \, \nabla ^H \phi ^1 \big )_x, \end{aligned}$$

for every $x \in U$, hence $\lambda ^{2 n} = \sqrt{\Lambda _0}$. $\square$

Lemma 23

For every subelliptic harmonic morphism $\phi : M^{2n+1} \rightarrow N^1$

$$\begin{aligned} \textbf{n}^\epsilon= & {} \frac{1}{\sqrt{1 + \epsilon ^2 \, f^2}} \; \Big \{ V_{2n} - \epsilon ^2 \, f \, T \Big \}, \;\;\;\; 0< \epsilon < 1, \nonumber \\ f\equiv & {} \frac{g_\theta \big ( V_{2n} \,, \, {{\mathscr {T}}} \big )}{\theta ({{\mathscr {T}}})} \in C^\infty (U), \end{aligned}$$

(83)

everywhere in $U = \phi ^{-1} (V)$, where $V_{2 n} \equiv \Lambda _0^{- 1/2} \; \nabla ^H \phi ^1$.

Proof

As $\textbf{n}^\epsilon \in {{\mathscr {H}}}^\phi _\epsilon \subset T(M) = H(M) \oplus {{\mathbb {R}}} T$

$$\begin{aligned} \textbf{n}^\epsilon = \sum _{a=1}^{2n} f_a \, V_a + f_0 \, T, \end{aligned}$$

with $f_j = 0$ [because of $g_\epsilon \big ( \textbf{n}^\epsilon \,, \, V_j \big ) = 0$] i.e. $\textbf{n}^\epsilon = \lambda \, V_{2 n} + f_0 \, T$ with $\lambda := f_{2 n}$. On the other hand

$$\begin{aligned} g_\epsilon (T, \, {{\mathscr {T}}}) = \frac{1}{u} \, g_\epsilon \big ( T, \, T^{{\mathscr {V}}} \big ) = \frac{1}{u} \Big \{ g_\theta \big ( T, \, T^{{\mathscr {V}}} \big ) + \Big ( \frac{1}{\epsilon ^2} - 1 \Big ) \, \theta \big ( T^{{\mathscr {V}}} \big ) \Big \} = \frac{u}{\epsilon ^2}, \end{aligned}$$

[because of $\theta \big ( T^{{\mathscr {V}}} \big ) = g_\theta \big ( T, \, T^{{\mathscr {V}}} \big ) = \Vert T^{{\mathscr {V}}} \Vert ^2 = u^2$]. Here $u = \theta ({{\mathscr {T}}})$. Then

$$\begin{aligned} 0= & {} g_\epsilon \big ( \textbf{n}^\epsilon \,, \, {{\mathscr {T}}} \big ) = \lambda \, g_\epsilon \big ( V_{2 n} \,, \, {{\mathscr {T}}} \big ) + f_0 \, \frac{u}{\epsilon ^2}\nonumber \\ \textbf{n}^\epsilon= & {} \lambda \, \Big \{ V_{2 n} - \epsilon ^2 \, f \; T \Big \}. \end{aligned}$$

(84)

Finally [by (84)]

$$\begin{aligned} 1= & {} g_\epsilon \big ( \textbf{n}^\epsilon \,, \, \textbf{n}^\epsilon \big ) = g_\theta \big ( \textbf{n}^\epsilon \,, \, \textbf{n}^\epsilon \big ) + \Big ( \frac{1}{\epsilon ^2} - 1 \Big ) \; \theta \big ( \textbf{n}^\epsilon \big )^2 \\= & {} \lambda ^2 \Big \{ 1 + \frac{\epsilon ^2}{u^2} \, g_\theta \big ( V_{2 n} \,, \, {{\mathscr {T}}} \big )^2 \Big \} = \lambda ^2 \, \big \{ 1 + \epsilon ^2 \, f^2 \big \}. \end{aligned}$$

$\square$

As a corollary of (83)

Proposition 2

(i) The function $f_\epsilon \in C^\infty ( \Omega )$ in (77) is given by

$$\begin{aligned} f_\epsilon = \frac{1}{1 + \epsilon ^2 \, f^2}. \end{aligned}$$

(ii) The vector field $\nu _\epsilon \in C^\infty \big ( \Omega , \, H(M) \big )$ is given by

$$\begin{aligned} \nu _\epsilon = \frac{1}{\sqrt{1 + \epsilon ^2 \, f^2}} \, V_{2 n} = \frac{1}{\sqrt{\big ( 1 + \epsilon ^2 \, f^2 \big )\, \Lambda _0}} \, \nabla ^H \phi ^1, \end{aligned}$$

everywhere on $U = \phi ^{-1} (V)$.

(iii) The horizontal normal $\textbf{n}^0$ is given by

$$\begin{aligned} \textbf{n}^0 = V_{2 n} = \frac{1}{\sqrt{\Lambda _0}} \, \nabla ^H \, \phi ^1, \end{aligned}$$

on U.

(iv) The horizontal mean curvature of ${\mathcal {F}}$ is given by

$$\begin{aligned} K_0 = \textrm{div} \Big ( \frac{1}{\sqrt{\Lambda _0}} \, \nabla ^H \phi ^1 \Big ) = - \frac{1}{\sqrt{\Lambda _0}} \, \Big \{ \Delta _b \phi ^1 + \big ( \nabla ^H \phi ^1 \big ) \, \log \, \sqrt{\Lambda _0} \Big \}, \end{aligned}$$

on U.

Proof

(i) By (83)

$$\begin{aligned} \theta \big ( \textbf{n}^\epsilon \big )= & {} - \frac{\epsilon ^2 \, f}{\sqrt{1 + \epsilon ^2 \, f^2}} \,, \\ g_\epsilon \big ( \textbf{n}^\epsilon \,, \, T \big )= & {} \frac{1}{\epsilon ^2} \, \theta \big ( \textbf{n}^\epsilon \big ), \;\;\;\; g_\epsilon (T, \, T) = \frac{1}{\epsilon ^2}, \end{aligned}$$

and then

$$\begin{aligned} f_\epsilon= & {} g_\epsilon \big ( \nu _\epsilon \,, \, \nu _\epsilon \big )= 1 - 2 \, \theta \big (\textbf{n}^\epsilon \big ) \, g_\epsilon \big ( \textbf{n}^\epsilon \,, \, T \big ) + \theta \big ( \textbf{n}^\epsilon \big )^2 \, g_\epsilon (T, \, T) \\= & {} \frac{1}{1 + \epsilon ^2 \, f^2}. \end{aligned}$$

The remainder of the section is devoted to the proof of Theorem 5. Statement (i) was proved in Proposition 2.

(ii) The horizontal mean curvature is given by

$$\begin{aligned} K_0 = \textrm{div} \big ( \textbf{n}^0 \big ) = \sum _{a=1}^{2 n} g_\theta \big ( \nabla _{V_a} \textbf{n}^0 \,, \, V_a \big ) =, \end{aligned}$$

[by $g_\theta \big ( \nabla _{V_{2n}} \textbf{n}^0 \,, \, V_{2 n} \big ) = 0$ and $\nabla g_\theta = 0$]

$$\begin{aligned} = - \sum _{j=1}^{2 n - 1} g_\theta \big ( \textbf{n}^0 \,, \, \nabla _{V_j} V_j \big ). \end{aligned}$$

On the other hand

$$\begin{aligned} 2 n \, \mu ^{{{\mathscr {V}}}^\phi } \big ( g_\theta \,, \, \nabla \big ) = \pi _{{{\mathscr {H}}}^\phi } \Big \{ \sum _{j=1}^{2n - 1} \nabla _{V_j} V_j + \nabla _{{\mathscr {T}}} {{\mathscr {T}}} \Big \}, \end{aligned}$$

hence [by taking the inner product with $\textbf{n}^0$]

$$\begin{aligned}{} & {} g_\theta \big ( \textbf{n}^0 \,, \, 2 n \, \mu ^{{{\mathscr {V}}}^\phi } ( g_\theta \,, \, \nabla ) - \pi _{{{\mathscr {H}}}^\phi } \, \nabla _{{\mathscr {T}}} {{\mathscr {T}}} \big ) = - K_0 - \sum _{j=1}^{2n-1} g_\theta \big ( \textbf{n}^0 \,, \, \pi _{{{\mathscr {V}}}^\phi } \, \nabla _{V_j} V_j \big ). \nonumber \\ \end{aligned}$$

(85)

Next [by (iii) in Proposition 2]

$$\begin{aligned} \textbf{n}^0 = \frac{1}{\sqrt{\Lambda _0}} \, \big \{ \nabla \phi ^1 - T(\phi ^1 ) \, T \big \}, \end{aligned}$$

(86)

and [by Lemma 1] $\nabla \phi ^1 \in {{\mathscr {H}}}^\phi$. Consequently [by (86)]

$$\begin{aligned} \pi _{{{\mathscr {V}}}^\phi } \, \textbf{n}^0 = - \frac{1}{\sqrt{\Lambda _0}} \, T(\phi ^1 ) \, T^{{\mathscr {V}}} = - \frac{\theta ({{\mathscr {T}}})}{\sqrt{\Lambda _0}} \, T(\phi ^1 ) \, {{\mathscr {T}}}. \end{aligned}$$

(87)

As $\dim _{{\mathbb {R}}} {{\mathscr {H}}}^\phi _x = 1$, there is a unique function $\varphi \in C^\infty ( \Omega )$ such that

$$\begin{aligned} T^{{\mathscr {H}}} = \varphi \, \nabla \phi ^1 = \varphi \, \big \{ \nabla ^H \phi ^1 + T(\phi ^1 ) \, T \big \}. \end{aligned}$$

(88)

To compute $\varphi$ one starts from $T^{{\mathscr {H}}} = T - T^{{\mathscr {V}}}$, yielding

$$\begin{aligned} \Vert T^{{\mathscr {H}}} \Vert ^2 = 1 - u^2. \end{aligned}$$

On the other hand [by (88)]

$$\begin{aligned} \Vert T^{{\mathscr {H}}} \Vert ^2 = \varphi ^2 \, \Big \{ \big \Vert \nabla ^H \phi ^1 \big \Vert ^2 + T(\phi ^1 )^2 \Big \}, \end{aligned}$$

so that [by the Levi conformality property]

$$\begin{aligned} \varphi ^2 \, \Big \{ \Lambda \, \big ( h^{11} \circ \phi \big ) + T(\phi ^1 )^2 \Big \} = 1 - \theta ({{\mathscr {T}}})^2. \end{aligned}$$

(89)

We may now compute the last term in (85) i.e.

$$\begin{aligned} \sum _{j=1}^{2n-1} g_\theta \big ( \textbf{n}^0 \,, \, \pi _{{{\mathscr {V}}}^\phi } \, \nabla _{V_j} V_j \big ) = \sum _{j=1}^{2n-1} g_\theta \big ( \pi _{{{\mathscr {V}}}^\phi } \, \textbf{n}^0 \,, \, \nabla _{V_j} V_j \big ) =, \end{aligned}$$

[by (88)]

$$\begin{aligned} = - \frac{1}{\sqrt{\Lambda _0}} \, T(\phi ^1 ) \, \sum _{j=1}^{2n-1} g_\theta \big ( T^{{\mathscr {V}}} \,, \, \nabla _{V_j} V_j \big ) =, \end{aligned}$$

[by $T^{{\mathscr {V}}} = T - T^{{\mathscr {H}}}$ and $T \; \bot \; H(M) \ni \nabla _{V_j} V_j$]

$$\begin{aligned} = \frac{1}{\sqrt{\Lambda _0}} \, T(\phi ^1 ) \, \sum _{j=1}^{2n-1} g_\theta \big ( T^{{\mathscr {H}}} \,, \, \nabla _{V_j} V_j \big ) =, \end{aligned}$$

[by (88)]

$$\begin{aligned}= & {} \frac{\varphi }{\sqrt{\Lambda _0}} \, T(\phi ^1 ) \, \sum _{j=1}^{2n-1} g_\theta \big ( \nabla \phi ^1 \,, \, \nabla _{V_j} V_j \big ) \\= & {} \frac{\varphi }{\sqrt{\Lambda _0}} \, T(\phi ^1 ) \, \sum _{j=1}^{2n-1} g_\theta \big ( \nabla ^H \phi ^1 \,, \, \nabla _{V_j} V_j \big ) =, \end{aligned}$$

[by $\nabla g_\theta = 0$]

$$\begin{aligned} = \frac{\varphi }{\sqrt{\Lambda _0}} \, T(\phi ^1 ) \, \sum _{j=1}^{2n-1} \Big \{ V_j \big ( g_\theta \big ( \nabla ^H \phi ^1 \,, \, V_j \big ) \big ) - g_\theta \big ( \nabla _{V_j} \nabla ^H \phi ^1 \,, \, V_j \big ) \Big \} =, \end{aligned}$$

[as $\nabla ^H \phi ^1 \in {{\mathscr {H}}}^\phi \; \bot \; {{\mathscr {V}}}^\phi \ni V_j$]

$$\begin{aligned} = - \frac{\varphi }{\sqrt{\Lambda _0}} \, T(\phi ^1 ) \, \sum _{j=1}^{2n-1} g_\theta \big ( \nabla _{V_j} \nabla ^H \phi ^1 \,, \, V_j \big ) =, \end{aligned}$$

[by (iii) in Proposition 2]

$$\begin{aligned} = - \frac{\varphi }{\sqrt{\Lambda _0}} \, T(\phi ^1 ) \, \Big \{ \textrm{div} \, \big ( \nabla ^H \phi ^1 \big ) - g_\theta \big ( \nabla _{V_{2n}} \big ( \sqrt{\Lambda _0} \, V_{2n} \big ) \,, \, V_{2n} \big ) \Big \} =, \end{aligned}$$

[as $g_\theta \big ( \nabla _{V_{2n}} V_{2n} \,, \, V_{2n} \big ) = 0$ and $g_\theta \big ( V_{2n} \,, \, V_{2n} \big ) = 1$]

$$\begin{aligned}= & {} \frac{\varphi }{\sqrt{\Lambda _0}} \, T(\phi ^1 ) \, \Big \{ \Delta _b \phi ^1 + V_{2n} \big ( \sqrt{\Lambda _0} \big ) \Big \}\\= & {} \frac{\varphi }{\sqrt{\Lambda _0}} \, T(\phi ^1 ) \, \Big \{ \Delta _b \phi ^1 + \big ( \nabla ^H \phi ^1 \big ) \, \log \, \sqrt{\Lambda _0} \Big \}, \end{aligned}$$

hence [by (iv) in Proposition 2]

$$\begin{aligned} \sum _{j=1}^{2n-1} g_\theta \big ( \textbf{n}^0 \, , \, \pi _{{{\mathscr {V}}}^\phi } \, \nabla _{V_j} V_j \big ) = - \varphi \, T(\phi ^1 ) \, K_0. \end{aligned}$$

(90)

Equation (85) becomes [by (90)]

$$\begin{aligned} g_\theta \big ( \textbf{n}^0 \, , \, 2 n \, \mu ^{{{\mathscr {V}}}^\phi } ( g_\theta \, , \, \nabla ) - \pi _{{{\mathscr {H}}}^\phi } \, \nabla _{{\mathscr {T}}} {{\mathscr {T}}} \big ) = \big \{ \varphi \, T(\phi ^1 ) - 1 \big \} \, K_0. \end{aligned}$$

(91)

This yields (4) in Theorem 5 because of

$$\begin{aligned} g_\theta \big ( \textbf{n}^0 \, , \, \pi _{{{\mathscr {H}}}^\phi } \, J {{\mathscr {T}}} \big ) = 0. \end{aligned}$$

(92)

Indeed, the identity ${{\mathscr {T}}} = u^{-1} \big \{ T - \varphi \, \nabla \phi ^1 \big \}$ implies

$$\begin{aligned} J {{\mathscr {T}}} = - \frac{\varphi }{u} \, \sqrt{\Lambda _0} \, J \textbf{n}^0. \end{aligned}$$

(93)

Also [by (87)]

$$\begin{aligned} \pi _{{{\mathscr {H}}}^\phi } \, \textbf{n}^0 = \textbf{n}^0 + \frac{u}{\sqrt{\Lambda _0}} \, T(\phi ^1 ) \, {{\mathscr {T}}} . \end{aligned}$$

(94)

Finally

$$\begin{aligned} g_\theta \big ( \textbf{n}^0 \,, \, \pi _{{{\mathscr {H}}}^\phi } \, J {{\mathscr {T}}} \big ) = g_\theta \big ( \pi _{{{\mathscr {H}}}^\phi } \, \textbf{n}^0 \,, \, J \, {{\mathscr {T}}} \big ) =, \end{aligned}$$

[by (94), as $g_\theta ({{\mathscr {T}}}, \, J {{\mathscr {T}}}) = 0$]

$$\begin{aligned}= & {} g_\theta \big ( \textbf{n}^0 \,, \, J \, {{\mathscr {T}}} \big ) = \;\;\;\; \text {[by (93)]}\\= & {} - \frac{\varphi }{u} \, \sqrt{\Lambda _0} \, g_\theta \big ( \textbf{n}^0 \,, \, J \textbf{n}^0 \big ) = 0, \end{aligned}$$

and (92) is proved.

Let us recall (61). This was stated for $m \ge 2$ yet it is easily seen to hold for any $m \ge 1$, everywhere in $\Omega$. Then [by (61) with $m = 1$]

$$\begin{aligned} \tau _b (\phi ) = \phi _*\, \nabla ^H \log \sqrt{\Lambda } - 2 n \, \phi _*\, \mu ^{{{\mathscr {V}}}^\phi }_{\textrm{hor}}, \end{aligned}$$

so that [by $\tau _b (\phi ) = 0$]

$$\begin{aligned} 2 n \, \mu ^{{{\mathscr {V}}}^\phi }_{\textrm{hor}} = \pi _{{{\mathscr {H}}}^\phi } \, \nabla ^H \, \log \sqrt{\Lambda }. \end{aligned}$$

(95)

Note that $\Lambda _0 = \Lambda \, \big ( h^{11} \big )^\phi$ yields

$$\begin{aligned} \nabla ^H \log \, \sqrt{\Lambda } = \nabla ^H \log \sqrt{\Lambda _0} + \left\{ \begin{array}{c} 1 \\ 11 \end{array} \right\} ^\phi \, \nabla ^H \phi ^1. \end{aligned}$$

(96)

Then [by (95) and (96)]

$$\begin{aligned} 2 n \, g_\theta \big ( \textbf{n}^0 \,, \, \mu ^{{{\mathscr {V}}}^\phi }_{\textrm{hor}} \big )= g_\theta \big ( \pi _{{{\mathscr {H}}}^\phi } \, \textbf{n}^0 \,, \, \nabla ^H \log \sqrt{\Lambda _0} \big ) + \left\{ \begin{array}{c} 1 \\ 11 \end{array} \right\} ^\phi \, g_\theta \big ( \pi _{{{\mathscr {H}}}^\phi } \, \textbf{n}^0 \,, \, \nabla ^H \phi ^1 \big ). \nonumber \\ \end{aligned}$$

(97)

The right hand side in (97) may be computed as follows

$$\begin{aligned}{} & {} g_\theta \big ( \pi _{{{\mathscr {H}}}^\phi } \, \textbf{n}^0 \,, \, \nabla ^H \log \sqrt{\Lambda _0} \big ) = \frac{1}{\sqrt{\Lambda _0}} \, \big [ 1 - \varphi \, T(\phi ^1 ) \big ] \, \big ( \nabla ^H \phi ^1 \big ) \, \log \sqrt{\Lambda _0} \,, \\{} & {} \quad g_\theta \big ( \pi _{{{\mathscr {H}}}^\phi } \, \textbf{n}^0 \,, \, \nabla ^H \phi ^1 \big ) = \sqrt{\Lambda _0} \, \big [ 1 - \varphi \, T(\phi ^1 ) \big ], \end{aligned}$$

hence (97) becomes

$$\begin{aligned} 2 n \, g_\theta \big ( \textbf{n}^0 \,, \, \mu ^{{{\mathscr {V}}}^\phi }_{\textrm{hor}} \big ) = \big \{ 1 - \varphi \, T(\phi ^1 ) \big \} \Big [ \frac{1}{\sqrt{\Lambda _0}} \, \big ( \nabla ^H \phi ^1 \big ) \, \ \log \sqrt{\Lambda _0} + \sqrt{\Lambda _0} \, \left\{ \begin{array}{c} 1 \\ 11 \end{array} \right\} ^\phi \Big ], \end{aligned}$$

or [by (3)]

$$\begin{aligned} 2 n \, g_\theta \big ( \textbf{n}^0 \,, \, \mu ^{{{\mathscr {V}}}^\phi }_{\textrm{hor}} \big ) = \big \{ \varphi \, T(\phi ^1 ) - 1 \big \} \, \Big [ K_0 + \frac{1}{\sqrt{\Lambda _0}} \, \Delta _b \phi ^1 - \sqrt{\Lambda _0} \, \left\{ \begin{array}{c} 1 \\ 11 \end{array} \right\} ^\phi \Big ], \end{aligned}$$

where $- \Delta _b \phi ^1 + \Lambda _0 \, \Big \{ \begin{array}{c} 1 \\ 11 \end{array} \Big \}^\phi = 0$ (as $\phi$ is also a subelliptic harmonic map). Therefore, unlike the case $m = 2$, the fundamental equation (61) for a subelliptic harmonic morphism is equivalent to (4) in Theorem 5 and implies no further restrictions on $K_0$.

8 Examples

8.1 Morphisms from the Heisenberg Group

Let us set $f (z, t) = |z|^2 - i \, t$, so that f is a CR function on ${{\mathbb {H}}}_n$ i.e. $\overline{L}_\alpha f = 0$ for any $1 \le \alpha \le n$.

Theorem 8

Let $\phi : {{\mathbb {H}}}_n \setminus \{ 0 \} \rightarrow {{\mathbb {R}}}$ be the $C^\infty$ map given by

$$\begin{aligned} \phi = 1 \big / \big ( f \, \overline{f} \big )^{n/2}. \end{aligned}$$

(98)

Then

(i) $\phi$ is a subelliptic harmonic morphism of the pseudohermitian manifold $\big ( {{\mathbb {H}}}_n {\setminus } \{ 0 \}, \, \theta _0 \big )$ into the Riemannian manifold $\big ( {{\mathbb {R}}}, \, d y^1 \otimes d y^1 \big )$.

(ii) $\textrm{Crit} (\phi ) = \emptyset$ and $S(\phi ) = \mathbb {H}_n {\setminus } \{ 0 \}$.

(iii) $\textrm{I}_1 (\phi )= {{\mathbb {C}}}^*\times \{ 0 \}$ where ${{\mathbb {C}}}^*= {{\mathbb {C}}} \setminus \{ 0 \}$.

(iv) $\phi$ is a subelliptic harmonic map of $({{\mathbb {H}}}_n {\setminus } \{ 0 \}, \, \theta _0 )$ into $({{\mathbb {R}}}, \, d y^1 \otimes d y^1 )$, and a Levi conformal map of square dilation

$$\begin{aligned} \Lambda (x) = \frac{2 n ^2 \, |z|^2}{|x|^{2 Q}} \, , \;\;\; x = (z, t) \in {{\mathbb {H}}}_n \, , \;\; x \ne 0. \end{aligned}$$

(99)

Consequently

$$\begin{aligned} \textrm{II}_1 (\phi )= \{ 0 \} \times {{\mathbb {R}}}^*\, , \;\;\; \textrm{III}_1 (\phi )= \mathbb {C}^*\times \mathbb {R}^*\, , \;\;\; {{\mathbb {R}}}^*= {{\mathbb {R}}} \setminus \{ 0 \} . \end{aligned}$$

(100)

(v) The horizontal mean curvature of the leaves of ${\mathscr {F}}$ is

$$\begin{aligned} K_0 = \frac{1}{2 \sqrt{2} \, |z|} \, \big ( f \, \overline{f} \big )^{- 1/2} \, \big [ f + \overline{f} - 2 \, Q \, |z|^2 \big ] = - \frac{(Q - 1) \, |z|}{\sqrt{2} \, |x|^2}. \end{aligned}$$

(101)

Here $Q = 2 n + 2$ (the homogeneous dimension of ${{\mathbb {H}}}_n$) and $|x| = \big ( |z|^4 + t^2\big )^{1/4}$ [the Heisenberg norm of $x = (z, t) \in {{\mathbb {H}}}_n$].

Proof

(i) By (98) $\phi (x) = |x|^{- Q + 2}$ for any $x \in {{\mathbb {H}}}_n$. Then $\phi (x)$ is the fundamental solution to ${{\mathscr {L}}}_0 = \Delta _b$ discovered by Folland (cf. [31]) i.e. there is a constant $c_0 \ne 0$ such that $\Delta _b \phi = c_0 \, \delta$, where $\delta$ is the Dirac distribution (concentrated in zero). In particular $\phi : {{\mathbb {H}}}_n {\setminus } \{ 0 \} \rightarrow {{\mathbb {R}}}$ is a subelliptic harmonic morphism of $\big ( {{\mathbb {H}}}_n {\setminus } \{ 0 \}, \; \theta _0 \big )$ into $\big ( {{\mathbb {R}}}, \; d y^1 \otimes d y^1 \big )$.

(ii) The Euclidean gradient of $\phi$ is

$$\begin{aligned} D \phi (x) = - \frac{(Q - 2) \, |z|^2}{|x|^{Q + 2}} \; \Big ( z, \, \frac{t}{2 |z|^2} \Big ) , \;\;\; x = (z, \, t) , \end{aligned}$$

(102)

so that $\textrm{Crit} (\phi ) = \emptyset$, and then $S(\phi ) = {{\mathbb {H}}}_n {\setminus } \{ 0 \}$.

(iii) Note that

$$\begin{aligned} T (\phi )_x = - \frac{n \, t}{|x|^{Q + 2}} \,, \;\;\; x = (z, \, t). \end{aligned}$$

Hence [by statement (i) in Lemma 3]

$$\begin{aligned} x \in \textrm{I}_1 (\phi ) \Longleftrightarrow (d_x \phi ) T_x = 0 \Longleftrightarrow t = 0. \end{aligned}$$

$\square$

(iv) The horizontal gradient of $\phi$ is

$$\begin{aligned} \nabla ^H \phi = - \frac{n}{|x|^{Q+2}} \sum _{\alpha = 1}^n \Big \{ z^\alpha \, f \, L_\alpha + \overline{z}_\alpha \, \overline{f} \, \overline{L}_\alpha \Big \} . \end{aligned}$$

(103)

Then [by (16)] the square dilation is

$$\begin{aligned} \Lambda (z, t) = G_\theta (\nabla ^H \phi , \nabla ^H \phi )_{(z, t)} = \frac{ 2 n^2 \, |z|^2}{\big ( |z|^4 + t^2\big )^{n+1}}, \end{aligned}$$

which is (99). Next [by (99) together with statements (ii) in Lemma 3 and (iii) in Lemma 1]

$$\begin{aligned} x \in \textrm{II}_1 (\phi ) \Longleftrightarrow \Lambda (x) = 0 \Longleftrightarrow z = 0. \end{aligned}$$

$\square$

(v) The use of the CR function $f (z, t) = |z|^2 - i t$ greatly simplifies calculations. One starts by rephrasing the square dilation as

$$\begin{aligned} \Lambda _0 = \Lambda = \frac{2 \, n^2 \, |z|^2}{\big ( f \, \overline{f} \big )^{Q/2}}, \end{aligned}$$

so that

$$\begin{aligned} L_\alpha \Big ( \log \sqrt{\Lambda _0} \Big ) = \frac{\overline{z}_\alpha }{2 \, f \, |z|^2} \, \Big ( f - Q \, |z|^2 \Big ), \end{aligned}$$

yielding [by (103) and statement (i) in Theorem 5] (101). $\square$

The sublaplacian $\Delta _b$ on $({{\mathbb {H}}}_n \,, \, \theta _0 )$ belongs to the family $\{ {{\mathscr {L}}}_\gamma \}_{\gamma \in {{\mathbb {C}}}}$ of Folland-Stein operators (cf. e.g. [29, p. 177])

$$\begin{aligned} {{\mathscr {L}}}_\gamma \equiv - \frac{1}{2} \sum _{\alpha = 1}^n \big ( L_\alpha \, \overline{L}_\alpha + \overline{L}_\alpha \, L_\alpha \big ) + i \, \gamma \, \frac{\partial }{\partial t} \,, \;\;\; \gamma \in {{\mathbb {C}}}. \end{aligned}$$

A $C^\infty$ map $\phi : {{\mathbb {H}}}_n \setminus \{ 0 \} \rightarrow N$ into a Riemannian manifold $(N, \, h)$ is a ${{\mathscr {L}}}_\gamma$-morphism if for any harmonic function $v: V \subset N \rightarrow {{\mathbb {R}}}$ [i.e. $\Delta _h v = 0$ in V] ${{\mathscr {L}}}_\gamma (v \circ \phi ) = 0$ in $U = \phi ^{-1} (V)$.

A complex number $\gamma \in {{\mathbb {C}}}$ is admissible if and only if $c_\gamma \ne 0$ where

$$\begin{aligned} c_\gamma = \frac{2^{2 - 2 n} \, \pi ^{n+1}}{\displaystyle {\Gamma \Big ( \frac{n + \gamma }{2} \Big ) \, \Gamma \Big ( \frac{n - \gamma }{2} \Big )}}. \end{aligned}$$

By a result of Folland and Stein (cf. [32], or [29, p. 179]) if $\gamma$ is admissible (equivalently $\gamma \in \big \{ \pm n, \; \pm (n + 2), \; \pm (n + 4), \; \ldots \; \}$)

$$\begin{aligned} \phi _\gamma = f^{- (n + \gamma )/2} \, \overline{f}^{- (n - \gamma )/2}, \end{aligned}$$

is a fundamental solution to ${{\mathscr {L}}}_\gamma$ i.e. ${{\mathscr {L}}}_\gamma \, \phi _\gamma = c_\gamma \, \delta$. In particular

$$\begin{aligned} \phi _{\pm p}: {{\mathbb {H}}}_n \setminus \{ 0 \} \rightarrow {{\mathbb {R}}}, \;\;\; p \in \big \{ n, \; n + 2, \; n + 4, \; \ldots \big \}, \end{aligned}$$

are ${{\mathscr {L}}}_{\pm p}$-morphisms, of ${{\mathbb {H}}}_n {\setminus } \{ 0 \}$ into $\big ( {{\mathbb {R}}}, \, d y^1 \otimes d y^1 \big )$. Also ${{\mathscr {L}}}_\gamma$ is hypoelliptic if and only if $\gamma$ is admissible. The study of ${{\mathscr {L}}}_\gamma$-morphisms into a general Riemannian manifold $(N, \, h)$ is an open problem.

8.2 Morphisms from Rossi Spheres

Let $S^2 = \big \{ \big ( y^1 \,, \, y^2 \,, \, y^3 \big ) \in {{\mathbb {R}}}^3 \,: \, \sum _{j=1}^3 (y^j )^2 = 1 \big \}$ and $S^3 = \big \{ (z, \, w) \in {{\mathbb {C}}}^2 \,: \, |z|^2 + |w|^2 = 1 \big \}$, and let $\pi : S^3 \rightarrow S^2$ be the Hopf fibration i.e. $\pi (z, \, w) = \big ( y^1 \,, \, y^2 \,, \, y^3 \big )$

$$\begin{aligned} {\left\{ \begin{array}{ll} y^1= |z|^2 - |w|^2 \, , \\ y^2 = z \, \overline{w} + \overline{z} \, w , \\ y^3 = - i \, \left( z \overline{w} - \overline{z} \, w \right) . \\ \end{array}\right. } \end{aligned}$$

(104)

Let $h_{S^N} = \textbf{j}^*\, g_0$ be the first fundamental form of $\textbf{j}: S^N \hookrightarrow {{\mathbb {R}}}^{N+1}$, where $g_0$ is the Euclidean metric on ${{\mathbb {R}}}^{N+1}$. Let $S^3$ be equipped with the standard CR structure $T_{1,0} (S^3)$ [induced by the complex structure of $\mathbb {C}^2$], and with the canonical contact form

$$\begin{aligned} \theta = \frac{i}{2} \big \{ - \overline{z} \, d z + z \, d \overline{z} - \overline{w} \, d w + w \, d\overline{w} \big \} \in {{\mathscr {P}}}_+ (S^3 ) . \end{aligned}$$

(105)

$T_{1,0} (S^3)$ is the span of $L = \overline{w} \, \big ( \partial / \partial z \big ) - \overline{z} \, \big ( \partial /\partial w \big )$. Let us set

$$\begin{aligned} L_t = L + t \, \overline{L}, \;\;\; |t| < 1, \end{aligned}$$

and let $H_t$ be CR structure on $S^3$ spanned by $L_t$ [$\{ (S^3 \,, \, H_t ) \}_{|t| < 1}$ are the Rossi spheres]. By a result in [57], the CR manifold $(S^3 \,, \, H_t )$ is globally embeddable in ${{\mathbb {C}}}^2$ if and only if $t = 0$.

Theorem 9

(i) The Hopf map $\pi : S^3 \rightarrow S^2$ is a subelliptic harmonic morphism of $\big ( S^3 \,, \, T_{1,0} (S^3 ), \, \theta \big )$ into $(S^2 \,, \, h_{S^2} \big )$.

(ii) $\pi$ is a subelliptic harmonic morphism of $\big ( S^3 \,, \, H_t \,, \, \theta \big )$ into $(S^2 \,, \, h_{S^2} \big )$ if and only if $t = 0$.

Note that $\theta$ is indeed a positively oriented contact form on each Rossi sphere $(S^3, \, H_t )$. The corresponding Levi form $G_\theta (t)$ is

$$\begin{aligned} G_\theta (t) \big ( L_t \,, \, \overline{L}_t \big ) =\frac{1 - t^2}{2} \,, \;\;\; |t| < 1. \end{aligned}$$

The Reeb vector field of $(S^3 \,, \, \theta )$ is

$$\begin{aligned} T= i \, \Big \{ z \, \frac{\partial }{\partial z} - \overline{z} \, \frac{\partial }{\partial \overline{z}} + w \, \frac{\partial }{\partial w} - \overline{w} \, \frac{\partial }{\partial \overline{w}} \Big \} . \end{aligned}$$

(106)

Also, the CR manifolds $\{ \big ( S^3 \,, \, H_t \big ) \}_{|t| < 1}$ have the same Levi distribution as $\big ( S^3 \,, \, H_0 \big )$ i.e. $H(S^3 ) = \textrm{Re} \{ H_0 \oplus \overline{H}_0 \}$

Proof of Theorem 9

(i) A calculation shows that $\pi _*\, T = 0$. Let $g_\theta$ be the Webster metric of $(S^3 \,, \, {{\mathscr {H}}}_0 \,, \, \theta )$. Then $g_\theta = g_{S^3}$ (cf. e.g. [29]). The Hopf map $\pi$ is an ordinary harmonic morphism of $\big ( S^3 \,, \, g_{S^3} \big )$ into $\big ( S^2 \,, \, g_{S^2} \big )$. Therefore [by (12)] $\pi$ is a subelliptic harmonic morphism of $\big ( S^3 \,, \, H_0 \,, \, \theta \big )$ into $\big ( S^2 \,, \, g_{S^2} \big )$.

(ii) By statement (i) in Theorem 9, the Hopf map $\pi$ is a $C^\infty$ submersion such that ${{\mathscr {V}}}^\pi = {{\mathbb {R}}} T$. In particular the Levi and horizontal distributions coincide i.e. $H(S^3 ) = {{\mathscr {H}}}^\pi$.

Let us assume that $\pi$ is Levi conformal, as a map of $(S^3 \,, \, H_t \,, \, \theta )$ into $(S^2 \,, \, h_{S^2} )$. The vector fields

$$\begin{aligned} X_1 (t)= & {} L_t + \overline{L}_t = (1+t) \, \big ( L + \overline{L} \big ), \\ X_2 (t)= & {} i \, \big ( L_t - \overline{L}_t \big ) = i \, (1 - t) \, \big ( L - \overline{L} \big ), \end{aligned}$$

span $H(S^3 )$ and

$$\begin{aligned} G_\theta (t) \left( X_a^t \,, \, X_a^t\right) = 1 - t^2 \,, \;\;\; G_\theta (t) \big ( X_1^t \,, \, X_2^t \big ) = 0, \;\;\; a \in \{ 1, \, 2 \}. \end{aligned}$$

Then [by (16)] for any $(z, w) \in S^3 {\setminus } \textrm{Crit} (\pi )$

$$\begin{aligned} 1 - t^2= & {} G_\theta (t) \big ( X_1 (t), \, X_1 (t) \big )_{(z, w)}= \Lambda (z, w) \, h^\pi _{S^2} \big ( \pi _*\, X_1 (t), \, \pi _*\, X_1 (t) \big )_{(z, w)}\\= & {} 4 (1 + t)^2 \, \Lambda (z, w), \end{aligned}$$

i.e.

$$\begin{aligned} 4 \, \Lambda (z, w) = \frac{1 - t}{1 + t}. \end{aligned}$$

On the other hand

$$\begin{aligned} 1 - t^2= & {} G_\theta (t) \big ( X_2 (t), \, X_2 (t) \big )_{(z, w)} \\= & {} \Lambda (z, w) \, h^\pi _{S^2} \big ( \pi _*\, X_2 (t), \, \pi _*\, X_2 (t) \big )_{(z, w)} = 4 (1 - t)^2 \, \Lambda (z, w), \end{aligned}$$

yields

$$\begin{aligned} 4 \, \Lambda (z, w) = \frac{1 + t}{1 - t}, \end{aligned}$$

hence $t=0$. $\square$

The Authors are grateful to the Referee for drawing their attention on the work by J. Ventura (cf. [62]) and the Ricci curvature calculations there vis-a-vis to a horizontally weakly conformal map $\Phi : {{\mathfrak {M}}} \rightarrow N^2$ from a 4-dimensional Lorentzian manifold ${\mathfrak {M}}$ to a surface $N^2$, together with the investigation (cf. op. cit.) of the behavior of Ricci curvature under biconformal deformations (7). It should be noticed that according to our Theorem 9 examples in that context are actually scarce (the only CR structure on $S^3$ in Rossi’s family, with respect to which the Hopf map $\pi : S^3 \rightarrow S^2$ is a subelliptic harmonic morphism, is the standard CR structure on the sphere). Several new space-time models are built in [62] starting from classical examples of space-times and harmonic morphisms (cf. [5]) and it is a natural question, asked by the Reviewer, whether and how biconformal changes of the metric affect our examples in Sect. 8. We leave that as an open problem.

We close with the observation that, in a simple context such as $M^3 = {{\mathbb {H}}}_1$ (the lowest dimensional Heisenberg group) and $\mathfrak {M}^4 = C \big ( {{\mathbb {H}}}_1 \big )$ equipped with the Fefferman metric $g_0 = F_{\theta _0}$ [associated to the canonical contact form $\theta _0$ in Sect. 7] looking for vacuum solutions to the gravitational field equations on $C({{\mathbb {H}}}_1 )$ by conformal or biconformal deformations of $g_0$, lacks a physical meaning. Indeed $g_0$ isn’t flat and its curvature corresponds, by the General Relativity and Gravitation Theory, to the content of matter and energy of the region $\Omega \subset C({{\mathbb {H}}}_1 )$ where gravitational effects are perceived. Said matter-energy content of $\Omega$ is described by an energy-momentum tensor $T_{\lambda \mu }$ that is by definition the traceless Ricci tensor associated to $g_0$. The linearized Einstein equations (in the presence of the matter distribution assimilated with the non flat character of $g_0$ i.e. involving $T_{\lambda \mu }$) were solved by Barletta et al. [11].