Foliated Hopf hypersurfaces in complex hyperbolic quadrics

Abstract

This paper deals with a limiting case motivated by contact geometry. The limiting case of a tensorial characterization of contact hypersurfaces in Kähler manifolds leads to Hopf hypersurfaces whose maximal complex subbundle of the tangent bundle is integrable. It is known that in non-flat complex space forms and in complex quadrics such real hypersurfaces do not exist, but the existence problem in other irreducible Kähler manifolds is open. In this paper we construct explicitly a one-parameter family of homogeneous Hopf hypersurfaces, whose maximal complex subbundle of the tangent bundle is integrable, in a Hermitian symmetric space of non-compact type and rank two. These are the first known examples of such real hypersurfaces in irreducible Kähler manifolds.

1 Introduction

We start with the motivation for this paper. A contact manifold is a smooth odd-dimensional manifold M together with a 1-form $\eta$ on M satisfying $\eta \wedge (d\eta )^{n-1} \ne 0$, where $\dim _\mathbb {R}(M) = 2n-1$. Such a 1-form $\eta$ is called a contact form. The kernel of $\eta$ defines a hyperplane distribution ${\mathcal {C}}$ on M, the so-called contact distribution. The contact condition $\eta \wedge (d\eta )^{n-1} \ne 0$ means that the maximal possible dimension of a submanifold of M all of whose tangent spaces are contained in ${\mathcal {C}}$ is equal to $n-1$. The contact condition therefore is a measure for maximal non-integrability of ${\mathcal {C}}$.

Let $\bar{M}$ be a Kähler manifold with Kähler structure J, Kähler metric g and $n = \dim _\mathbb {C}(\bar{M}) \ge 2$. Let M be a real hypersurface in $\bar{M}$ and $(\phi ,\xi ,\eta ,g)$ be the induced almost contact metric structure on M (see Sect. 2). The subbundle ${\mathcal {C}}= \ker (\eta ) = TM \cap J(TM)$ of TM is the maximal complex subbundle of the tangent bundle TM. The real hypersurface M is said to be a contact hypersurface if there exists an everywhere non-zero smooth function $f: M \rightarrow \mathbb {R}$ so that $d\eta = 2f \omega$, where $\omega$ is the fundamental 2-form on M defined by $\omega (X,Y) = g(\phi X,Y)$ for all $X,Y \in {\mathfrak {X}}(M)$. The fundamental 2-form $\omega$ is always closed, which implies $\eta \wedge d\eta ^{n-1} = (2f)^{n-1}(\eta \wedge \omega ^{n-1}) \ne 0$ if M is a contact hypersurface. Thus every contact hypersurface in a Kähler manifold is a contact manifold. In this situation the maximal complex subbundle ${\mathcal {C}}$ of the tangent bundle of the contact hypersurface coincides with the contact distribution. A natural problem is to determine the contact hypersurfaces in Kähler manifolds.

The first systematic study of contact hypersurfaces in Kähler manifolds was carried out by Okumura [14]. Okumura proved the following very useful characterization of contact hypersurfaces in Kähler manifolds: A real hypersurface M in a Kähler manifold $\bar{M}$ is a contact hypersurface if and only if there exists an every non-zero smooth function $f: M \rightarrow \mathbb {R}$ so that that the shape operator A of M and the structure tensor field $\phi$ satisfy $A\phi + \phi A = 2f \phi$. It is not difficult to prove that the function f is constant when $n > 2$ (see [3], Proposition 3.5.4). Starting from Okumura’s work, contact hypersurfaces were classified in various Hermitian symmetric spaces (see [3] for an overview). The motivation for this paper is to understand the limiting case $f = 0$. We will show (see Proposition 2.2) that the limiting case $f = 0$ characterizes Hopf hypersurfaces in Kähler manifolds for which the maximal complex subbundle ${\mathcal {C}}$ is integrable. For the concept of Hopf hypersurfaces see Sect. 2.

The totally geodesic real hypersurface $\mathbb {R}^{2n-1}$ in the complex Euclidean space $\mathbb {C}^n$ is an elementary example of a Hopf hypersurface whose maximal complex subbundle ${\mathcal {C}}$ is integrable. In contrast, it is quite remarkable and not obvious that in non-flat complex space forms there are no Hopf hypersurfaces whose maximal complex subbundle ${\mathcal {C}}$ is integrable. This is not difficult to prove for the complex projective space $\mathbb {C}P^n(c)$ with the Fubini-Study metric of constant holomorphic sectional curvature $c > 0$, but the proof is quite involved for the complex hyperbolic space $\mathbb {C}H^n(c)$ with the Bergman metric of constant holomorphic sectional curvature $c < 0$. A detailed discussion of these two cases can be found in Section 2 of [13]. These non-existence results raise the existence question for other irreducible Kähler manifolds. In [3], the geometry of real hypersurfaces in some irreducible Hermitian symmetric spaces of rank 2 was investigated. One of these Hermitian symmetric spaces is the Grassmann manifold $SO_{2+n}/(SO_2\times SO_n)$ of oriented 2-planes in $\mathbb {R}^{2+n}$, which is isometric to the complex quadric $Q^n$ in $\mathbb {C}P^{n+1}(c)$ (with a suitable normalization of the metric). From the investigations in [3], Section 6.4, we can conclude that there are no Hopf hypersurfaces in this Hermitian symmetric space for which ${\mathcal {C}}$ is integrable.

In this paper we investigate the existence question in the dual Hermitian symmetric space of non-compact type, the complex hyperbolic quadric ${Q^n}^* = SO^o_{2,n}/(SO_2\times SO_n)$. Surprisingly, we can construct a one-parameter family of pairwise non-congruent homogeneous Hopf hypersurfaces in ${Q^n}^*$ whose maximal complex subbundle ${\mathcal {C}}$ is integrable.

Theorem 1.1

There exists a one-parameter family $M^{2n-1}_\alpha$, $0 \le \alpha < \infty$, of (pairwise non-congruent) homogeneous Hopf hypersurfaces, whose maximal complex subbundle of the tangent bundle is integrable, in the Hermitian symmetric space ${Q^n}^* = SO^o_{2,n}/(SO_2\times SO_n)$, $n \ge 3$.

We give a brief geometric description of these real hypersurfaces. We normalize the Riemannian metric on ${Q^n}^*$ so that the minimum of the sectional curvature is equal to $-4$. The complex hyperbolic quadric ${Q^n}^*$ is equipped with a circle bundle ${\mathfrak {A}}_0$ of real structures (see Sect. 3). This circle bundle determines a maximal ${\mathfrak {A}}_0$-invariant subbundle ${\mathcal {Q}}$ of the tangent bundle TM of M. The maximal Satake compactification of ${Q^n}^* = SO^o_{2,n}/(SO_2\times SO_n)$ has two boundary components of rank 1, namely a complex hyperbolic line $B_1 \cong \mathbb {C}H^1(-4)$ of constant (holomorphic) sectional curvature $-4$ and a real hyperbolic space $B_2 \cong \mathbb {R}H^{n-2}(-2)$ of constant sectional curvature $-2$. It is an interesting fact that all non-zero tangent vectors of $B_1$ are singular tangent vectors of ${Q^n}^*$ of a particular type (that is, tangent vectors that are contained in more than one maximal flat of ${Q^n}^*$). In [4], we developed a technique, the so-called canonical extension method, for extending isometric actions on boundary components of irreducible Riemannian symmetric spaces of non-compact type to isometric actions on the entire symmetric space. This method can be used to extend submanifolds in boundary components. By extending a point in the boundary component $B_1 \cong \mathbb {C}H^1(-4)$ we obtain an isometric embedding $P^{n-1}$ of the complex hyperbolic space $\mathbb {C}H^{n-1}(-4)$ into ${Q^n}^*$ as a homogeneous complex hypersurface. This construction will be explained in detail in Sect. 4, where we will also investigate the geometry of this homogeneous complex hypersurface.

This homogeneous complex hypersurface $P^{n-1}$ will appear as the integral manifolds of the integrable distribution ${\mathcal {C}}$ in our examples. The Langlands decomposition of the parabolic subgroup of $SO^o_{2,n}$ with boundary component $B_1 \cong \mathbb {C}H^1(-4)$ induces a horospherical decomposition $B_1 \times \mathbb {R}\times H^{2n-3}$ of ${Q^n}^*$, where $H^{2n-3}$ is the $(2n-3)$-dimensional Heisenberg group with 1-dimensional center. The product $\mathbb {R}\times H^{2n-3}$ corresponds to the homogeneous complex hypersurface $P^{n-1} \cong \mathbb {C}H^{n-1}(-4)$. Now take any complete curve $\gamma$ in the boundary component $B_1 \cong \mathbb {C}H^1(-4)$ with constant geodesic curvature $\alpha \ge 0$. The curve $\gamma$ is a geodesic in $\mathbb {C}H^1(-4)$ if $\alpha = 0$, an equidistant curve to a geodesic in $\mathbb {C}H^1(-4)$ if $0< \alpha < 2$, a horocycle if $\alpha = 2$, or a closed circle in $\mathbb {C}H^1(-4)$ if $2< \alpha < \infty$. Sliding the homogeneous complex hypersurface $P^{n-1} \cong \mathbb {C}H^{n-1}(-4)$ along the curve $\gamma$ in a suitable way, we obtain a homogeneous Hopf hypersurface $M^{2n-1}_\alpha$ in ${Q^n}^*$ whose maximal complex subbundle ${\mathcal {C}}$ is integrable. We will see that the homogeneous real hypersurface $M^{2n-1}_\alpha$ has constant principal curvatures $\alpha$, 0, $+1$, $-1$ with multiplicities 1, 2, $n-1$, $n-1$, respectively. The principal curvature space $T_\alpha$ is equal to the orthogonal complement ${\mathcal {C}}^\perp$ of ${\mathcal {C}}$ in TM. The principal curvature space $T_0$ is equal to the orthogonal complement ${\mathcal {C}}\ominus {\mathcal {Q}}$ of ${\mathcal {Q}}$ in ${\mathcal {C}}$. The principal curvature spaces $T_1$ and $T_{-1}$ span ${\mathcal {Q}}$ are mapped into each other by the structure tensor field $\phi$ and are equal to the $\pm 1$-eigenspaces of the restriction to ${\mathcal {Q}}$ of a suitable real structure in ${\mathfrak {A}}_0$. The hypersurfaces $M^{2n-1}_\alpha$ will in fact be constructed through an algebraic process, and the “sliding” description is a geometric interpretation of this algebraic construction, which will be explained thoroughly during the construction process. The homogeneous real hypersurface $M^{2n-1}_\alpha$ is diffeomorphic to $\mathbb {R}^{2n-1}$ for $0 \le \alpha \le 2$ and diffeomorphic to $S^1 \times \mathbb {R}^{2n-2}$ for $2< \alpha < \infty$.

We point out that none of the homogeneous real hypersurfaces $M^{2n-1}_\alpha$ in Theorem 1.1 arises as a limit of contact hypersurfaces in ${Q^n}^*$. The classification of contact hypersurfaces in ${Q^n}^*$ can be found in Section 7.8 of [3]. For every real number $f > 0$ there exists, up to isometric congruence, a unique connected complete contact hypersurface $\tilde{M}^{2n-1}_f$ in ${Q^n}^*$ satisfying $A\phi + \phi A = 2f\phi$. This family $\tilde{M}^{2n-1}_f$ of contact hypersurfaces collapses to a totally geodesic complex embedding of the complex hyperbolic quadric ${Q^{n-1}}^*$ into ${Q^n}^*$ when taking the limit $f \rightarrow 0$, and so $\lim _{f\rightarrow 0} \tilde{M}^{2n-1}_f = {Q^{n-1}}^*$ is not a real hypersurface.

The paper is organized as follows. In Sect. 2 we introduce basic concepts from almost contact metric geometry in Kähler manifolds and provide characterizations of Hopf hypersurfaces and of real hypersurfaces satisfying $A\phi + \phi A = 0$. In Sect. 3 we present two models for the complex hyperbolic quadric ${Q^n}^* = SO^o_{2,n}/(SO_2\times SO_n)$. The first one is the standard symmetric space model, and the second one is the solvable Lie group model originating from an Iwasawa decomposition of $SO^o_{2,n}$. The interplay between both models allows us to switch between geometric and algebraic interpretations of relevant concepts. In Sect. 4 we construct the isometric embedding of the complex hyperbolic space $\mathbb {C}H^{n-1}(-4)$ as a homogeneous complex hypersurface $P^{n-1}$ in ${Q^n}^*$ and discuss aspects of the geometry of this embedding. The homogeneous real hypersurfaces $M^{2n-1}_\alpha$ ($2 < \alpha$) will be constructed in Sect. 5 as the tubes around the homogeneous complex hypersurface $P^{n-1}$ in ${Q^n}^*$. In Sect. 6 we use the theory of parabolic subalgebras of real semisimple Lie algebras for the construction of the minimal homogeneous real hypersurface $M^{2n-1}_0$. In Sect. 7 we construct the homogeneous real hypersurfaces $M^{2n-1}_\alpha$ ($0< \alpha < 2$) as the equidistant hypersurfaces to $M^{2n-1}_0$. The homogeneous real hypersurface $M^{2n-1}_2$ will be constructed in Sect. 8 as the canonical extension of a horocycle in the boundary component $B_1 \cong \mathbb {C}H^1(-4)$. We will also investigate the geometry of the homogeneous real hypersurfaces $M^{2n-1}_\alpha$ in the corresponding sections. In Sect. 9 we investigate the curvature of the homogeneous real hypersurfaces $M^{2n-1}_\alpha$.

2 The maximal complex subbundle of the tangent bundle

Let $\bar{M}$ be a Kähler manifold with Kähler structure J and Kähler metric g. We always assume $n = \dim _\mathbb {C}(\bar{M}) \ge 2$. Let M be a real hypersurface in $\bar{M}$. We will denote the induced Riemannian metric on M also by g. The Levi Civita covariant derivative of $\bar{M}$ and M is denoted by $\bar{\nabla }$ and $\nabla$, respectively. The Lie algebra of smooth vector fields on M is denoted by ${\mathfrak {X}}(M)$.

Let $\zeta$ be a (local) unit normal vector field on M. We denote by $A = A_\zeta$ the shape operator of M with respect to $\zeta$. The unit vector field

$$\begin{aligned} \xi = -J\zeta \end{aligned}$$

is the Reeb vector field on M. The flow of the Reeb vector field $\xi$ is the Reeb flow on M. We define a 1-form $\eta$ on M by

$$\begin{aligned} \eta (X) = g(X,\xi ) \end{aligned}$$

for all $X \in {\mathfrak {X}}(M)$ and a skew-symmetric tensor field $\phi$ on M by decomposing JX into its tangential component $\phi X$ and its normal component $g(JX,\zeta )\zeta$, that is,

$$\begin{aligned} JX = \phi X + g(JX,\zeta )\zeta = \phi X + \eta (X)\zeta \end{aligned}$$

for all $X \in {\mathfrak {X}}(M)$. The 1-form $\eta$ is the almost contact form on M, and the skew-symmetric tensor field $\phi$ is the structure tensor field on M. The quadruple $(\phi ,\xi ,\eta ,g)$ is the induced almost contact metric structure on M. Note that

$$\begin{aligned} \eta (\xi ) = 1,\ \phi \xi = 0 \text{ and } \phi ^2 X = -X + \eta (X)\xi \end{aligned}$$

for all $X \in {\mathfrak {X}}(M)$. Using the Kähler property $\bar{\nabla }J = 0$ and the Weingarten formula we obtain

$$\begin{aligned} 0 = (\bar{\nabla }_XJ)\zeta = \bar{\nabla }_X J\zeta - J\bar{\nabla }_X\zeta = -\bar{\nabla }_X\xi + JAX \end{aligned}$$

for all $X \in {\mathfrak {X}}(M)$. The tangential component of this equation induces the useful equation

$$\begin{aligned} \nabla _X\xi = \phi AX \end{aligned}$$

for all $X \in {\mathfrak {X}}(M)$.

The subbundle

$$\begin{aligned} {\mathcal {C}}= \ker (\eta ) = TM \cap J(TM) \end{aligned}$$

of the tangent bundle TM of M is the maximal complex subbundle of TM. We denote by $\Gamma ({\mathcal {C}})$ the set of all vector fields X on M with values in ${\mathcal {C}}$, that is,

$$\begin{aligned} \Gamma ({\mathcal {C}})&= \{ X \in {\mathfrak {X}}(M) : X_p \in {\mathcal {C}}_p \text{ for } \text{ all } p \in M\} \\&= \{ X \in {\mathfrak {X}}(M) : \eta (X) = 0 \} . \end{aligned}$$

The real hypersurface M is called a Hopf hypersurface if the Reeb flow on M is a geodesic flow, that is, if the integral curves of the Reeb vector field $\xi$ are geodesics in M. We have the following characterization of Hopf hypersurfaces.

Proposition 2.1

Let M be a real hypersurface in a Kähler manifold $\bar{M}$ with induced almost contact metric structure $(\phi ,\xi ,\eta ,g)$. The following statements are equivalent:

• (i) M is a Hopf hypersurface in $\bar{M}$;

• (ii) $\nabla _\xi \xi = 0$;

• (iii) The Reeb vector field $\xi$ is a principal curvature vector of M at every point;

• (iv) The maximal complex subbundle ${\mathcal {C}}$ of TM is invariant under the shape operator A of M, that is, $A{\mathcal {C}}\subseteq {\mathcal {C}}$.

Proof

Let $p \in M$ and $c: I \rightarrow M$ be an integral curve of the Reeb vector field $\xi$ with $0 \in I$ and $c(0) = p$. Then we have $\nabla _{\xi _p}\xi = \nabla _{\dot{c}(0)}\xi = (\xi \circ c)'(0) = \dot{c}'(0)$. If M is a Hopf hypersurface, then we have $\dot{c}'(0) = 0$ by definition and therefore $\nabla _{\xi _p}\xi = 0$. Since this holds at any point $p \in M$, we obtain $\nabla _\xi \xi = 0$. Conversely, if $\nabla _\xi \xi = 0$, then $\dot{c}' = \nabla _{\dot{c}}\xi = \nabla _{\xi \circ c}\xi = 0$ for any integral curve c of $\xi$. Thus any integral curve of $\xi$ is a geodesic in M and hence M is a Hopf hypersurface. This establishes the equivalence of (i) and (ii)

The kernel $\ker (\phi )$ of the structure tensor field $\phi$ is spanned by the Reeb vector field, that is, $\ker (\phi ) = \mathbb {R}\xi$. Since $\nabla _\xi \xi = \phi A\xi$, we therefore see that $\nabla _\xi \xi = 0$ if and only if $A\xi \in \mathbb {R}\xi$, which shows that (ii) and (iii) are equivalent.

We have the orthogonal decomposition $TM = {\mathcal {C}}\oplus \mathbb {R}\xi$. Since the shape operator A is self-adjoint, the equivalence of (iii) and (iv) is obvious. $\square$

The next result provides a characterization of real hypersurfaces when taking the limit $f \rightarrow 0$ in Okumura’s characterization $A\phi + \phi A = 2f\phi$ of contact hypersurfaces in Kähler manifolds.

Proposition 2.2

Let M be a real hypersurface in a Kähler manifold $\bar{M}$ with induced almost contact metric structure $(\phi ,\xi ,\eta ,g)$. The following statements are equivalent:

• (i) The almost contact form $\eta$ is closed, that is, $d\eta = 0$.

• (ii) The shape operator A of M and the structure tensor field $\phi$ satisfy

  $$\begin{aligned} A \phi + \phi A = 0. \end{aligned}$$

• (iii) The real hypersurface M is a Hopf hypersurface, and the maximal complex subbundle ${\mathcal {C}}$ of TM is integrable.

Proof

Using the equation $\nabla _X\xi = \phi AX$, the exterior derivative $d\eta$ of $\eta$ is

$$\begin{aligned} d\eta (X,Y)&= d(\eta (Y))(X) - d(\eta (X))(Y) - \eta ([X,Y]) \\&= Xg(Y,\xi ) - Yg(X,\xi ) - g([X,Y],\xi ) \\&= g(\nabla _XY,\xi ) + g(Y,\nabla _X\xi ) - g(\nabla _YX,\xi ) - g(X,\nabla _Y\xi ) - g([X,Y],\xi ) \\&= g(Y,\phi A X) - g(X, \phi A Y) \\&= g((A\phi + \phi A)X,Y) \end{aligned}$$

for all $X,Y \in {\mathfrak {X}}(M)$. It follows that $\eta$ is closed if and only if $A\phi + \phi A = 0$, which shows that (i) and (ii) are equivalent.

The above calculations imply that for $X,Y \in \Gamma ({\mathcal {C}})$ we have

$$\begin{aligned} \eta ([X,Y]) = - d\eta (X,Y) = -g((A\phi + \phi A)X,Y). \end{aligned}$$

It follows that the distribution ${\mathcal {C}}$ is involutive if and only if $g((A\phi + \phi A)X,Y) = 0$ holds for all $X,Y \in \Gamma ({\mathcal {C}})$. We have $g((A\phi + \phi A)\xi ,Y) = g(\phi A\xi , Y) = 0$ for all $Y \in \Gamma ({\mathcal {C}})$ if and only if $A\xi \in \mathbb {R}\xi$, that is, if and only if M is a Hopf hypersurface. We always have $g((A\phi + \phi A)\xi ,\xi ) = 0$. Using Frobenius Theorem we can now conclude the equivalence of (ii) and (iii). $\square$

3 The complex hyperbolic quadric

The complex hyperbolic quadric is the Riemannian symmetric space

$$\begin{aligned} {Q^n}^* = SO^o_{2,n}/(SO_2\times SO_n),\ n \ge 1, \end{aligned}$$

where $SO^o_{2,n}$ denotes the identity component of the indefinite special orthogonal group $SO_{2,n}$ and $SO_2\times SO_n$ is embedded canonically into $SO^o_{2,n}$. The complex hyperbolic quadric $SO^o_{2,n}/(SO_2\times SO_n)$ is the non-compact dual symmetric space of the complex quadric $SO_{2+n}/(SO_2\times SO_n)$. We put $G = SO^o_{2,n}$, $K = SO_2\times SO_n$ and denote by $o \in {Q^n}^*$ the “base point” $I_{2+n}K$ of the homogeneous space G/K, where $I_{2+n} \in G$ is the identity $((2+n) \times (2+n))$-matrix. Then K is the isotropy group of G at o. We now describe the construction of the complex hyperbolic quadric as a Riemannian symmetric space in some more detail.

We denote by $M_{2,n}(\mathbb {R})$ the real vector space of $(2 \times n)$-matrices with real coefficients. Let

$$\begin{aligned} {\mathfrak {g}}= {\mathfrak {s}}{\mathfrak {o}}_{2,n} = \left\{ \begin{pmatrix} A_1 &{} B \\ B^\top &{} A_2 \end{pmatrix}: A_1 \in {\mathfrak {s}}{\mathfrak {o}}_2,\ A_2\in {\mathfrak {s}}{\mathfrak {o}}_n,\ B \in M_{2,n}(\mathbb {R}) \right\} \end{aligned}$$

be the Lie algebra of $G = SO^o_{2,n}$ and

$$\begin{aligned} {\mathfrak {k}}= {\mathfrak {s}}{\mathfrak {o}}_2 \oplus {\mathfrak {s}}{\mathfrak {o}}_n = \left\{ \begin{pmatrix} A_1 &{} 0_{2,n} \\ 0_{n,2} &{} A_2 \end{pmatrix}: A_1 \in {\mathfrak {s}}{\mathfrak {o}}_2,\ A_2\in {\mathfrak {s}}{\mathfrak {o}}_n \right\} \end{aligned}$$

be the Lie algebra of $K = SO_2\times SO_n$. Let

$$\begin{aligned} B: {\mathfrak {g}}\times {\mathfrak {g}}\rightarrow \mathbb {R},\,\ (X,Y) \mapsto \mathrm{{tr}}(\mathrm{{ad}}(X)\mathrm{{ad}}(Y)) = n\mathrm{{tr}}(XY) \end{aligned}$$

be the Killing form of ${\mathfrak {g}}$ and

$$\begin{aligned} {\mathfrak {p}}= \left\{ \begin{pmatrix} 0_{2,2} &{} B \\ B^\top &{} 0_{n,n} \end{pmatrix}: B \in M_{2,n}(\mathbb {R}) \right\} \end{aligned}$$

be the orthogonal complement of ${\mathfrak {k}}$ in ${\mathfrak {g}}$ with respect to B. The resulting decomposition ${\mathfrak {g}}= {\mathfrak {k}}\oplus {\mathfrak {p}}$ is a Cartan decomposition of ${\mathfrak {g}}$. We identify the tangent space $T_o{Q^n}^*$ of ${Q^n}^*$ at o with ${\mathfrak {p}}$ in the usual way.

The Cartan involution $\theta \in \mathrm{{Aut}}({\mathfrak {g}})$ on ${\mathfrak {g}}$ is given by

$$\begin{aligned} \theta (X) = I_{2,n} X I_{2,n} \text{ with } I_{2,n} = \begin{pmatrix} -I_2 &{} 0_{2,n} \\ 0_{n,2} &{} I_n \end{pmatrix}, \end{aligned}$$

where $I_2$ and $I_n$ are the identity $(2 \times 2)$-matrix and $(n \times n)$-matrix, respectively. Then

$$\begin{aligned} B_\theta : {\mathfrak {g}}\times {\mathfrak {g}}\rightarrow \mathbb {R},\,\ (X,Y) = -B(X,\theta (Y)) \end{aligned}$$

is a positive definite $\mathrm{{Ad}}(K)$-invariant inner product on ${\mathfrak {g}}$. The Cartan decomposition ${\mathfrak {g}}= {\mathfrak {k}}\oplus {\mathfrak {p}}$ is orthogonal with respect to $B_\theta$. The restriction of $B_\theta$ to ${\mathfrak {p}}\times {\mathfrak {p}}$ induces a G-invariant Riemannian metric $g_{B_\theta }$ on ${Q^n}^*$, which is often referred to as the standard homogeneous metric on ${Q^n}^*$. The complex hyperbolic quadric $({Q^n}^*,g_{B_\theta })$ is an Einstein manifold with Einstein constant $-\frac{1}{2}$ (see [18] and use duality between Riemannian symmetric spaces of compact type and of non-compact type). We renormalize the standard homogeneous metric $g_{B_\theta }$ so that the Einstein constant of the renormalized Riemannian metric g is equal to $-2n$, that is,

$$\begin{aligned} g_{B_\theta } = 4ng. \end{aligned}$$

This renormalization implies that the minimum of the sectional curvature of $({Q^n}^*,g)$ is equal to $-4$. Note that $({Q^1}^*,g)$ is isometric to the complex hyperbolic line $\mathbb {C}H^1(-4)$ and $({Q^2}^*,g)$ is isometric to the Riemannian product $\mathbb {C}H^1(-4) \times \mathbb {C}H^1(-4)$ of two complex hyperbolic lines. For $n \ge 3$, $({Q^n}^*,g)$ is an irreducible Riemannian symmetric space of non-compact type and rank 2. We assume $n \ge 3$ in the following.

The Lie algebra ${\mathfrak {k}}$ decomposes orthogonally into ${\mathfrak {k}}= {\mathfrak {s}}{\mathfrak {o}}_2 \oplus {\mathfrak {s}}{\mathfrak {o}}_n$. The first factor ${\mathfrak {s}}{\mathfrak {o}}_2$ is the 1-dimensional center of ${\mathfrak {k}}$. The adjoint action of

$$\begin{aligned} Z = \begin{pmatrix} 0 &{} -1 &{} 0 &{} \cdots &{} 0 \\ 1 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 1 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 1 \\ \end{pmatrix} \in SO_2 \subset SO_2\times SO_n = K \end{aligned}$$

on ${\mathfrak {p}}$ induces a Kähler structure J on ${Q^n}^*$. In this way $({Q^n}^*,g,J)$ becomes a Hermitian symmetric space.

We define

$$\begin{aligned} c_0 = \begin{pmatrix} 1 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} -1 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 1 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 1 \\ \end{pmatrix} \in O_2\times SO_n. \end{aligned}$$

Note that $c_0 \not \in K$, but $c_0$ is in the isotropy group at o of the full isometry group of $({Q^n}^*,g)$. The adjoint transformation $\mathrm{{Ad}}(c_0)$ leaves ${\mathfrak {p}}$ invariant and $C_0 = \mathrm{{Ad}}(c_0)|_{\mathfrak {p}}$ is an anti-linear involution on ${\mathfrak {p}}\cong T_o{Q^n}^*$ satisfying $C_0J + JC_0 = 0$. In other words, $C_0$ is a real structure on $T_o{Q^n}^*$. The involution $C_0$ commutes with $\mathrm{{Ad}}(g)$ for all $g \in SO_n \subset K$ but not for all $g \in K$. More precisely, for $g = (g_1,g_2) \in K$ with $g_1 \in SO_2$ and $g_2 \in SO_n$, say $g_1 = \left( {\begin{matrix} \cos (\varphi ) &{} -\sin (\varphi ) \\ \sin (\varphi ) &{} \cos (\varphi ) \end{matrix}} \right)$ with $\varphi \in \mathbb {R}$, so that $\mathrm{{Ad}}(g_1)$ corresponds to multiplication with the complex number $\mu = e^{i\varphi }$, we have

$$\begin{aligned} C_0 \circ \mathrm{{Ad}}(g) = \mu ^{-2} \mathrm{{Ad}}(g) \circ C_0. \end{aligned}$$

It follows that we have a circle of real structures

$$\begin{aligned} \{\cos (\varphi )C_0 + \sin (\varphi )JC_0: \varphi \in \mathbb {R}\}. \end{aligned}$$

This set is $\mathrm{{Ad}}(K)$-invariant and therefore generates an $\mathrm{{Ad}}(G)$-invariant $S^1$-subbundle ${\mathfrak {A}}_0$ of the endomorphism bundle $\mathrm{{End}}(T{Q^n}^*)$, consisting of real structures (or conjugations) on the tangent spaces of ${Q^n}^*$. This $S^1$-bundle naturally extends to an $\mathrm{{Ad}}(G)$-invariant vector subbundle ${\mathfrak {A}}$ of $\mathrm{{End}}(T{Q^n}^*)$ with $\mathrm{{rk}}({\mathfrak {A}}) = 2$, which is parallel with respect to the induced connection on $\mathrm{{End}}(T{Q^n}^*)$. For any real structure $C \in {\mathfrak {A}}_0$ the tangent line to the fiber of ${\mathfrak {A}}$ through C is spanned by JC. For every $p \in {Q^n}^*$ and real structure $C \in {\mathfrak {A}}_p$ we have an orthogonal decomposition

$$\begin{aligned} T_p{Q^n}^* = V(C) \oplus JV(C) \end{aligned}$$

into two totally real subspaces of $T_p{Q^n}^*$. Here V(C) and JV(C) are the $(+1)$- and $(-1)$-eigenspaces of C, respectively. By construction, we have

$$\begin{aligned} V(C_0) = \left\{ \begin{pmatrix} 0 &{} 0 &{} u_1 &{} \cdots &{} u_n \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ u_1 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ u_n &{} 0 &{} 0 &{} \cdots &{} 0 \end{pmatrix}: u \in \mathbb {R}^n \right\} \end{aligned}$$

and

$$\begin{aligned} JV(C_0) = \left\{ \begin{pmatrix} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} v_1 &{} \cdots &{} v_n \\ 0 &{} v_1 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} v_n &{} 0 &{} \cdots &{} 0 \end{pmatrix}: v \in \mathbb {R}^n \right\} . \end{aligned}$$

For

$$\begin{aligned} C = \cos (\varphi )C_0 + \sin (\varphi )JC_0 \end{aligned}$$

and $u \in V(C_0)$ we have

$$\begin{aligned}&C(\cos (\varphi /2)u + \sin (\varphi /2)Ju) \\&= \cos (\varphi /2)Cu + \sin (\varphi /2)CJu \\&= \cos (\varphi /2)Cu - \sin (\varphi /2)JCu \\&= \cos (\varphi /2)(\cos (\varphi )C_0 + \sin (\varphi )JC_0)u - \sin (\varphi /2)J(\cos (\varphi )C_0 + \sin (\varphi )JC_0)u \\&= (\cos (\varphi /2)\cos (\varphi ) + \sin (\varphi /2)\sin (\varphi ))u + (\cos (\varphi /2)\sin (\varphi ) - \sin (\varphi /2)\cos (\varphi ))Ju \\&= \cos (\varphi /2)u + \sin (\varphi /2)Ju. \end{aligned}$$

It follows that

$$\begin{aligned} V(C) = \{ \cos (\varphi /2)u + \sin (\varphi /2)Ju: u \in V(C_0) \}. \end{aligned}$$

Geometrically this tells us that, if we rotate a real structure by angle $\varphi$, then the $\pm 1$-eigenspaces rotate by angle $\varphi /2$.

The Riemannian metric g, the Kähler structure J and a real structure C on ${Q^n}^*$ can be used to give an explicit expression of the Riemannian curvature tensor $\bar{R}$ of $({Q^n}^*,g)$ (see [15] and use duality). More precisely, we have

$$\begin{aligned} \bar{R}(X,Y)Z&= g(X,Z)Y - g(Y,Z)X + g(JX,Z)JY - g(JY,Z)JX + 2g(JX,Y)JZ \\&\qquad + g(CX,Z)CY - g(CY,Z)CX + g(JCX,Z)JCY - g(JCY,Z)JCX \end{aligned}$$

for all $X,Y,Z \in {\mathfrak {X}}({Q^n}^*)$, where C is an arbitrary real structure in ${\mathfrak {A}}_0$.

For every non-zero tangent vector $v \in {\mathfrak {p}}\cong T_o{Q^n}^*$ there exists a maximal abelian subspace ${\mathfrak {a}}\subset {\mathfrak {p}}$ with $v \in {\mathfrak {a}}$. If ${\mathfrak {a}}$ is unique, then v is said to be a regular tangent vector, otherwise v is said to be a singular tangent vector. From the explicit expression of the Riemannian curvature tensor it is straightforward to find the singular tangent vectors of ${Q^n}^*$. There are exactly two types of singular tangent vectors $v \in T_o{Q^n}^*$, which can be characterized as follows:

• (i) If there exists a real structure $C \in {\mathfrak {A}}_0$ such that $v \in V(C)$, then v is singular. Such a singular tangent vector is called ${\mathfrak {A}}$-principal.

• (ii) If there exist a real structure $C \in {\mathfrak {A}}_0$ and orthonormal vectors $u,w \in V(C)$ such that $\frac{v}{||v||} = \frac{1}{\sqrt{2}}(u+Jw)$, then v is singular. Such a singular tangent vector is called ${\mathfrak {A}}$-isotropic.

For every unit tangent vector $v \in T_o{Q^n}^*$ there exist a real structure $C \in {\mathfrak {A}}_0$ and orthonormal vectors $u,w \in V(C)$ such that

$$\begin{aligned} v = \cos (t)u + \sin (t)Jw \end{aligned}$$

for some $t \in [0,\frac{\pi }{4}]$. The singular tangent vectors correspond to the boundary values $t = 0$ and $t = \frac{\pi }{4}$.

Let v be a unit tangent vector of ${Q^n}^*$ and consider the Jacobi operator $\bar{R}_v$ defined by

$$\begin{aligned} \bar{R}_v X = \bar{R}(X,v)v. \end{aligned}$$

We have

$$\begin{aligned} \bar{R}_v X = -X + g(X,v)v - 3g(X,Jv)Jv + g(X,Cv)Cv - g(Cv,v)CX + g(X,JCv)JCv. \end{aligned}$$

By a straightforward computation we obtain the eigenvalues and eigenspaces of $\bar{R}_v$ (see also [15]). The eigenvalues are

$$\begin{aligned} 0,-1+\cos (2t),-1-\cos (2t),-2+2\sin (2t),-2-2\sin (2t) \end{aligned}$$

with corresponding eigenspaces

$$\begin{aligned} E_0&= \mathbb {R}u \oplus \mathbb {R}w \cong \mathbb {R}^2, \\ E_{-1+\cos (2t)}&= V(C) \ominus (\mathbb {R}u \oplus \mathbb {R}w) \cong \mathbb {R}^{n-2}, \\ E_{-1-\cos (2t)}&= JV(C) \ominus J(\mathbb {R}u \oplus \mathbb {R}w) \cong \mathbb {R}^{n-2} ,\\ E_{-2+2\sin (2t)}&= \mathbb {R}(Ju+w) \cong \mathbb {R}, \\ E_{-2-2\sin (2t)}&= \mathbb {R}(Ju-w) \cong \mathbb {R}, \end{aligned}$$

where C is a suitable real structure, and $u,w \in V(C)$ are orthonormal vectors such that

$$\begin{aligned} v = \cos (t)u + \sin (t)Jw \end{aligned}$$

for some $t \in [0,\frac{\pi }{4}]$. The five eigenvalues are distinct unless $t \in \{0,\tan ^{-1}(\frac{1}{2}),\frac{\pi }{4}\}$.

If $t = 0$, then $Cv = v$ and hence v is ${\mathfrak {A}}$-principal. In this case $\bar{R}_v$ has two eigenvalues $0,-2$ with corresponding eigenspaces

$$\begin{aligned} E_0&= \mathbb {R}v \oplus J(V(C)\ominus \mathbb {R}v) \cong \mathbb {R}^n,\\ E_{-2}&= \mathbb {R}Jv \oplus (V(C)\ominus \mathbb {R}v) \cong \mathbb {R}^n. \end{aligned}$$

If $t = \frac{\pi }{4}$, then $v = \frac{1}{\sqrt{2}}(u+Jw)$ and hence v is ${\mathfrak {A}}$-isotropic. In this case $\bar{R}_v$ has three eigenvalues $0,-1,-4$ with corresponding eigenspaces

$$\begin{aligned} E_0&= \mathbb {R}v \oplus \mathbb {R}Cv \oplus \mathbb {R}JCv = \mathbb {R}v \oplus \mathbb {C}Cv \cong \mathbb {R}\oplus \mathbb {C},\\ E_{-1}&= {\mathfrak {p}}\ominus (\mathbb {C}v \oplus \mathbb {C}Cv) \cong \mathbb {C}^{n-2}, \\ E_{-4}&= \mathbb {R}Jv \cong \mathbb {R}. \end{aligned}$$

If $t = \tan ^{-1}(\frac{1}{2})$, then $\cos (t) = \frac{2}{\sqrt{5}}$, $\sin (t) = \frac{1}{\sqrt{5}}$, and hence $\cos (2t) = \frac{3}{5}$ and $\sin (2t) = \frac{4}{5}$. In this case $\bar{R}_v$ has four eigenvalues $0,-\frac{2}{5},-\frac{8}{5},-\frac{18}{5}$.

Let ${\mathfrak {a}}$ be a maximal abelian subspace of ${\mathfrak {p}}$ and ${\mathfrak {a}}^*$ be the dual vector space of ${\mathfrak {a}}$. For each $\alpha \in {\mathfrak {a}}^*$ we define

$$\begin{aligned} {\mathfrak {g}}_\alpha = \{ X \in {\mathfrak {g}}: \mathrm{{ad}}(H)X = \alpha (H)X \text{ for } \text{ all } H \in {\mathfrak {a}}\}. \end{aligned}$$

If $\alpha \ne 0$ and ${\mathfrak {g}}_\alpha \ne \{0\}$, then $\alpha$ is a restricted root and ${\mathfrak {g}}_\alpha$ is a restricted root space. Let $\Sigma \subset {\mathfrak {a}}^*$ be the set of restricted roots. The restricted root spaces provide a restricted root space decomposition

$$\begin{aligned} {\mathfrak {g}}= {\mathfrak {g}}_0 \oplus \left( \bigoplus _{\alpha \in \Sigma } {\mathfrak {g}}_\alpha \right) \end{aligned}$$

of ${\mathfrak {g}}$, where ${\mathfrak {g}}_0 = {\mathfrak {k}}_0 \oplus {\mathfrak {a}}$ and ${\mathfrak {k}}_0 \cong {\mathfrak {s}}{\mathfrak {o}}_{n-2}$ is the centralizer of ${\mathfrak {a}}$ in ${\mathfrak {k}}$. The restricted root spaces ${\mathfrak {g}}_\alpha$ and ${\mathfrak {g}}_0$ are pairwise orthogonal with respect to $B_\theta$. The corresponding restricted root system is of type $B_2$. We choose a set $\Lambda = \{\alpha _1,\alpha _2\}$ of simple roots of $\Sigma$ such that $\alpha _1$ is the longer root of the two simple roots and denote by $\Sigma ^+$ the resulting set of positive restricted roots. If we write, as usual, $\alpha _1 = \epsilon _1 - \epsilon _2$ and $\alpha _2 = \epsilon _2$, the positive restricted roots are

$$\begin{aligned} \alpha _1 = \epsilon _1 - \epsilon _2,\ \alpha _2 = \epsilon _2,\ \alpha _1 + \alpha _2 = \epsilon _1,\ \alpha _1 + 2\alpha _2 = \epsilon _1 + \epsilon _2. \end{aligned}$$

The multiplicities of the two long roots $\alpha _1$ and $\alpha _1 + 2\alpha _2$ are equal to 1, and the multiplicities of the two short roots $\alpha _2$ and $\alpha _1 + \alpha _2$ are equal to $n-2$, respectively. Explicitly, the positive restricted root spaces and ${\mathfrak {g}}_0$ are:

$$\begin{aligned} {\mathfrak {g}}_0= & {} \left\{ \begin{pmatrix} 0 &{} 0 &{} a_1 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} a_2 &{} 0 &{} \cdots &{} 0 \\ a_1 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} a_2 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} &{} &{} \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} &{} B &{} \\ 0 &{} 0 &{} 0 &{} 0 &{} &{} &{} \end{pmatrix}: a_1,a_2 \in \mathbb {R},\ B \in {\mathfrak {s}}{\mathfrak {o}}_{n-2} \right\} \cong \mathbb {R}^2 \oplus {\mathfrak {s}}{\mathfrak {o}}_{n-2},\\ {\mathfrak {g}}_{\alpha _1+\alpha _2}= & {} \left\{ \begin{pmatrix} 0 &{} 0 &{} 0 &{} 0 &{} v_1 &{} \cdots &{} v_{n-2} \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} v_1 &{} \cdots &{} v_{n-2} \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ v_1 &{} 0 &{} -v_1 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{n-2} &{} 0 &{} -v_{n-2} &{} 0 &{} 0 &{} \cdots &{} 0 \end{pmatrix}: v \in \mathbb {R}^{n-2} \right\} \cong \mathbb {R}^{n-2}, \\ {\mathfrak {g}}_{\alpha _2}= & {} \left\{ \begin{pmatrix} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} w_1 &{} \cdots &{} w_{n-2} \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} w_1 &{} \cdots &{} w_{n-2} \\ 0 &{} w_1 &{} 0 &{} -w_1 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} w_{n-2} &{} 0 &{} -w_{n-2} &{} 0 &{} \cdots &{} 0 \end{pmatrix}: w \in \mathbb {R}^{n-2} \right\} \cong \mathbb {R}^{n-2}, \\ {\mathfrak {g}}_{\alpha _1}= & {} \left\{ \begin{pmatrix} 0 &{} x &{} 0 &{} x &{} 0 &{} \cdots &{} 0 \\ -x &{} 0 &{} x &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} x &{} 0 &{} x &{} 0 &{} \cdots &{} 0 \\ x &{} 0 &{} -x &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \end{pmatrix}: x \in \mathbb {R}\right\} \cong \mathbb {R},\\ {\mathfrak {g}}_{\alpha _1+2\alpha _2}= & {} \left\{ \begin{pmatrix} 0 &{} y &{} 0 &{} -y &{} 0 &{} \cdots &{} 0 \\ -y &{} 0 &{} y &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} y &{} 0 &{} -y &{} 0 &{} \cdots &{} 0 \\ -y &{} 0 &{} y &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \end{pmatrix}: y \in \mathbb {R}\right\} \cong \mathbb {R}. \end{aligned}$$

The negative restricted root spaces can be computed easily from the positive restricted root spaces using the fact that ${\mathfrak {g}}_{-\alpha } = \theta ({\mathfrak {g}}_\alpha )$.

For each $\alpha \in \Sigma$ we define

$$\begin{aligned} {\mathfrak {k}}_\alpha = {\mathfrak {k}}\cap ({\mathfrak {g}}_\alpha \oplus {\mathfrak {g}}_{-\alpha }),\ {\mathfrak {p}}_\alpha = {\mathfrak {p}}\cap ({\mathfrak {g}}_\alpha \oplus {\mathfrak {g}}_{-\alpha }). \end{aligned}$$

Then we have ${\mathfrak {p}}_\alpha = {\mathfrak {p}}_{-\alpha }$, ${\mathfrak {k}}_\alpha = {\mathfrak {k}}_{-\alpha }$ and ${\mathfrak {p}}_\alpha \oplus {\mathfrak {k}}_\alpha = {\mathfrak {g}}_\alpha \oplus {\mathfrak {g}}_{-\alpha }$ for all $\alpha \in \Sigma$.

We define a nilpotent subalgebra ${\mathfrak {n}}$ of ${\mathfrak {g}}$ by

$$\begin{aligned} {\mathfrak {n}}&= {\mathfrak {g}}_{\alpha _1} \oplus {\mathfrak {g}}_{\alpha _2} \oplus {\mathfrak {g}}_{\alpha _1 + \alpha _2} \oplus {\mathfrak {g}}_{\alpha _1 + 2\alpha _2} \\&= \left\{ \begin{pmatrix} 0 &{} x+y &{} 0 &{} x-y &{} v_1 &{} \cdots &{} v_{n-2} \\ -x-y &{} 0 &{} x+y &{} 0 &{} w_1 &{} \cdots &{} w_{n-2} \\ 0 &{} x+y &{} 0 &{} x-y &{} v_1 &{} \cdots &{} v_{n-2} \\ x-y &{} 0 &{} -x+y &{} 0 &{} w_1 &{} \cdots &{} w_{n-2} \\ v_1 &{} w_1 &{} -v_1 &{} -w_1 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{n-2} &{} w_{n-2} &{} -v_{n-2} &{} -w_{n-2} &{} 0 &{} \cdots &{} 0 \end{pmatrix} : \begin{array}{l} x,y \in \mathbb {R},\\ v,w \in \mathbb {R}^{n-2} \end{array} \right\} . \end{aligned}$$

Then ${\mathfrak {g}}= {\mathfrak {k}}\oplus {\mathfrak {a}}\oplus {\mathfrak {n}}$ is an Iwasawa decomposition of ${\mathfrak {g}}$, which induces a corresponding Iwasawa decomposition $G = KAN$ of G. Here, A and N are the connected closed subgroups of G with Lie algebras ${\mathfrak {a}}$ and ${\mathfrak {n}}$, respectively.

The subalgebra

$$\begin{aligned} {\mathfrak {a}}\oplus {\mathfrak {n}}= \left\{ \begin{pmatrix} 0 &{} x+y &{} a_1 &{} x-y &{} v_1 &{} \cdots &{} v_{n-2} \\ -x-y &{} 0 &{} x+y &{} a_2 &{} w_1 &{} \cdots &{} w_{n-2} \\ a_1 &{} x+y &{} 0 &{} x-y &{} v_1 &{} \cdots &{} v_{n-2} \\ x-y &{} a_2 &{} -x+y &{} 0 &{} w_1 &{} \cdots &{} w_{n-2} \\ v_1 &{} w_1 &{} -v_1 &{} -w_1 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{n-2} &{} w_{n-2} &{} -v_{n-2} &{} -w_{n-2} &{} 0 &{} \cdots &{} 0 \end{pmatrix}: \begin{array}{l} a_1,a_2,x,y \in \mathbb {R},\\ v,w \in \mathbb {R}^{n-2} \end{array} \right\} \end{aligned}$$

of ${\mathfrak {g}}$ is solvable and the corresponding connected closed subgroup AN of G with Lie algebra ${\mathfrak {a}}\oplus {\mathfrak {n}}$ is solvable, simply connected and acts simply transitively on ${Q^n}^*$. Then $({Q^n}^*,g)$ is isometric to the solvable Lie group AN equipped with the left-invariant Riemannian metric $\langle \cdot , \cdot \rangle$ defined by

$$\begin{aligned} \langle H_1 + \hat{X}_1 , H_2 + \hat{X}_2 \rangle&= -\frac{1}{4n}B(H_1,\theta (H_2)) - \frac{1}{8n}B(\hat{X}_1,\theta (\hat{X}_2)) \\&= -\frac{1}{4}\mathrm{{tr}}(H_1\theta (H_2)) - \frac{1}{8}\mathrm{{tr}}(\hat{X}_1\theta (\hat{X}_2)) \\&= \frac{1}{4}\mathrm{{tr}}(H_1H_2) - \frac{1}{8}\mathrm{{tr}}(\hat{X}_1\theta (\hat{X}_2)) \end{aligned}$$

with $H_1,H_2 \in {\mathfrak {a}}$ and $\hat{X}_1,\hat{X}_2 \in {\mathfrak {n}}$. For each $\hat{X} \in {\mathfrak {n}}$, the orthogonal projection X onto ${\mathfrak {p}}$ with respect to $B_\theta$ is

$$\begin{aligned} X = \frac{1}{2}(\hat{X} - \theta (\hat{X})) \in {\mathfrak {p}}. \end{aligned}$$

By construction, we have $\langle \hat{X}, \hat{X} \rangle = g(X,X)$ and

$$\begin{aligned} \langle H_1 + \hat{X}_1, H_2 + \hat{X}_2 \rangle = g(H_1 + X_1,H_2 + X_2). \end{aligned}$$

Let $H^1,H^2 \in {\mathfrak {a}}$ be the dual basis of $\alpha _1,\alpha _2 \in {\mathfrak {a}}^*$ defined by $\alpha _\nu (H^\mu ) = \delta _{\nu \mu }$. Since $\alpha _1 = \epsilon _1 - \epsilon _2$ and $\alpha _2 = \epsilon _2$, we have

$$\begin{aligned} H^1 = \begin{pmatrix} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \end{pmatrix},\ H^2 = \begin{pmatrix} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} \cdots &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \end{pmatrix}. \end{aligned}$$

Note that

$$\begin{aligned} \langle H^1, H^1 \rangle = \frac{1}{4}\mathrm{{tr}}(H^1H^1) = \frac{1}{2},\ \langle H^2, H^2 \rangle = \frac{1}{4}\mathrm{{tr}}(H^2H^2) = 1. \end{aligned}$$

For each $\alpha$ in $\Sigma$ we define the root vector $H_\alpha \in {\mathfrak {a}}$ of $\alpha$ by $\langle H_\alpha , H \rangle = \alpha (H)$ for all $H \in {\mathfrak {a}}$. Note that

$$\begin{aligned}{}[H,X_\alpha ] = \mathrm{{ad}}(H)X_\alpha = \alpha (H)X_\alpha = \langle H_\alpha , H \rangle X_\alpha \end{aligned}$$

for all $H \in {\mathfrak {a}}$ and $X_\alpha \in {\mathfrak {g}}_\alpha$. If we put

$$\begin{aligned} H_{\alpha } = \begin{pmatrix} 0 &{} 0 &{} x_1 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} x_2 &{} 0 &{} \cdots &{} 0 \\ x_1 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} x_2 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \end{pmatrix},\ H = \begin{pmatrix} 0 &{} 0 &{} a_1 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} a_2 &{} 0 &{} \cdots &{} 0 \\ a_1 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} a_2 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \end{pmatrix}, \end{aligned}$$

then

$$\begin{aligned} \langle H_\alpha , H \rangle = \frac{1}{4}\mathrm{{tr}}(H_\alpha H) = \frac{1}{2}(x_1a_1 + x_2a_2). \end{aligned}$$

It follows that

$$\begin{aligned} \begin{matrix} H_{\alpha _1} &{} = &{} \begin{pmatrix} 0 &{} 0 &{} 2 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} -2 &{} 0 &{} \cdots &{} 0 \\ 2 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} -2 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \end{pmatrix} &{}, &{} H_{\alpha _1+2\alpha _2} &{} = &{} \begin{pmatrix} 0 &{} 0 &{} 2 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 2 &{} 0 &{} \cdots &{} 0 \\ 2 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 2 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \end{pmatrix} &{},\\ &{} &{} &{} &{} &{} &{} &{} \\ H_{\alpha _2} &{} = &{} \begin{pmatrix} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 2 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 2 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \end{pmatrix} &{}, &{} H_{\alpha _1+\alpha _2} &{} = &{} \begin{pmatrix} 0 &{} 0 &{} 2 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 2 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \end{pmatrix}&. \end{matrix} \end{aligned}$$

We have

$$\begin{aligned} \langle H_{\alpha _1} , H_{\alpha _1} \rangle&= \frac{1}{4}\mathrm{{tr}}(H_{\alpha _1}H_{\alpha _1}) = 4,\\ \langle H_{\alpha _1+2\alpha _2} , H_{\alpha _1+2\alpha _2} \rangle&= \frac{1}{4}\mathrm{{tr}}(H_{\alpha _1+2\alpha _2}H_{\alpha _1+2\alpha _2}) = 4,\\ \langle H_{\alpha _2} , H_{\alpha _2} \rangle&= \frac{1}{4}\mathrm{{tr}}(H_{\alpha _2}H_{\alpha _2}) = 2,\\ \langle H_{\alpha _1+\alpha _2} , H_{\alpha _1+\alpha _2} \rangle&= \frac{1}{4}\mathrm{{tr}}(H_{\alpha _1+\alpha _2} H_{\alpha _1+\alpha _2} ) = 2, \end{aligned}$$

and

$$\begin{aligned} 2H^1 = H_{\alpha _1+\alpha _2} \qquad \text{ and } \qquad 2H^2 = H_{\alpha _1+2\alpha _2}. \end{aligned}$$

4 The homogeneous complex hypersurface

In this section we construct a homogeneous complex hypersurface $P^{n-1} \cong \mathbb {C}H^{n-1}(-4)$ in $({Q^n}^*,g)$ and compute its shape operator. We define

$$\begin{aligned} {\mathfrak {h}}^{2n-3} = {\mathfrak {g}}_{\alpha _2} \oplus {\mathfrak {g}}_{\alpha _1+\alpha _2} \oplus {\mathfrak {g}}_{\alpha _1+2\alpha _2}. \end{aligned}$$

It is easy to verify that ${\mathfrak {h}}^{2n-3}$ is a nilpotent subalgebra of ${\mathfrak {n}}$ and isomorphic to the $(2n-3)$-dimensional Heisenberg algebra with 1-dimensional center.

We have

$$\begin{aligned}{}[H^2,\hat{X}] = {\left\{ \begin{array}{ll} \hat{X} &{}, \text{ if } \hat{X} \in {\mathfrak {g}}_{\alpha _2} \oplus {\mathfrak {g}}_{\alpha _1+\alpha _2}, \\ 2\hat{X} &{}, \text{ if } \hat{X} \in {\mathfrak {g}}_{\alpha _1+2\alpha _2}. \end{array}\right. } \end{aligned}$$

It follows that

$$\begin{aligned} {\mathfrak {d}}= \mathbb {R}H^2 \oplus {\mathfrak {h}}^{2n-3} = \mathbb {R}H_{\alpha _1+2\alpha _2} \oplus {\mathfrak {g}}_{\alpha _2} \oplus {\mathfrak {g}}_{\alpha _1+\alpha _2} \oplus {\mathfrak {g}}_{\alpha _1+2\alpha _2} \end{aligned}$$

is a solvable subalgebra of ${\mathfrak {a}}\oplus {\mathfrak {n}}$. (Note that $\mathbb {R}H^2$ denotes here the real span of $H^2$ and not the real hyperbolic plane!) In fact, this subalgebra is the standard solvable extension of the Heisenberg algebra ${\mathfrak {h}}^{2n-3}$ and isomorphic to the solvable Lie algebra of the solvable part of the Iwasawa decomposition of the isometry group of the complex hyperbolic space $\mathbb {C}H^{n-1}(-4)$ (see, e.g., [5] or [17]).

This construction leads to an isometric embedding $\hat{P}^{n-1}$ of the $(n-1)$-dimensional complex hyperbolic space $\mathbb {C}H^{n-1}(-4)$ with constant holomorphic sectional curvature $-4$ into $(AN,\langle \cdot , \cdot \rangle )$. By construction, $\hat{P}^{n-1}$ is a homogeneous submanifold of $(AN,\langle \cdot , \cdot \rangle )$. Let $\hat{J}$ be the complex structure on $(AN,\langle \cdot , \cdot \rangle )$ corresponding to the complex structure J on $({Q^n}^*,g)$. We have $\hat{J}{\mathfrak {g}}_{\alpha _2} = {\mathfrak {g}}_{\alpha _1+\alpha _2}$ and $\hat{J}H^2 \in {\mathfrak {g}}_{\alpha _1+2\alpha _2}$, which shows that the tangent space

$$\begin{aligned} T_o\hat{P}^{n-1} = \mathbb {R}H_{\alpha _1+2\alpha _2} \oplus {\mathfrak {g}}_{\alpha _2} \oplus {\mathfrak {g}}_{\alpha _1+\alpha _2} \oplus {\mathfrak {g}}_{\alpha _1+2\alpha _2} \end{aligned}$$

is a complex subspace of $T_oAN$. Since AN is contained in the identity component $SO^o_{2,n}$ of the full isometry group of ${Q^n}^*$, it consists of holomorphic isometries, which implies that $\hat{P}^{n-1}$ is a complex submanifold of $(AN,\langle \cdot , \cdot \rangle )$.

Altogether we conclude that the solvable subalgebra

$$\begin{aligned} {\mathfrak {d}}= \mathbb {R}H_{\alpha _1+2\alpha _2} \oplus {\mathfrak {g}}_{\alpha _2} \oplus {\mathfrak {g}}_{\alpha _1+\alpha _2} \oplus {\mathfrak {g}}_{\alpha _1+2\alpha _2} \end{aligned}$$

of ${\mathfrak {a}}\oplus {\mathfrak {n}}$ induces an isometric embedding $\hat{P}^{n-1}$ of the complex hyperbolic space $\mathbb {C}H^{n-1}(-4)$ with constant holomorphic sectional curvature $-4$ into $(AN,\langle \cdot , \cdot \rangle )$ as a homogeneous complex hypersurface. This induces an isometric embedding $P^{n-1}$ of the complex hyperbolic space $\mathbb {C}H^{n-1}(-4)$ with constant holomorphic sectional curvature $-4$ into $({Q^n}^*,g)$ as a homogeneous complex hypersurface.

Remark 4.1

Smyth [16] proved that every homogeneous complex hypersurface in the complex hyperbolic space $\mathbb {C}H^n$ is a complex hyperbolic hyperplane $\mathbb {C}H^{n-1}$ embedded in $\mathbb {C}H^n$ as a totally geodesic submanifold. As we have just seen, up to congruency, there are at least two homogeneous complex hypersurfaces in the complex hyperbolic quadric ${Q^n}^*$, namely the complex hyperbolic quadric ${Q^{n-1}}^*$ and the complex hyperbolic space $P^{n-1} \cong \mathbb {C}H^{n-1}(-4)$. The first one is totally geodesic (see [8, 11] and use duality between Riemannian symmetric spaces of compact type and of non-compact type), the second one is not. The classification of the homogeneous complex hypersurfaces in the complex hyperbolic quadric ${Q^n}^*$ remains an open problem.

We now compute the shape operator $\hat{A}$ of $\hat{P}^{n-1} \cong \mathbb {C}H^{n-1}(-4)$ in $(AN,\langle \cdot , \cdot \rangle )$. Let

$$\begin{aligned} \hat{\zeta }\in ({\mathfrak {a}}\ominus \mathbb {R}H^2) \oplus {\mathfrak {g}}_{\alpha _1} = \mathbb {R}H_{\alpha _1} \oplus {\mathfrak {g}}_{\alpha _1} \end{aligned}$$

be a unit normal vector of $\hat{P}^{n-1}$ at o. The Weingarten equation tells us that

$$\begin{aligned} \langle \hat{A}_{\hat{\zeta }} \hat{X}, \hat{Y} \rangle = - \langle \hat{\nabla }_{\hat{X}}\hat{\zeta }, \hat{Y} \rangle , \end{aligned}$$

where $\hat{\nabla }$ is the Levi Civita covariant derivative of $(AN,\langle \cdot , \cdot \rangle )$ and $\hat{X},\hat{Y} \in {\mathfrak {d}}$. We consider $\hat{\zeta },\hat{X},\hat{Y}$ as left-invariant vector fields. Since $\langle \cdot , \cdot \rangle$ is a left-invariant Riemannian metric, the Koszul formula for $\hat{\nabla }$ implies

$$\begin{aligned} 2\langle \hat{A}_{\hat{\zeta }}\hat{X}, \hat{Y} \rangle = 2\langle \hat{\nabla }_{\hat{X}} \hat{Y}, \hat{\zeta } \rangle = \langle [\hat{X},\hat{Y}],\hat{\zeta } \rangle + \langle [\hat{\zeta },\hat{X}],\hat{Y} \rangle + \langle [\hat{\zeta },\hat{Y}],\hat{X} \rangle . \end{aligned}$$

Since ${\mathfrak {d}}$ is a subalgebra of ${\mathfrak {a}}\oplus {\mathfrak {n}}$, we have $[\hat{X},\hat{Y}] \in {\mathfrak {d}}$ and hence $\langle [\hat{X},\hat{Y}],\hat{\zeta } \rangle = 0$. Moreover, since $\mathrm{{ad}}(\hat{\zeta })^* = -\mathrm{{ad}}(\theta (\hat{\zeta }))$, we have

$$\begin{aligned} \langle [\hat{\zeta },\hat{Y}],\hat{X} \rangle = -\langle [\theta (\hat{\zeta }),\hat{X}],\hat{Y} \rangle . \end{aligned}$$

Altogether this implies

$$\begin{aligned} 2\langle \hat{A}_{\hat{\zeta }} \hat{X}, \hat{Y} \rangle = \langle [\hat{\zeta }-\theta (\hat{\zeta }),\hat{X}],\hat{Y} \rangle . \end{aligned}$$

Thus, the shape operator $\hat{A}_{\hat{\zeta }}$ of $\hat{P}^{n-1}$ is given by

$$\begin{aligned} \hat{A}_{\hat{\zeta }} \hat{X} = [\zeta ,\hat{X}]_{{\mathfrak {d}}}, \end{aligned}$$

where

$$\begin{aligned} \zeta = \frac{1}{2}(\hat{\zeta }- \theta (\hat{\zeta })) \in {\mathfrak {p}}\end{aligned}$$

is the orthogonal projection of $\hat{\zeta }$ onto ${\mathfrak {p}}$ and $[\, \cdot \ ]_{\mathfrak {d}}$ is the orthogonal projection onto ${\mathfrak {d}}$.

The normal space $\nu _o\hat{P}^{n-1}$ of $\hat{P}^{n-1}$ at the point o is given by

$$\begin{aligned} \nu _o\hat{P}^{n-1} = \mathbb {R}H_{\alpha _1} \oplus {\mathfrak {g}}_{\alpha _1} = \left\{ \begin{pmatrix} 0 &{} x &{} a &{} x &{} 0 &{} \cdots &{} 0 \\ -x &{} 0 &{} x &{} -a &{} 0 &{} \cdots &{} 0 \\ a &{} x &{} 0 &{} x &{} 0 &{} \cdots &{} 0 \\ x &{} -a &{} -x &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \end{pmatrix}: a,x \in \mathbb {R}\right\} , \end{aligned}$$

and the tangent space $T_o\hat{P}^{n-1}$ of $\hat{P}^{n-1}$ at the point o is given by

$$\begin{aligned} T_o\hat{P}^{n-1} = {\mathfrak {d}}= \left\{ \begin{pmatrix} 0 &{} y &{} b &{} -y &{} v_1 &{} \cdots &{} v_{n-2} \\ -y &{} 0 &{} y &{} b &{} w_1 &{} \cdots &{} w_{n-2} \\ b &{} y &{} 0 &{} -y &{} v_1 &{} \cdots &{} v_{n-2} \\ -y &{} b &{} y &{} 0 &{} w_1 &{} \cdots &{} w_{n-2} \\ v_1 &{} w_1 &{} -v_1 &{} -w_1 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{n-2} &{} w_{n-2} &{} -v_{n-2} &{} -w_{n-2} &{} 0 &{} \cdots &{} 0 \end{pmatrix}: \begin{array}{l} b,y \in \mathbb {R},\\ v,w \in \mathbb {R}^{n-2} \end{array} \right\} . \end{aligned}$$

The vector $\hat{\zeta }= \frac{1}{2}H_{\alpha _1} \in {\mathfrak {a}}$ is a unit normal vector of $\hat{P}^{n-1}$ at o. We have

$$\begin{aligned} \theta (\hat{\zeta }) = \frac{1}{2}\theta (H_{\alpha _1}) = -\frac{1}{2}H_{\alpha _1} = - \hat{\zeta }\end{aligned}$$

and thus

$$\begin{aligned} \zeta = \frac{1}{2}(\hat{\zeta }- \theta (\hat{\zeta })) = \hat{\zeta }. \end{aligned}$$

A straightforward matrix computation gives

$$\begin{aligned}&\left[ \left( {\begin{matrix} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} -1 &{} 0 &{} \cdots &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} -1 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \end{matrix}} \right) , \left( {\begin{matrix} 0 &{} y &{} b &{} -y &{} v_1 &{} \cdots &{} v_{n-2} \\ -y &{} 0 &{} y &{} b &{} w_1 &{} \cdots &{} w_{n-2} \\ b &{} y &{} 0 &{} -y &{} v_1 &{} \cdots &{} v_{n-2} \\ -y &{} b &{} y &{} 0 &{} w_1 &{} \cdots &{} w_{n-2} \\ v_1 &{} w_1 &{} -v_1 &{} -w_1 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{n-2} &{} w_{n-2} &{} -v_{n-2} &{} -w_{n-2} &{} 0 &{} \cdots &{} 0 \end{matrix}} \right) \right] \\&= \left( {\begin{matrix} 0 &{} 0 &{} 0 &{} 0 &{} v_1 &{} \cdots &{} v_{n-2} \\ 0 &{} 0 &{} 0 &{} 0 &{} -w_1 &{} \cdots &{} -w_{n-2} \\ 0 &{} 0 &{} 0 &{} 0 &{} v_1 &{} \cdots &{} v_{n-2} \\ 0 &{} 0 &{} 0 &{} 0 &{} -w_1 &{} \cdots &{} -w_{n-2} \\ v_1 &{} -w_1 &{} -v_1 &{} w_1 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{n-2} &{} -w_{n-2} &{} -v_{n-2} &{} w_{n-2} &{} 0 &{} \cdots &{} 0 \end{matrix}} \right) \in {\mathfrak {d}}. \end{aligned}$$

Since the latter matrix is in ${\mathfrak {d}}$, we conclude that

$$\begin{aligned} \hat{A}_{\hat{\zeta }} \hat{X} = \begin{pmatrix} 0 &{} 0 &{} 0 &{} 0 &{} v_1 &{} \cdots &{} v_{n-2} \\ 0 &{} 0 &{} 0 &{} 0 &{} -w_1 &{} \cdots &{} -w_{n-2} \\ 0 &{} 0 &{} 0 &{} 0 &{} v_1 &{} \cdots &{} v_{n-2} \\ 0 &{} 0 &{} 0 &{} 0 &{} -w_1 &{} \cdots &{} -w_{n-2} \\ v_1 &{} -w_1 &{} -v_1 &{} w_1 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{n-2} &{} -w_{n-2} &{} -v_{n-2} &{} w_{n-2} &{} 0 &{} \cdots &{} 0 \end{pmatrix} \end{aligned}$$

with

$$\begin{aligned} \hat{X} = \begin{pmatrix} 0 &{} y &{} b &{} -y &{} v_1 &{} \cdots &{} v_{n-2} \\ -y &{} 0 &{} y &{} b &{} w_1 &{} \cdots &{} w_{n-2} \\ b &{} y &{} 0 &{} -y &{} v_1 &{} \cdots &{} v_{n-2} \\ -y &{} b &{} y &{} 0 &{} w_1 &{} \cdots &{} w_{n-2} \\ v_1 &{} w_1 &{} -v_1 &{} -w_1 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{n-2} &{} w_{n-2} &{} -v_{n-2} &{} -w_{n-2} &{} 0 &{} \cdots &{} 0 \end{pmatrix} \in T_o\hat{P}^{n-1}. \end{aligned}$$

It follows that the principal curvatures of $\hat{P}^{n-1}$ with respect to the unit normal vector $\hat{\zeta }$ are 0, 1 and $-1$, with corresponding principal curvature spaces

$$\begin{aligned} \hat{T}^{\hat{\zeta }}_0 = \mathbb {R}H_{\alpha _1+2\alpha _2} \oplus {\mathfrak {g}}_{\alpha _1+2\alpha _2}\,\ \hat{T}^{\hat{\zeta }}_1 = {\mathfrak {g}}_{\alpha _1+\alpha _2}\,\ \hat{T}^{\hat{\zeta }}_{-1} = {\mathfrak {g}}_{\alpha _2}. \end{aligned}$$

We now compute the shape operator of $\hat{P}^{n-1}$ at o for other unit normal vectors. Since $\nu _o\hat{P}^{n-1}$ is $\hat{J}$-invariant, the vector $\hat{J}\hat{\zeta }\in {\mathfrak {g}}_{\alpha _1}$ is a unit normal vector of $\hat{P}^{n-1}$ at o. Moreover, $\hat{\zeta },\hat{J}\hat{\zeta }$ is an orthonormal basis of the normal space $\nu _o\hat{P}^{n-1}$. Using a well-known formula for the shape operator of a complex submanifold of a Kähler manifold (see, e.g., [7], Lemma 7.4), we have

$$\begin{aligned} \hat{A}_{\hat{J}\hat{\zeta }} = \hat{J}\hat{A}_{\hat{\zeta }}. \end{aligned}$$

Since every unit normal vector of $\hat{P}^{n-1}$ at o is of the form

$$\begin{aligned} \cos (\varphi )\hat{\zeta }+ \sin (\varphi )\hat{J}\hat{\zeta }, \end{aligned}$$

the shape operator $\hat{A}_{\hat{\zeta }}$ therefore completely determines the shape operator for every other unit normal vector of $\hat{P}^{n-1}$ at o. More precisely, we have

$$\begin{aligned} \hat{A}_{\cos (\varphi )\hat{\zeta }+ \sin (\varphi )\hat{J}\hat{\zeta }} = \cos (\varphi )\hat{A}_{\hat{\zeta }} + \sin (\varphi )J\hat{A}_{\hat{\zeta }}. \end{aligned}$$

This readily implies that the principal curvatures of $\hat{P}^{n-1}$ with respect to the unit normal vector $\cos (\varphi )\hat{\zeta }+ \sin (\varphi )\hat{J}\hat{\zeta }$ are 0, 1 and $-1$, with corresponding principal curvature spaces

$$\begin{aligned} \hat{T}^{\cos (\varphi )\hat{\zeta }+ \sin (\varphi )\hat{J}\hat{\zeta }}_0&= \mathbb {R}H_{\alpha _1+2\alpha _2} \oplus {\mathfrak {g}}_{\alpha _1+2\alpha _2},\\ \hat{T}^{\cos (\varphi )\hat{\zeta }+ \sin (\varphi )\hat{J}\hat{\zeta }}_1&= \{ \cos \left( \textstyle {\frac{\varphi }{2}}\right) \hat{X} + \sin \left( \textstyle {\frac{\varphi }{2}}\right) J\hat{X} : \hat{X} \in {\mathfrak {g}}_{\alpha _1+\alpha _2}\},\\ \hat{T}^{\cos (\varphi )\hat{\zeta }+ \sin (\varphi )\hat{J}\hat{\zeta }}_{-1}&= \{\sin \left( \textstyle {\frac{\varphi }{2}}\right) \hat{X} - \cos \left( \textstyle {\frac{\varphi }{2}}\right) J\hat{X} : \hat{X} \in {\mathfrak {g}}_{\alpha _1+\alpha _2}\}. \end{aligned}$$

Using orthogonal projections onto ${\mathfrak {p}}$ we obtain the corresponding description of the shape operator A of $P^{n-1}$ at o.

Recall that

$$\begin{aligned} \nu _o\hat{P}^{n-1} = \mathbb {R}H_{\alpha _1} \oplus {\mathfrak {g}}_{\alpha _1}. \end{aligned}$$

The orthogonal projection of $\nu _o\hat{P}^{n-1}$ onto ${\mathfrak {p}}$ is

$$\begin{aligned} \nu _oP^{n-1} = \mathbb {C}H_{\alpha _1} = \mathbb {R}H_{\alpha _1} \oplus {\mathfrak {p}}_{\alpha _1} = \left\{ \begin{pmatrix} 0 &{} 0 &{} a &{} x &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} x &{} -a &{} 0 &{} \cdots &{} 0 \\ a &{} x &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ x &{} -a &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \end{pmatrix}: a,x \in \mathbb {R}\right\} . \end{aligned}$$

The complex line $\mathbb {C}H_{\alpha _1}$ is a Lie triple system in ${\mathfrak {p}}$ and therefore determines a totally geodesic complex submanifold $B_1$ of ${Q^n}^*$. The (non-zero) tangent vectors of $B_1$ are ${\mathfrak {A}}$-isotropic, which implies that the sectional curvature of $B_1$ is equal to $-4$. Thus $B_1$ is isometric to the complex hyperbolic line $\mathbb {C}H^1(-4)$ of constant (holomorphic) sectional curvature $-4$. We will encounter $B_1$ again later, where it appears in a horospherical decomposition of the complex hyperbolic quadric.

We now apply the standard real structure $C_0$ to the normal space $\nu _oP^{n-1} = T_oB_1$,

$$\begin{aligned} C_0(T_oB_1)&= \left\{ \begin{pmatrix} 0 &{} 0 &{} a &{} x &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} -x &{} a &{} 0 &{} \cdots &{} 0 \\ a &{} -x &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ x &{} a &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \end{pmatrix} : a,x \in \mathbb {R}\right\} \\&= \mathbb {R}H_{\alpha _1+2\alpha _2} \oplus {\mathfrak {p}}_{\alpha _1+2\alpha _2} = \mathbb {C}H_{\alpha _1+2\alpha _2} . \end{aligned}$$

Note that $C_0(T_oB_1) = C(T_oB_1)$ for any real structure C at o and therefore the construction is independent of the choice of real structure. The complex line $\mathbb {C}H_{\alpha _1+2\alpha _2}$ is also a Lie triple system in ${\mathfrak {p}}$ and determines a totally geodesic complex submanifold $\Sigma _1$ of ${Q^n}^*$. The (non-zero) tangent vectors of $\Sigma _1$ are also ${\mathfrak {A}}$-isotropic, which implies that the sectional curvature of $\Sigma _1$ is equal to $-4$. Thus $\Sigma _1$ is isometric to the complex hyperbolic line $\mathbb {C}H^1(-4)$ of constant (holomorphic) sectional curvature $-4$. The tangent space $T_o\Sigma _1$ is the kernel of the shape operator of the homogeneous complex hypersurface $P^{n-1} \cong \mathbb {C}H^{n-1}(-4)$. Since $\mathbb {R}H_{\alpha _1+2\alpha _2} \oplus {\mathfrak {g}}_{\alpha _1+2\alpha _2}$ is a subalgebra of ${\mathfrak {d}}$, this implies geometrically that $\hat{P}^{n-1}$, and hence also $P^{n-1}$, is foliated by totally geodesic complex hyperbolic lines $\mathbb {C}H^1(-4)$ whose tangent spaces are obtained by rotating the normal spaces of $\hat{P}^{n-1}$ (resp. $P^{n-1}$) via a real structure $\hat{C}$ (resp. C).

The Riemannian product $B_1 \times \Sigma _1 \cong \mathbb {C}H^1(-4) \times \mathbb {C}H^1(-4)$ is isometric to the complex hyperbolic quadric ${Q^2}^*$ and describes the standard isometric embedding of ${Q^2}^*$ into ${Q^n}^*$.

We have

$$\begin{aligned}{}[{\mathfrak {p}}_{\alpha _1+\alpha _2} \oplus {\mathfrak {p}}_{\alpha _2},{\mathfrak {p}}_{\alpha _1+\alpha _2} \oplus {\mathfrak {p}}_{\alpha _2}] \subset {\mathfrak {k}}_0 \oplus {\mathfrak {k}}_{\alpha _1} \oplus {\mathfrak {k}}_{\alpha _1+2\alpha _2} \end{aligned}$$

and

$$\begin{aligned}{}[{\mathfrak {k}}_0,{\mathfrak {p}}_{\alpha _1+\alpha _2}]&\subset {\mathfrak {p}}_{\alpha _1+\alpha _2},\\ [{\mathfrak {k}}_{\alpha _1},{\mathfrak {p}}_{\alpha _1+\alpha _2}]&\subset {\mathfrak {p}}_{\alpha _2},\\ [{\mathfrak {k}}_{\alpha _1+2\alpha _2},{\mathfrak {p}}_{\alpha _1+\alpha _2}]&\subset {\mathfrak {p}}_{\alpha _2},\\ [{\mathfrak {k}}_0,{\mathfrak {p}}_{\alpha _2}]&\subset {\mathfrak {p}}_{\alpha _2},\\ [{\mathfrak {k}}_{\alpha _1},{\mathfrak {p}}_{\alpha _2}]&\subset {\mathfrak {p}}_{\alpha _1+\alpha _2},\\ [{\mathfrak {k}}_{\alpha _1+2\alpha _2},{\mathfrak {p}}_{\alpha _2}]&\subset {\mathfrak {p}}_{\alpha _1+\alpha _2}. \end{aligned}$$

Altogether we conclude that ${\mathfrak {p}}_{\alpha _1+\alpha _2} \oplus {\mathfrak {p}}_{\alpha _2}$ is a Lie triple system. It is easy to see that this Lie triple system is J-invariant. The complex totally geodesic submanifold of ${Q^n}^*$ generated by this Lie triple system is isometric to ${Q^{n-2}}^*$. However, the only complex totally geodesic submanifolds of a complex hyperbolic space are again complex hyperbolic spaces (see [19] and use duality). It follows that there exists a totally geodesic submanifold $\Sigma ^{n-2} \cong \mathbb {C}H^{n-2}(-4)$ of $P \cong \mathbb {C}H^{n-1}(-4)$ with $T_o\Sigma ^{n-2} = {\mathfrak {p}}_{\alpha _1+\alpha _2} \oplus {\mathfrak {p}}_{\alpha _2}$. We have

$$\begin{aligned} T_oP^{n-1} = T_o\Sigma _1 \oplus T_o\Sigma ^{n-2},\ \nu _oP^{n-1} = T_oB_1. \end{aligned}$$

The tangent space $T_o\Sigma ^{n-2} = {\mathfrak {p}}_{\alpha _1+\alpha _2} \oplus {\mathfrak {p}}_{\alpha _2}$ and the normal space $\nu _o\Sigma ^{n-2} = {\mathfrak {a}}\oplus {\mathfrak {p}}_{\alpha _1+2\alpha _2} \oplus {\mathfrak {p}}_{\alpha _1}$ are Lie triple systems in ${\mathfrak {p}}$.

We summarize the previous discussion in the following theorem.

Theorem 4.2

There exists a homogeneous complex hypersurface $P^{n-1}$ in $({Q^n}^*,g)$ which is isometric to the complex hyperbolic space $\mathbb {C}H^{n-1}(-4)$ of constant holomorphic sectional curvature $-4$. In terms of root spaces and root vectors, the tangent space and normal space of $P^{n-1}$ at o is

$$\begin{aligned} T_oP^{n-1} = \mathbb {R}H_{\alpha _1+2\alpha _2} \oplus {\mathfrak {p}}_{\alpha _1+2\alpha _2} \oplus {\mathfrak {p}}_{\alpha _1+\alpha _2} \oplus {\mathfrak {p}}_{\alpha _2},\ \nu _oP^{n-1} = \mathbb {R}H_{\alpha _1} \oplus {\mathfrak {p}}_{\alpha _1}. \end{aligned}$$

The normal space $\nu _oP^{n-1}$ is a Lie triple system, and the totally geodesic submanifold $B_1$ of ${Q^n}^*$ generated by this Lie triple system is isometric to a complex hyperbolic line $\mathbb {C}H^1(-4)$ of constant (holomorphic) sectional curvature $-4$. The (non-zero) tangent vectors of $B_1$ are ${\mathfrak {A}}$-isotropic. In particular, the (non-zero) normal vectors of $P^{n-1}$ are ${\mathfrak {A}}$-isotropic singular tangent vectors of ${Q^n}^*$.

The tangent space $T_oP^{n-1}$ decomposes orthogonally into

$$\begin{aligned} T_oP^{n-1} = C(\nu _oP^{n-1}) \oplus ({\mathfrak {p}}_{\alpha _1+\alpha _2} \oplus {\mathfrak {p}}_{\alpha _2}), \end{aligned}$$

where C is any real structure in ${\mathfrak {A}}_0$ at o. The subspace $C(\nu _oP^{n-1})$ is a Lie triple system, and the totally geodesic submanifold $\Sigma _1$ of ${Q^n}^*$ generated by this Lie triple system is isometric to a complex hyperbolic line $\mathbb {C}H^1(-4)$ of constant (holomorphic) sectional curvature $-4$. The (non-zero) tangent vectors of $\Sigma _1$ are ${\mathfrak {A}}$-isotropic. The subspace ${\mathfrak {p}}_{\alpha _1+\alpha _2} \oplus {\mathfrak {p}}_{\alpha _2}$ is a Lie triple system in ${\mathfrak {p}}$ and a complex subspace of $T_oP^{n-1}$. The totally geodesic submanifold of ${Q^n}^*$ generated by this Lie triple system is isometric to the complex hyperbolic quadric ${Q^{n-2}}^*$, and the totally geodesic submanifold of $P^{n-1} \cong \mathbb {C}H^{n-1}(-4)$ generated by this complex subspace is isometric to the complex hyperbolic space $\mathbb {C}H^{n-2}(-4)$.

Let $\zeta \in \nu _oP^{n-1}$ be a unit normal vector of $P^{n-1}$. Then, $\zeta$ is of the form

$$\begin{aligned} \zeta = \frac{1}{2}\cos (\varphi )H_{\alpha _1} + \frac{1}{2}\sin (\varphi )JH_{\alpha _1} \end{aligned}$$

and the principal curvatures of $P^{n-1}$ with respect to $\zeta$ are $0,1,-1$ with corresponding principal curvature spaces

$$\begin{aligned} T^\zeta _0&= C(\nu _oP^{n-1}) = T_o\Sigma _1,\\ T^\zeta _1&= \{ \cos \left( \textstyle {\frac{\varphi }{2}}\right) X + \sin \left( \textstyle {\frac{\varphi }{2}}\right) JX : X \in {\mathfrak {p}}_{\alpha _1+\alpha _2}\} \subset V(C), \\ T^\zeta _{-1}&= \{\sin \left( \textstyle {\frac{\varphi }{2}}\right) X- \cos \left( \textstyle {\frac{\varphi }{2}}\right) JX : X \in {\mathfrak {p}}_{\alpha _1+\alpha _2}\} \subset JV(C), \end{aligned}$$

where $C = \cos (\varphi )C_0 + \sin (\varphi )JC_0$. The 0-eigenspace is independent of the choice of unit normal vector $\zeta$ and coincides with the kernel $T_0$ of the shape operator of $P^{n-1}$.

Let M be a submanifold of a Riemannian manifold $\bar{M}$ and $\zeta \in \nu _pM$ be a normal vector of M. Consider the Jacobi operator $\bar{R}_\zeta = \bar{R}(\cdot ,\zeta )\zeta : T_p\bar{M} \rightarrow T_p\bar{M}$. If $\bar{R}_\zeta (T_pM) \subseteq T_pM$, then the restriction ${\mathcal {K}}_\zeta$ of $\bar{R}_\zeta$ to $T_pM$ is a self-adjoint endomorphism of $T_pM$, the so-called normal Jacobi operator of M with respect to $\zeta$. The family ${\mathcal {K}}= ({\mathcal {K}}_\zeta )_{\zeta \in \nu M}$ is called the normal Jacobi operator of M.

A submanifold M of a Riemannian manifold $\bar{M}$ is curvature-adapted if for every normal vector $\zeta \in \nu _pM$, $p \in M$, the following two conditions are satisfied:

• (i) $\bar{R}_\zeta (T_pM) \subseteq T_pM$;

• (ii) the normal Jacobi operator ${\mathcal {K}}_\zeta$ and the shape operator $A_\zeta$ of M are simultaneously diagonalizable, that is,

  $$\begin{aligned} {\mathcal {K}}_\zeta A_\zeta = A_\zeta {\mathcal {K}}_\zeta . \end{aligned}$$

Since $\bar{R}_{\lambda \zeta } = \lambda ^2 \bar{R}_\zeta$ for all $\lambda > 0$, it suffices to check conditions (i) and (ii) only for unit normal vectors. Curvature-adapted submanifolds were introduced in [6]. They were also studied by Gray in [10] using the notion of compatible submanifolds. Curvature-adapted submanifolds form a very useful class of submanifolds in the context of focal sets and tubes.

Corollary 4.3

The homogeneous complex hypersurface $P^{n-1} \cong \mathbb {C}H^{n-1}(-4)$ in $({Q^n}^*,g)$ is curvature-adapted.

Proof

Let $\zeta$ be a unit normal vector of $P^{n-1}$ at o. Then $\zeta$ is an ${\mathfrak {A}}$-isotropic singular tangent vector of ${Q^n}^*$. We already computed the eigenvalues and eigenspaces of the Jacobi operator $\bar{R}_\zeta$ in Sect. 3. It follows from this that $\bar{R}_\zeta$ has three eigenvalues $0,-1,-4$ with corresponding eigenspaces

$$\begin{aligned} E^\zeta _0 = \mathbb {R}\zeta \oplus \mathbb {C}C_0\zeta, \,\ E^\zeta _{-1} = {\mathfrak {p}}\ominus (\mathbb {C}\zeta \oplus \mathbb {C}C_0\zeta ),\,\ E^\zeta _{-4} = \mathbb {R}J\zeta . \end{aligned}$$

Note that $E^\zeta _{-1}$ is independent of the choice of the unit normal vector $\zeta$ and hence we can denote this space by $E_{-1}$. The tangent space $T_oP^{n-1}$ is given by

$$\begin{aligned} T_oP^{n-1} = C_0(\nu _oP^{n-1}) \oplus E_{-1}. \end{aligned}$$

From Theorem 4.2 we see that $T_0^\zeta = T_0 = C_0(\nu _oP^{n-1}) = \mathbb {C}C_0\zeta \subset E^\zeta _0$ and $E_{-1} = T^\zeta _1 \oplus T^\zeta _{-1}$, which implies that ${\mathcal {K}}_\zeta$ and $A_\zeta$ commute. Since this holds for all unit normal vectors $\zeta$, it follows that $P^{n-1}$ is curvature-adapted. $\square$

5 Tubes around the homogeneous complex hypersurface

In this section we discuss the geometry of the tubes around the homogeneous complex hypersurface $P^{n-1} \cong \mathbb {C}H^{n-1}(-4)$ in $({Q^n}^*,g)$. We first observe that the tubes around $P^{n-1}$ are homogeneous real hypersurfaces in ${Q^n}^*$. In fact, the connected closed subgroup H of $G = SO^0_{2,n}$ with Lie algebra

$$\begin{aligned} {\mathfrak {h}}= {\mathfrak {k}}_{\alpha _1} \oplus {\mathfrak {d}}= {\mathfrak {k}}_{\alpha _1} \oplus \mathbb {R}H_{\alpha _1+2\alpha _2} \oplus {\mathfrak {g}}_{\alpha _2} \oplus {\mathfrak {g}}_{\alpha _1+\alpha _2} \oplus {\mathfrak {g}}_{\alpha _1+2\alpha _2} \end{aligned}$$

acts on ${Q^n}^*$ with cohomogeneity one (see [2], Theorem 8). By construction, the orbit of H containing o is the homogeneous complex hypersurface $P^{n-1}$ and the principal orbits are the tubes around $P^{n-1}$. We denote by $P^{2n-1}_r$ the tube with radius $r \in \mathbb {R}_+$ around $P^{n-1}$ in ${Q^n}^*$. Note that $P^{2n-1}_r$ is a homogeneous real hypersurface in ${Q^n}^*$ and hence $\dim _\mathbb {R}(P^{2n-1}_r) = 2n-1.$

By Corollary 4.3, the homogeneous complex hypersurface $P^{n-1}$ is curvature-adapted. Since tubes around curvature-adapted submanifolds in Riemannian symmetric spaces are again curvature-adapted (see [10], Theorem 6.14, or [6], Theorem 6), Corollary 4.3 implies:

Proposition 5.1

The tube $P^{2n-1}_r$ with radius $r \in \mathbb {R}_+$ around the homogeneous complex hypersurface $P^{n-1} \cong \mathbb {C}H^{n-1}(-4)$ in $({Q^n}^*,g)$ is curvature-adapted.

We can therefore use Jacobi field theory to compute the principal curvatures and principal curvature spaces of $P^{2n-1}_r$ (see, e.g., [1], Section 10.2.3, for a detailed description of the methodology). Since $P^{2n-1}_r$ is a homogeneous real hypersurface in ${Q^n}^*$, it suffices to compute the principal curvatures and principal curvature spaces at one point. Let $\zeta \in \nu _oP^{n-1}$ be a unit normal vector and $\gamma : \mathbb {R}\rightarrow {Q^n}^*$ the geodesic in ${Q^n}^*$ with $\gamma (0) = o$ and $\dot{\gamma }(0) = \zeta$. Then $p = \gamma (r) \in P^{2n-1}_r$ and $\zeta _r = \dot{\gamma }(r)$ is a unit normal vector of $P^{2n-1}_r$ at o. Since $\zeta$ is ${\mathfrak {A}}$-isotropic, also $\zeta _r$ is ${\mathfrak {A}}$-isotropic. Thus the normal bundle of $P^{2n-1}_r$ consists of ${\mathfrak {A}}$-isotropic singular tangent vectors of ${Q^n}^*$.

We denote by $\gamma ^\perp$ the parallel subbundle of the tangent bundle of ${Q^n}^*$ along $\gamma$ that is defined by the orthogonal complements of $\mathbb {R}\dot{\gamma }(t)$ in $T_{\gamma (t)}{Q^n}^*$, $t \in \mathbb {R}$, and put

$$\begin{aligned} \bar{R}^\perp _\gamma = \bar{R}_\gamma |_{\gamma ^\perp } = \bar{R}(\cdot ,\dot{\gamma })\dot{\gamma }|_{\gamma ^\perp }. \end{aligned}$$

Let D be the $\mathrm{End}(\gamma ^\perp )$-valued tensor field along $\gamma$ solving the Jacobi equation

$$\begin{aligned} D^{\prime \prime } + \bar{R}^\perp _\gamma \circ D = 0,\ D(0) = \begin{pmatrix} \mathrm{{id}}_{T_oP^{n-1}} &{} 0 \\ 0 &{} 0 \end{pmatrix},\ D^\prime (0) = \begin{pmatrix} -A_\zeta &{} 0 \\ 0 &{} \mathrm{{id}}_{\mathbb {R}J\zeta } \end{pmatrix}, \end{aligned}$$

where the decomposition of the matrices is with respect to the decomposition $\gamma ^\perp (0) = T_oP^{n-1} \oplus \mathbb {R}J\zeta$ and $A_\zeta$ is the shape operator of $P^{n-1}$ with respect to $\zeta$. If $v \in T_oP^{n-1}$ and $B_v$ is the parallel vector field along $\gamma$ with $B_v(0) = v$, then $Z_v = DB_v$ is the Jacobi field along $\gamma$ with initial values $Z_v(0) = v$ and $Z_v^\prime (0) = -A_\zeta v$. If $v \in \mathbb {R}J\zeta$ and $B_v$ is the parallel vector field along $\gamma$ with $B_v(0) = v$, then $Z_v = DB_v$ is the Jacobi field along $\gamma$ with initial values $Z_v(0) = 0$ and $Z_v^\prime (0) = v$. We decompose $T_oP^{n-1}$ orthogonally into $T_oP^{n-1} = T^\zeta _0 \oplus T^\zeta _1 \oplus T^\zeta _{-1}$ (see Theorem 4.2).

Since $\zeta$ is ${\mathfrak {A}}$-isotropic, the Jacobi operator $\bar{R}^\perp _\gamma$ at o is of matrix form

$$\begin{aligned} \begin{pmatrix} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} -1 &{} 0 \\ 0 &{} 0 &{} 0 &{} -4 \end{pmatrix} \end{aligned}$$

with respect to the decomposition $T^\zeta _0 \oplus T^\zeta _1 \oplus T^\zeta _{-1} \oplus \mathbb {R}J\zeta$. Since $({Q^n}^*,g)$ is a Riemannian symmetric space, the Jacobi operator $\bar{R}^\perp _\gamma$ is parallel along $\gamma$. By solving the above second-order initial value problem explicitly we obtain

$$\begin{aligned} D(r) = \begin{pmatrix} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} e^{-r} &{} 0 &{} 0 \\ 0 &{} 0 &{} e^r &{} 0 \\ 0 &{} 0 &{} 0 &{} \frac{1}{2}\sinh (2r) \end{pmatrix} \end{aligned}$$

with respect to the parallel translate of the decomposition $T^\zeta _0 \oplus T^\zeta _1 \oplus T^\zeta _{-1} \oplus \mathbb {R}J\zeta$ along $\gamma$ from o to $\gamma (r)$. The shape operator $A^r_{\zeta _r}$ of $P^{2n-1}_r$ with respect to the unit normal vector $\zeta _r = \dot{\gamma }(r)$ satisfies the equation

$$\begin{aligned} A^r_{\zeta _r} = - D^\prime (r) \circ D^{-1}(r). \end{aligned}$$

The matrix representation of $A^r_{\zeta _r}$ with respect to the parallel translate of the decomposition $T^\zeta _0 \oplus T^\zeta _1 \oplus T^\zeta _{-1} \oplus \mathbb {R}J\zeta$ along $\gamma$ from o to $\gamma (r)$ therefore is

$$\begin{aligned} A^r_{\zeta _r} = \begin{pmatrix} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} -1 &{} 0 \\ 0 &{} 0 &{} 0 &{} -2\coth (2r) \end{pmatrix}. \end{aligned}$$

It is remarkable that the principal curvatures of the tubes $P^{2n-1}_r$, corresponding to the maximal complex subbundle ${\mathcal {C}}$, are the same as those for the focal set $P^{n-1}$. The only additional principal curvature comes from the circles in $P^{2n-1}_r$ generated by the unit normal bundle of $P^{n-1}$, which in fact is the Hopf principal curvature function $\alpha$. We change the orientation of the unit normal vector field of $P^{2n-1}_r$ so that $\alpha$ becomes positive, that is, $\alpha = 2\coth (2r)$.

Since the Kähler structure J is parallel along $\gamma$, the condition $JT^\zeta _1 = T^\zeta _{-1}$ is preserved by parallel translation along $\gamma$. From this we easily see that the shape operator $A^r_{\zeta _r}$ of $P^{2n-1}_r$ satisfies $A^r_{\zeta _r}\phi + \phi A^r_{\zeta _r} = 0$. We summarize the previous discussion in the following result.

Theorem 5.2

Let $P^{2n-1}_r$ be the tube with radius $r \in \mathbb {R}_+$ around the homogeneous complex hypersurface $P^{n-1} \cong \mathbb {C}H^{n-1}(-4)$ in $({Q^n}^*,g)$. The normal bundle of $P^{2n-1}_r$ consists of ${\mathfrak {A}}$-isotropic singular tangent vectors of $({Q^n}^*,g)$. The homogeneous real hypersurface $P^{2n-1}_r$ has four distinct constant principal curvatures

$$\begin{aligned} 0,1,-1,2\coth (2r) \end{aligned}$$

with multiplicities $2,n-2,n-2,1$, respectively, with respect to a suitable orientation of the unit normal vector field $\zeta _r$ of $P^{2n-1}_r$. In particular, the mean curvature of $P^{2n-1}_r$ is equal to $2\coth (2r)$. The corresponding principal curvature spaces are

$$\begin{aligned} T^{\zeta _r}_0 = \mathbb {C}C\zeta _r = {\mathcal {C}}\ominus {\mathcal {Q}},\,\ T^{\zeta _r}_{2\coth (2r)} = \mathbb {R}J\zeta _r, \end{aligned}$$

where C is an arbitrary real structure on ${Q^n}^*$. The principal curvature spaces $T^{\zeta _r}_1$ and $T^{\zeta _r}_{-1}$ are mapped into each other by the complex structure J (or equivalently, by the structure tensor field $\phi$) and are contained in the $\pm 1$-eigenspaces of a suitable real structure C. Moreover, the shape operator $A^r$ and the structure tensor field $\phi$ of $P^{2n-1}_r$ satisfy

$$\begin{aligned} A^r \phi + \phi A^r = 0. \end{aligned}$$

To put this into the context of Theorem 1.1, we define $M_\alpha ^{2n-1} = P_r^{2n-1}$ with $\alpha = 2\coth (2r)$. Recall that the normal space $\nu _oP^{n-1}$ is a Lie triple system and the totally geodesic submanifold $B_1$ of ${Q^n}^*$ generated by this Lie triple system is isometric to a complex hyperbolic line $\mathbb {C}H^1(-4)$ of constant holomorphic sectional curvature $-4$. The same is true for all the other normal spaces of $P^{n-1}$. It follows that, by construction, the integral curves of the Reeb vector field $\xi = -J\zeta$ are circles of radius r in a complex hyperbolic line of constant sectional curvature $-4$. Such a circle has constant geodesic curvature $\alpha = 2\coth (2r)$. We thus see that the integral curves of the Reeb vector field are circles with radius r in a complex hyperbolic line $\mathbb {C}H^1(-4)$. This clarifies the geometric construction discussed in the introduction.

6 The minimal homogeneous Hopf hypersurface

In this section we construct the minimal homogeneous real hypersurface $M^{2n-1}_0$ in $({Q^n}^*,g)$. The construction is a special case of the canonical extension technique developed by the author and Tamaru in [4].

We start by defining the reductive subalgebra

$$\begin{aligned} {\mathfrak {l}}_1 = {\mathfrak {g}}_{-\alpha _1} \oplus {\mathfrak {g}}_0 \oplus {\mathfrak {g}}_{\alpha _1} \cong {\mathfrak {s}}{\mathfrak {u}}_{1,1} \oplus \mathbb {R}\oplus {\mathfrak {s}}{\mathfrak {o}}_{n-2} \end{aligned}$$

and the nilpotent subalgebra

$$\begin{aligned} {\mathfrak {n}}_1 = {\mathfrak {g}}_{\alpha _2} \oplus {\mathfrak {g}}_{\alpha _1 + \alpha _2} \oplus {\mathfrak {g}}_{\alpha _1 + 2\alpha _2} \cong {\mathfrak {h}}^{2n-3} \end{aligned}$$

of ${\mathfrak {g}}= {\mathfrak {s}}{\mathfrak {o}}_{2,n}$. Here, ${\mathfrak {h}}^{2n-3}$ is the $(2n-3)$-dimensional Heisenberg algebra with 1-dimensional center. Note that ${\mathfrak {n}}_1$ already appeared in the construction of the homogeneous complex hypersurface $P^{n-1}$ in Sect. 4 as part of the subalgebra ${\mathfrak {d}}= \mathbb {R}H_{\alpha _1+2\alpha _2} \oplus {\mathfrak {n}}_1$. We define

$$\begin{aligned} {\mathfrak {a}}_1 = \ker (\alpha _1) = \mathbb {R}H_{\alpha _1+2\alpha _2},\ {\mathfrak {a}}^1 = \mathbb {R}H_{\alpha _1}, \end{aligned}$$

which gives an orthogonal decomposition of ${\mathfrak {a}}$ into ${\mathfrak {a}}= {\mathfrak {a}}_1\oplus {\mathfrak {a}}^1$. The reductive subalgebra ${\mathfrak {l}}_1$ is the centralizer and the normalizer of ${\mathfrak {a}}_1$ in ${\mathfrak {g}}$. Since $[{\mathfrak {l}}_1,{\mathfrak {n}}_1] \subseteq {\mathfrak {n}}_1$,

$$\begin{aligned} {\mathfrak {q}}_1 = {\mathfrak {l}}_1 \oplus {\mathfrak {n}}_1 \end{aligned}$$

is a subalgebra of ${\mathfrak {g}}$, the so-called parabolic subalgebra of ${\mathfrak {g}}$ associated with the simple root $\alpha _1$. The subalgebra ${\mathfrak {l}}_1 = {\mathfrak {q}}_1 \cap \theta ({\mathfrak {q}}_1)$ is a reductive Levi subalgebra of ${\mathfrak {q}}_1$ and ${\mathfrak {n}}_1$ is the unipotent radical of ${\mathfrak {q}}_1$. Therefore the decomposition ${\mathfrak {q}}_1 = {\mathfrak {l}}_1 \oplus {\mathfrak {n}}_1$ is a semidirect sum of the Lie algebras ${\mathfrak {l}}_1$ and ${\mathfrak {n}}_1$. The decomposition ${\mathfrak {q}}_1 = {\mathfrak {l}}_1 \oplus {\mathfrak {n}}_1$ is the Chevalley decomposition of the parabolic subalgebra ${\mathfrak {q}}_1$.

Next, we define a reductive subalgebra ${\mathfrak {m}}_1$ of ${\mathfrak {g}}$ by

$$\begin{aligned} {\mathfrak {m}}_1 = {\mathfrak {g}}_{-\alpha _1} \oplus {\mathfrak {a}}^1 \oplus {\mathfrak {g}}_{\alpha _1} \oplus {\mathfrak {k}}_0 \cong {\mathfrak {s}}{\mathfrak {u}}_{1,1} \oplus {\mathfrak {s}}{\mathfrak {o}}_{n-2}. \end{aligned}$$

The subalgebra ${\mathfrak {m}}_1$ normalizes ${\mathfrak {a}}_1 \oplus {\mathfrak {n}}_1$. The decomposition

$$\begin{aligned} {\mathfrak {q}}_1 = {\mathfrak {m}}_1 \oplus {\mathfrak {a}}_1 \oplus {\mathfrak {n}}_1 \end{aligned}$$

is the Langlands decomposition of the parabolic subalgebra ${\mathfrak {q}}_1$. We define a subalgebra ${\mathfrak {k}}_1$ of ${\mathfrak {k}}$ by

$$\begin{aligned} {\mathfrak {k}}_1 = {\mathfrak {q}}_1 \cap {\mathfrak {k}}= {\mathfrak {l}}_1 \cap {\mathfrak {k}}= {\mathfrak {m}}_1 \cap {\mathfrak {k}}= {\mathfrak {k}}_{\alpha _1} \oplus {\mathfrak {k}}_0 \cong {\mathfrak {s}}{\mathfrak {o}}_2 \oplus {\mathfrak {s}}{\mathfrak {o}}_{n-2}. \end{aligned}$$

Next, we define the semisimple subalgebra

$$\begin{aligned} {\mathfrak {g}}_1 = {\mathfrak {g}}_{-\alpha _1} \oplus {\mathfrak {a}}^1 \oplus {\mathfrak {g}}_{\alpha _1} \cong {\mathfrak {s}}{\mathfrak {u}}_{1,1}. \end{aligned}$$

It is easy to see that the subspaces

$$\begin{aligned} {\mathfrak {a}}\oplus {\mathfrak {p}}_{\alpha _1} = {\mathfrak {l}}_1\cap {\mathfrak {p}},\ {\mathfrak {a}}^1 \oplus {\mathfrak {p}}_{\alpha _1} = {\mathfrak {m}}_1 \cap {\mathfrak {p}}= {\mathfrak {g}}_1 \cap {\mathfrak {p}}\end{aligned}$$

are Lie triple systems in ${\mathfrak p}$. Then ${\mathfrak {g}}_1 = {\mathfrak {k}}_{\alpha _1} \oplus ({\mathfrak {a}}^1 \oplus {\mathfrak {p}}_{\alpha _1})$ is a Cartan decomposition of the semisimple subalgebra ${\mathfrak {g}}_1$ of ${\mathfrak {g}}$ and ${\mathfrak {a}}^1$ is a maximal abelian subspace of ${\mathfrak {a}}^1 \oplus {\mathfrak {p}}_{\alpha _1}$. Moreover, ${\mathfrak {g}}_1 = ({\mathfrak {k}}_{\alpha _1} \oplus {\mathfrak {a}}^1) \oplus {\mathfrak {g}}_{-\alpha _1} \oplus {\mathfrak {g}}_{\alpha _1}$ is the restricted root space decomposition of ${\mathfrak {g}}_1$ with respect to ${\mathfrak {a}}^1$ and $\{\pm \alpha _1\}$ is the corresponding set of restricted roots.

We now relate these algebraic constructions to the geometry of the complex hyperbolic quadric ${Q^n}^*$. We denote by $A_1 \cong \mathbb {R}$ the connected abelian subgroup of G with Lie algebra ${\mathfrak {a}}_1$ and by $N_1 \cong H^{2n-3}$ the connected nilpotent subgroup of G with Lie algebra ${\mathfrak {n}}_1 \cong {\mathfrak {h}}^{2n-3}$. Here, $H^{2n-3}$ is the $(2n-3)$-dimensional Heisenberg group with 1-dimensional center. The centralizer $L_1 = Z_G({\mathfrak {a}}_1) \cong SU_{1,1} \times \mathbb {R}\times SO_{n-2}$ of ${\mathfrak {a}}_1$ in G is a reductive subgroup of G with Lie algebra ${\mathfrak {l}}_1$. The subgroup $A_1$ is contained in the center of $L_1$. The subgroup $L_1$ normalizes $N_1$ and $Q_1 = L_1 N_1$ is a subgroup of G with Lie algebra ${\mathfrak {q}}_1$. The subgroup $Q_1$ coincides with the normalizer $N_G({\mathfrak {l}}_1 \oplus {\mathfrak {n}}_1)$ of ${\mathfrak {l}}_1 \oplus {\mathfrak {n}}_1$ in G and hence $Q_1$ is a closed subgroup of G. The subgroup $Q_1$ is the parabolic subgroup of G associated with the simple root $\alpha _1$.

Let $G_1 \cong SU_{1,1}$ be the connected subgroup of G with Lie algebra ${\mathfrak {g}}_1 \cong {\mathfrak {s}}{\mathfrak {u}}_{1,1}$. The intersection $K_1$ of $L_1$ and K, i.e., $K_1 = L_1 \cap K \cong SO_2 \times SO_{n-2}$, is a maximal compact subgroup of $L_1$ and ${\mathfrak {k}}_1$ is the Lie algebra of $K_1$. The adjoint group $\mathrm{{Ad}}(L_1)$ normalizes ${\mathfrak {g}}_1$, and consequently $M_1 = K_1 G_1 \cong SU_{1,1} \times SO_{n-2}$ is a subgroup of $L_1$. The Lie algebra of $M_1$ is ${\mathfrak {m}}_1$ and $L_1$ is isomorphic to the Lie group direct product $M_1 \times A_1$, i.e., $L_1 = M_1 \times A_1 \cong (SU_{1,1} \times SO_{n-2}) \times \mathbb {R}$. The parabolic subgroup $Q_1$ acts transitively on ${Q^n}^*$ and the isotropy subgroup at o is $K_1$, that is, ${Q^n}^* \cong Q_1/K_1$.

Since ${\mathfrak {g}}_1 = {\mathfrak {k}}_{\alpha _1} \oplus ({\mathfrak {a}}^1 \oplus {\mathfrak {p}}_{\alpha _1})$ is a Cartan decomposition of the semisimple subalgebra ${\mathfrak {g}}_1$, we have $[{\mathfrak {a}}^1 \oplus {\mathfrak {p}}_{\alpha _1},{\mathfrak {a}}^1 \oplus {\mathfrak {p}}_{\alpha _1}] = {\mathfrak {k}}_{\alpha _1}$. Thus $G_1 \cong SU_{1,1}$ is the connected closed subgroup of G with Lie algebra $[{\mathfrak {a}}^1 \oplus {\mathfrak {p}}_{\alpha _1},{\mathfrak {a}}^1 \oplus {\mathfrak {p}}_{\alpha _1}] \oplus ({\mathfrak {a}}^1 \oplus {\mathfrak {p}}_{\alpha _1})$. Since ${\mathfrak {a}}^1 \oplus {\mathfrak {p}}_{\alpha _1}$ is a Lie triple system in ${\mathfrak {p}}$, the orbit $B_1 = G_1 \cdot o$ of the $G_1$-action on ${Q^n}^*$ containing o is a connected totally geodesic submanifold of ${Q^n}^*$ with $T_oB_1 = {\mathfrak {a}}^1 \oplus {\mathfrak {p}}_{\alpha _1}$. Moreover, $B_1$ is a Riemannian symmetric space of non-compact type and rank 1, and

$$\begin{aligned} B_1 = G_1 \cdot o = G_1/(G_1\cap K_1) \cong SU_{1,1}/SO_2 \cong \mathbb {C}H^1(-4), \end{aligned}$$

where $\mathbb {C}H^1(-4)$ is a complex hyperbolic line of constant (holomorphic) sectional curvature $-4$. The submanifold $B_1$ is a boundary component of ${Q^n}^*$ in the context of the maximal Satake compactification of ${Q^n}^*$. This boundary component coincides with the totally geodesic submanifold $B_1$ that we constructed in Sect. 4.

Clearly, ${\mathfrak {a}}_1$ is a Lie triple system and the corresponding totally geodesic submanifold is a Euclidean line $\mathbb {R}= A_1 \cdot o$. Since the action of $A_1$ on M is free and $A_1$ is simply connected, we can identify $\mathbb {R}$ and $A_1$ canonically.

Finally, ${\mathfrak {f}}_1 = {\mathfrak {a}}\oplus {\mathfrak {p}}_{\alpha _1}$ is a Lie triple system and the corresponding totally geodesic submanifold $F_1$ is the symmetric space

$$\begin{aligned} F_1 = L_1 \cdot o = L_1/K_1= (M_1 \times A_1)/K_1 = B_1 \times \mathbb {R}\cong \mathbb {C}H^1(-4) \times \mathbb {R}. \end{aligned}$$

The submanifolds $F_1$ and $B_1$ have a natural geometric interpretation. Denote by $\bar{C}^+(\Lambda ) \subset {\mathfrak {a}}$ the closed positive Weyl chamber that is determined by the two simple roots $\alpha _1$ and $\alpha _2$. Let Z be non-zero vector in $\bar{C}^+(\Lambda )$ such that $\alpha _1(Z) = 0$ and $\alpha _2(Z) > 0$, and consider the geodesic $\gamma _Z(t) = \mathrm{{Exp}}(tZ) \cdot o$ in ${Q^n}^*$ with $\gamma _Z(0) = o$ and $\dot{\gamma }_Z(0) = Z$. The totally geodesic submanifold $F_1$ is the union of all geodesics in ${Q^n}^*$ parallel to $\gamma _Z$, and $B_1$ is the semisimple part of $F_1$ in the de Rham decomposition of $F_1$ (see, e.g., [9], Proposition 2.11.4 and Proposition 2.20.10).

The parabolic group $Q_1$ is diffeomorphic to the product $M_1 \times A_1 \times N_1$. This analytic diffeomorphism induces an analytic diffeomorphism between

$$\begin{aligned} B_1 \times \mathbb {R}\times N_1 \cong \mathbb {C}H^1(-4) \times \mathbb {R}\times H^{2n-3} \end{aligned}$$

and ${Q^n}^*$, giving a horospherical decomposition of the complex hyperbolic quadric ${Q^n}^*$,

$$\begin{aligned} \mathbb {C}H^1(-4) \times \mathbb {R}\times H^{2n-3} \cong {Q^n}^*. \end{aligned}$$

The factor $\mathbb {R}\times H^{2n-3}$ corresponds to the homogeneous complex hypersurface $P^{n-1} \cong \mathbb {C}H^{n-1}(-4)$ that we discussed in Sect. 4.

We have $\mathbb {R}H_{\alpha _1} = {\mathfrak {a}}^1 \subset {\mathfrak {g}}_1$ and $G_1 \cdot o = B_1$. It follows from Theorem 4.2 that ${\mathfrak {a}}^1$ consists of ${\mathfrak {A}}$-isotropic tangent vectors of ${Q^n}^*$. Let $A^1 \cong \mathbb {R}$ be the abelian subalgebra of ${\mathfrak {a}}$ with Lie algebra ${\mathfrak {a}}^1$. Then the orbit $A^1 \cdot o$ is the path of an ${\mathfrak {A}}$-isotropic geodesic $\gamma$ (determined by the root vector $H_{\alpha _1}$) in the complex hyperbolic quadric ${Q^n}^*$. Moreover, by construction, this geodesic is contained in the boundary component $B_1 \cong \mathbb {C}H^1(-4)$. The action of $A^1$ on $\mathbb {C}H^1(-4)$ is of cohomogeneity one. The orbit containing o is the geodesic $\gamma$, and the other orbits are the equidistant curves to $\gamma$.

The canonical extension of the cohomogeneity one action of $A^1$ on the boundary component $B_1 \cong \mathbb {C}H^1(-4)$ is defined as follows. We first define the solvable subalgebra

$$\begin{aligned} {\mathfrak {s}}_1&= {\mathfrak {a}}^1 \oplus {\mathfrak {a}}_1 \oplus {\mathfrak {n}}_1 = {\mathfrak {a}}\oplus {\mathfrak {n}}_1 \\&= \left\{ \begin{pmatrix} 0 &{} y &{} a_1 &{} -y &{} v_1 &{} \cdots &{} v_{n-2}\\ -y &{} 0 &{} y &{} a_2 &{} w_1 &{} \cdots &{} w_{n-2} \\ a_1 &{} y &{} 0 &{} -y &{} v_1 &{} \cdots &{} v_{n-2}\\ -y &{} a_2 &{} y &{} 0 &{} w_1 &{} \cdots &{} w_{n-2} \\ v_1 &{} w_1 &{} -v_1 &{} -w_1 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{n-2} &{} w_{n-2} &{} -v_{n-2} &{} -w_{n-2} &{} 0 &{} \cdots &{} 0 \end{pmatrix} : \begin{array}{l} a_1,a_2,y \in \mathbb {R},\\ v,w \in \mathbb {R}^{n-2} \end{array} \right\} \end{aligned}$$

of ${\mathfrak {a}}\oplus {\mathfrak {n}}$. Let $S_1$ be the connected solvable subgroup of AN with Lie algebra ${\mathfrak {s}}_1$. Then the action of $S_1$ on AN (resp. ${Q^n}^*$) is of cohomogeneity one (see [4]). By construction, all orbits of the $S_1$-action on AN (resp. ${Q^n}^*$) are homogeneous real hypersurfaces in $(AN,\langle \cdot , \cdot \rangle )$ (resp. $({Q^n}^*,g)$). Let $\hat{M}^{2n-1}_0$ (resp. $M^{2n-1}_0$) be the orbit containing the point o. Geometrically, we can describe this orbit as the canonical extension of an ${\mathfrak {A}}$-isotropic geodesic in the boundary component $B_1 \cong \mathbb {C}H^1(-4)$.

We will now compute the shape operator of the homogeneous real hypersurface $\hat{M}^{2n-1}_0$ in $(AN,\langle \cdot , \cdot \rangle )$. Since $\hat{M}^{2n-1}_0$ is homogeneous, it suffices to make the computations at the point o. We define

$$\begin{aligned} \hat{\zeta }= \begin{pmatrix} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} \cdots &{} 0 \\ -1 &{} 0 &{} 1 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} \cdots &{} 0 \\ 1 &{} 0 &{} -1 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \end{pmatrix} \in {\mathfrak {g}}_{\alpha _1} \subset {\mathfrak {n}}. \end{aligned}$$

Then

$$\begin{aligned} \theta (\hat{\zeta }) = \begin{pmatrix} 0 &{} 1 &{} 0 &{} -1 &{} 0 &{} \cdots &{} 0 \\ -1 &{} 0 &{} -1 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} -1 &{} 0 &{} 1 &{} 0 &{} \cdots &{} 0 \\ -1 &{} 0 &{} -1 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \end{pmatrix} \in {\mathfrak {g}}_{-\alpha _1} \end{aligned}$$

and

$$\begin{aligned} \langle \hat{\zeta }, \hat{\zeta }\rangle&= - \frac{1}{8}\mathrm{{tr}}(\hat{\zeta }\theta (\hat{\zeta })) = 1. \end{aligned}$$

Thus $\hat{\zeta }$ is a unit normal vector of $\hat{M}^{2n-1}_0$ at o. Let $\hat{A}$ be the shape operator of $\hat{M}^{2n-1}_0$ in $(AN,\langle \cdot , \cdot \rangle )$ with respect to $\hat{\zeta }$. As in Sect. 4, using arguments involving the Weingarten and Koszul formulas, we can show that

$$\begin{aligned} \hat{A} \hat{X} = [\zeta ,\hat{X}]_{{\mathfrak {s}}_1} \end{aligned}$$

for all $\hat{X} \in {\mathfrak {s}}_1$, where $\zeta = \frac{1}{2}(\hat{\zeta }- \theta (\hat{\zeta }))$ is the orthogonal projection of $\hat{\zeta }$ onto ${\mathfrak {p}}$ and $[\, \cdot \ ]_{{\mathfrak {s}}_1}$ is the orthogonal projection onto ${\mathfrak {s}}_1$.

We have

$$\begin{aligned} \zeta = \frac{1}{2}(\hat{\zeta }- \theta (\hat{\zeta })) = \begin{pmatrix} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \end{pmatrix} \in {\mathfrak {p}}_{\alpha _1}. \end{aligned}$$

For

$$\begin{aligned} \hat{X} = \begin{pmatrix} 0 &{} y &{} a_1 &{} -y &{} v_1 &{} \cdots &{} v_{n-2}\\ -y &{} 0 &{} y &{} a_2 &{} w_1 &{} \cdots &{} w_{n-2} \\ a_1 &{} y &{} 0 &{} -y &{} v_1 &{} \cdots &{} v_{n-2}\\ -y &{} a_2 &{} y &{} 0 &{} w_1 &{} \cdots &{} w_{n-2} \\ v_1 &{} w_1 &{} -v_1 &{} -w_1 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{n-2} &{} w_{n-2} &{} -v_{n-2} &{} -w_{n-2} &{} 0 &{} \cdots &{} 0 \end{pmatrix} \in {\mathfrak {s}}_1 \end{aligned}$$

we then compute

$$\begin{aligned}{}[\zeta ,\hat{X}] = \begin{pmatrix} 0 &{} a_2-a_1 &{} 0 &{} 0 &{} w_1 &{} \cdots &{} w_{n-2} \\ a_1-a_2 &{} 0 &{} 0 &{} 0 &{} v_1 &{} \cdots &{} v_{n-2} \\ 0 &{} 0 &{} 0 &{} a_2-a_1 &{} w_1 &{} \cdots &{} w_{n-2}\\ 0 &{} 0 &{} a_1-a_2 &{} 0 &{} v_1 &{} \cdots &{} v_{n-2} \\ w_1 &{} v_1 &{} -w_1 &{} -v_1 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ w_{n-2} &{} v_{n-2} &{} -w_{n-2} &{} -v_{n-2} &{} 0 &{} \cdots &{} 0 \end{pmatrix}. \end{aligned}$$

The orthogonal projection of $[\zeta ,\hat{X}]$ onto ${\mathfrak {s}}_1$ is

$$\begin{aligned}{}[\zeta ,\hat{X}]_{{\mathfrak {s}}_1} = \begin{pmatrix} 0 &{} 0 &{} 0 &{} 0 &{} w_1 &{} \cdots &{} w_{n-2} \\ 0 &{} 0 &{} 0 &{} 0 &{} v_1 &{} \cdots &{} v_{n-2} \\ 0 &{} 0 &{} 0 &{} 0 &{} w_1 &{} \cdots &{} w_{n-2}\\ 0 &{} 0 &{} 0 &{} 0 &{} v_1 &{} \cdots &{} v_{n-2} \\ w_1 &{} v_1 &{} -w_1 &{} -v_1 &{} 0 &{} \cdots &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ w_{n-2} &{} v_{n-2} &{} -w_{n-2} &{} -v_{n-2} &{} 0 &{} \cdots &{} 0 \end{pmatrix}. \end{aligned}$$

We conclude that the shape operator $\hat{A}$ of $\hat{M}^{2n-1}_0$ in $(AN,\langle \cdot , \cdot \rangle )$ with respect to $\hat{\zeta }$ is given by

$$\begin{aligned} \hat{A} \hat{X} = \begin{pmatrix} 0 &{} 0 &{} 0 &{} 0 &{} w_1 &{} \cdots &{} w_{n-2} \\ 0 &{} 0 &{} 0 &{} 0 &{} v_1 &{} \cdots &{} v_{n-2} \\ 0 &{} 0 &{} 0 &{} 0 &{} w_1 &{} \cdots &{} w_{n-2}\\ 0 &{} 0 &{} 0 &{} 0 &{} v_1 &{} \cdots &{} v_{n-2} \\ w_1 &{} v_1 &{} -w_1 &{} -v_1 &{} 0 &{} \cdots &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ w_{n-2} &{} v_{n-2} &{} -w_{n-2} &{} -v_{n-2} &{} 0 &{} \cdots &{} 0 \end{pmatrix} \end{aligned}$$

with

$$\begin{aligned} \hat{X} = \begin{pmatrix} 0 &{} y &{} a_1 &{} -y &{} v_1 &{} \cdots &{} v_{n-2}\\ -y &{} 0 &{} y &{} a_2 &{} w_1 &{} \cdots &{} w_{n-2} \\ a_1 &{} y &{} 0 &{} -y &{} v_1 &{} \cdots &{} v_{n-2}\\ -y &{} a_2 &{} y &{} 0 &{} w_1 &{} \cdots &{} w_{n-2} \\ v_1 &{} w_1 &{} -v_1 &{} -w_1 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{n-2} &{} w_{n-2} &{} -v_{n-2} &{} -w_{n-2} &{} 0 &{} \cdots &{} 0 \end{pmatrix} \in {\mathfrak {s}}_1. \end{aligned}$$

From this we deduce that 0 is a principal curvature of $\hat{M}^{2n-1}_0$ with multiplicity 3 and corresponding principal curvature space

$$\begin{aligned} \hat{T}_0&= {\mathfrak {a}}\oplus {\mathfrak {g}}_{\alpha _1\oplus 2\alpha _2}. \end{aligned}$$

On the orthogonal complement ${\mathfrak {g}}_{\alpha _1+\alpha _2} \oplus {\mathfrak {g}}_{\alpha _2}$ the shape operator is of the form

$$\begin{aligned} \hat{A} = \begin{pmatrix} 0 &{} 1 \\ 1 &{} 0 \end{pmatrix} \end{aligned}$$

with respect to the orthogonal decomposition ${\mathfrak {g}}_{\alpha _1+\alpha _2} \oplus {\mathfrak {g}}_{\alpha _2}$. The characteristic polynomial of this matrix is $x^2 - 1$, and hence the eigenvalues of $\hat{A}$ restricted to ${\mathfrak {g}}_{\alpha _1+\alpha _2} \oplus {\mathfrak {g}}_{\alpha _2}$ are 1 and $-1$. The corresponding eigenspaces are

$$\begin{aligned} \hat{T}_1 = \left\{ \begin{pmatrix} 0 &{} 0 &{} 0 &{} 0 &{} u_1 &{} \cdots &{} u_{n-2}\\ 0 &{} 0 &{} 0 &{} 0 &{} u_1 &{} \cdots &{} u_{n-2} \\ 0 &{} 0 &{} 0 &{} 0 &{} u_1 &{} \cdots &{} u_{n-2}\\ 0 &{} 0 &{} 0 &{} 0 &{} u_1 &{} \cdots &{} u_{n-2} \\ u_1 &{} u_1 &{} -u_1 &{} -u_1 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ u_{n-2} &{} u_{n-2} &{} -u_{n-2} &{} -u_{n-2} &{} 0 &{} \cdots &{} 0 \end{pmatrix} \right\} \cong \mathbb {R}^{n-2} \end{aligned}$$

and

$$\begin{aligned} \hat{T}_{-1} = \left\{ \begin{pmatrix} 0 &{} 0 &{} 0 &{} 0 &{} u_1 &{} \cdots &{} u_{n-2}\\ 0 &{} 0 &{} 0 &{} 0 &{} -u_1 &{} \cdots &{} -u_{n-2} \\ 0 &{} 0 &{} 0 &{} 0 &{} u_1 &{} \cdots &{} u_{n-2}\\ 0 &{} 0 &{} 0 &{} 0 &{} -u_1 &{} \cdots &{} -u_{n-2} \\ u_1 &{} -u_1 &{} -u_1 &{} u_1 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ u_{n-2} &{} -u_{n-2} &{} -u_{n-2} &{} u_{n-2} &{} 0 &{} \cdots &{} 0 \end{pmatrix} \right\} \cong \mathbb {R}^{n-2}. \end{aligned}$$

All of the above calculations are with respect to the metric $\langle \cdot , \cdot \rangle$ on AN. We now switch to the Riemannian metric g on ${Q^n}^*$ and the Cartan decomposition ${\mathfrak {g}}= {\mathfrak {k}}\oplus {\mathfrak {p}}$. Recall that, by construction, $(AN,\langle \cdot , \cdot \rangle )$ and $({Q^n}^*,g)$ are isometric and the metrics are related by

$$\begin{aligned} \langle H_1 + \hat{X}_1, H_2 + \hat{X}_2 \rangle = g(H_1,H_2) + g(X_1,X_2) \end{aligned}$$

with $H_1,H_2 \in {\mathfrak {a}}$ and $\hat{X}_1,\hat{X}_2 \in {\mathfrak {n}}$.

Since $\hat{\zeta }$ is a unit vector in ${\mathfrak {g}}_{\alpha _1}$, the vector $\zeta = \frac{1}{2}(\hat{\zeta }- \theta (\hat{\zeta }))$ is a unit vector in ${\mathfrak {p}}_{\alpha _1}$. Since ${\mathfrak {p}}_{\alpha _1} \subset T_oB_1$ and all (non-zero) tangent vectors of the boundary component $B_1$ are ${\mathfrak {A}}$-isotropic (see Theorem 4.2), we conclude that the normal bundle of $M^{2n-1}_0$ consists of ${\mathfrak {A}}$-isotropic singular tangent vectors of $({Q^n}^*,g)$.

Let A be the shape operator of $M^{2n-1}_0$ in $({Q^n}^*,g)$ with respect to $\zeta$. The above calculations imply that

$$\begin{aligned} A X = \begin{pmatrix} 0 &{} 0 &{} 0 &{} 0 &{} w_1 &{} \cdots &{} w_{n-2} \\ 0 &{} 0 &{} 0 &{} 0 &{} v_1 &{} \cdots &{} v_{n-2} \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ w_1 &{} v_1 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ w_{n-2} &{} v_{n-2} &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \end{pmatrix} \end{aligned}$$

with

$$\begin{aligned} X = \begin{pmatrix} 0 &{} 0 &{} a_1 &{} -y &{} v_1 &{} \cdots &{} v_{n-2}\\ 0 &{} 0 &{} y &{} a_2 &{} w_1 &{} \cdots &{} w_{n-2} \\ a_1 &{} y &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ -y &{} a_2 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ v_1 &{} w_1 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{n-2} &{} w_{n-2} &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \end{pmatrix} \in T_oM^{2n-1}_0 \subset {\mathfrak {p}}. \end{aligned}$$

From this, we easily deduce the following result.

Theorem 6.1

Let $M^{2n-1}_0$ be the homogeneous real hypersurface in $({Q^n}^*,g)$ obtained by canonical extension of the geodesic that is tangent to the root vector $H_{\alpha _1}$ in the boundary component $B_1 \cong \mathbb {C}H^1(-4)$ of $({Q^n}^*,g)$. The normal bundle of $M^{2n-1}_0$ consists of ${\mathfrak {A}}$-isotropic singular tangent vectors of $({Q^n}^*,g)$ and $M^{2n-1}_0$ has three distinct constant principal curvatures 0, 1, $-1$ with multiplicities 3, $n-2$, $n-2$, respectively. The principal curvature spaces $T_0$, $T_1$ and $T_{-1}$ are

$$\begin{aligned} T_0&= {\mathfrak {a}}\oplus {\mathfrak {p}}_{\alpha _1+2\alpha _2} = \mathbb {R}J\zeta \oplus ({\mathcal {C}}\ominus {\mathcal {Q}}),\\ T_1&= \{ X-JX : X \in {\mathfrak {p}}_{\alpha _2}\} = \{ X+JX : X \in {\mathfrak {p}}_{\alpha _1+\alpha _2}\},\\ T_{-1}&= \{ X+JX : X \in {\mathfrak {p}}_{\alpha _2}\} = \{ X-JX : X \in {\mathfrak {p}}_{\alpha _1+\alpha _2}\}. \end{aligned}$$

We have $T_1 \oplus T_{-1} = {\mathcal {Q}}$ and $JT_1 = T_{-1}$. The shape operator A of $M^{2n-1}_0$ satisfies

$$\begin{aligned} A\phi + \phi A = 0. \end{aligned}$$

Note that

$$\begin{aligned} T_1 \subset V\left( \frac{1}{\sqrt{2}}(C_0 + JC_0) \right) ,\ T_{-1} \subset JV\left( \frac{1}{\sqrt{2}}(C_0 + JC_0) \right) . \end{aligned}$$

We immediately see from Theorem 6.1 that $\mathrm{{tr}}(A) = 0$.

Corollary 6.2

The homogeneous Hopf hypersurface $M^{2n-1}_0$ in $({Q^n}^*,g)$ is minimal.

The eigenspaces $T_0$, $T_1$ and $T_{-1}$ of the shape operator A and the eigenspaces $E_0$, $E_{-1}$ and $E_{-4}$ of the normal Jacobi operator ${\mathcal {K}}_\zeta$ satisfy

$$\begin{aligned} T_0 = E_0 \oplus E_{-4},\,\ T_{-1} \oplus T_1 = E_{-1}. \end{aligned}$$

It follows that A and ${\mathcal {K}}= {\mathcal {K}}_\zeta$ are simultaneously diagonalizable and hence $A{\mathcal {K}}= {\mathcal {K}}A$. This implies that $M^{2n-1}_0$ is curvature-adapted.

Corollary 6.3

The homogeneous Hopf hypersurface $M^{2n-1}_0$ in $({Q^n}^*,g)$ is curvature-adapted.

We finally relate this construction to the discussion in the introduction. The subalgebra

$$\begin{aligned} {\mathfrak {s}}_1 = {\mathfrak {a}}\oplus {\mathfrak {n}}_1 = {\mathfrak {a}}\oplus {\mathfrak {g}}_{\alpha _2} \oplus {\mathfrak {g}}_{\alpha _1 + \alpha _2} \oplus {\mathfrak {g}}_{\alpha _1 + 2\alpha _2} \end{aligned}$$

of ${\mathfrak {a}}\oplus {\mathfrak {n}}$ contains the subalgebra

$$\begin{aligned} {\mathfrak {d}}= \mathbb {R}H_{\alpha _1+2\alpha _2} \oplus {\mathfrak {g}}_{\alpha _2} \oplus {\mathfrak {g}}_{\alpha _1+\alpha _2} \oplus {\mathfrak {g}}_{\alpha _1+2\alpha _2}. \end{aligned}$$

The subalgebra ${\mathfrak {d}}$ induces the homogeneous complex hypersurface $\hat{P}^{n-1} \cong \mathbb {C}H^{n-1}(-4)$, as discussed in Sect. 4. Since the construction is left-invariant, it follows that the homogeneous real hypersurface $\hat{M}^{2n-1}_0$ in $(AN,\langle \cdot , \cdot \rangle )$ is foliated by isometric copies of the homogeneous complex hypersurface $\hat{P}^{n-1} \cong \mathbb {C}H^{n-1}(-4)$. This implies that the homogeneous complex hypersurface $M^{2n-1}_0$ in $({Q^n}^*,g)$ is foliated by isometric copies of the homogeneous complex hypersurface $P^{n-1} \cong \mathbb {C}H^{n-1}(-4)$. The normal space $\nu _oP^{n-1}$ is a Lie triple system, and the totally geodesic submanifold $B_1$ of ${Q^n}^*$ generated by this Lie triple system is a complex hyperbolic line $\mathbb {C}H^1(-4)$ of constant holomorphic sectional curvature $-4$. The same is true for all the normal spaces of $P^{n-1}$ at other points. It follows that, by construction, the integral curves of the Reeb vector field $\xi = -J\zeta$ are geodesics in a complex hyperbolic line of constant (holomorphic) sectional curvature $-4$. Such a geodesic has constant geodesic curvature 0. This clarifies the geometric construction explained in the introduction.

7 Equidistant real hypersurfaces

In this section we compute the shape operator of the other orbits of the cohomogeneity one action on $({Q^n}^*,g)$ that we discussed in Sect. 6. Recall that $M^{2n-1}_0$ is the orbit of this action containing o. Since the action is isometric, the other orbits are the equidistant real hypersurfaces to $M^{2n-1}_0$. For $r \in \mathbb {R}_+$ we denote by $M^{2n-1}_\alpha$ the equidistant real hypersurface to $M^{2n-1}_0$ at oriented distance $r \in \mathbb {R}_+$, where we put $\alpha = 2\tanh (2r)$.

From Corollary 6.3 we know that $M^{2n-1}_0$ is a curvature-adapted real hypersurface in ${Q^n}^*$. We can therefore use Jacobi field theory to compute the principal curvatures and principal curvature spaces of $M^{2n-1}_\alpha$ (see, e.g., [1], Section 10.2.2). Since $M^{2n-1}_\alpha$ is a homogeneous real hypersurface in ${Q^n}^*$, it suffices to compute the principal curvatures and principal curvature spaces at one point. Let $\zeta \in \nu _oM^{2n-1}_0$ be the unit normal vector of $M^{2n-1}_0$ as defined in Sect. 6 and $A_\zeta$ be the shape operator of $M^{2n-1}_0$ at o with respect to $\zeta$. We denote by $T_0$, $T_1$ and $T_{-1}$ the principal curvature spaces as in Theorem 6.1. Let $\gamma : \mathbb {R}\rightarrow {Q^n}^*$ be the geodesic in ${Q^n}^*$ with $\gamma (0) = o$ and $\dot{\gamma }(0) = \zeta$. Then $p = \gamma (r) \in M^{2n-1}_\alpha$ and $\zeta _r = \dot{\gamma }(r)$ is a unit normal vector of $M^{2n-1}_\alpha$ at p. We denote by $\gamma ^\perp$ the parallel subbundle of the tangent bundle of ${Q^n}^*$ along $\gamma$ that is defined by the orthogonal complements of $\mathbb {R}\dot{\gamma }(t)$ in $T_{\gamma (t)}{Q^n}^*$, and put

$$\begin{aligned} \bar{R}^\perp _\gamma = \bar{R}_\gamma |_{\gamma ^\perp } = \bar{R}(\cdot ,\dot{\gamma })\dot{\gamma }|_{\gamma ^\perp }. \end{aligned}$$

Let D be the $\mathrm{End}(\gamma ^\perp )$-valued tensor field along $\gamma$ solving the Jacobi equation

$$\begin{aligned} D^{\prime \prime } + \bar{R}^\perp _\gamma \circ D = 0,\ D(0) = \mathrm{{id}}_{T_oM^{2n-1}_0},\ D^\prime (0) = -A_\zeta . \end{aligned}$$

If $v \in T_oM^{2n-1}_0$ and $B_v$ is the parallel vector field along $\gamma$ with $B_v(0) = v$, then $Y = DB_v$ is the Jacobi field along $\gamma$ with initial values $Y(0) = v$ and $Y^\prime (0) = -A_\zeta v$.

Since $\zeta$ is ${\mathfrak {A}}$-isotropic, the Jacobi operator $\bar{R}^\perp _\gamma$ at o is of matrix form

$$\begin{aligned} \begin{pmatrix} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} -1 &{} 0 \\ 0 &{} 0 &{} 0 &{} -4 \end{pmatrix} \end{aligned}$$

with respect to the decomposition $\mathbb {C}C\zeta \oplus T_1 \oplus T_{-1} \oplus \mathbb {R}J\zeta$. Note that $T_0 = \mathbb {C}C\zeta \oplus \mathbb {R}J\zeta$. Since $({Q^n}^*,g)$ is a Riemannian symmetric space, the Jacobi operator $\bar{R}^\perp _\gamma$ is parallel along $\gamma$. By solving the above second-order initial value problem explicitly we obtain

$$\begin{aligned} D(r) = \begin{pmatrix} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} e^{-r} &{} 0 &{} 0 \\ 0 &{} 0 &{} e^r &{} 0 \\ 0 &{} 0 &{} 0 &{} \cosh (2r) \end{pmatrix} \end{aligned}$$

with respect to the parallel translate of the decomposition $T_0 \oplus T_1 \oplus T_{-1} \oplus \mathbb {R}J\zeta$ along $\gamma$ from o to $\gamma (r)$. The shape operator $A^\alpha _{\zeta _r}$ of $M^{2n-1}_\alpha$ with respect to $\zeta _r = \dot{\gamma }(r)$ satisfies the equation

$$\begin{aligned} A^\alpha _{\zeta _r} = - D^\prime (r) \circ D^{-1}(r). \end{aligned}$$

The matrix representation of $A^\alpha _{\zeta _r}$ with respect to the parallel translate of the decomposition $T_0 \oplus T_1 \oplus T_{-1} \oplus \mathbb {R}J\zeta$ along $\gamma$ from o to $\gamma (r)$ therefore is

$$\begin{aligned} A^\alpha _{\zeta _r} = \begin{pmatrix} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} -1 &{} 0 \\ 0 &{} 0 &{} 0 &{} -2\tanh (2r) \end{pmatrix}. \end{aligned}$$

It is remarkable that the principal curvatures of the equidistant real hypersurfaces to $M^{2n-1}_0$ are preserved along the parallel translate of the maximal complex subspace ${\mathcal {C}}_o \subset T_oM_0^{2n-1}$. The only additional principal curvature arises in direction of the Reeb vector field, which is the Hopf principal curvature. We change the orientation of the unit normal vector field $\zeta _r$ so that the Hopf principal curvature is positive, that is, is equal to $\alpha$. Thus we have proved:

Theorem 7.1

Let $M^{2n-1}_0$ be the minimal homogeneous Hopf hypersurface in $({Q^n}^*,g)$ as in Sect. 6and $M^{2n-1}_\alpha$ be the equidistant real hypersurface at oriented distance $r \in \mathbb {R}_+$ from $M^{2n-1}_0$, where $\alpha = 2\tanh (2r)$. Then $M^{2n-1}_\alpha$ is a homogeneous Hopf hypersurface with four distinct constant principal curvatures 0, 1, $-1$, $2\tanh (2r)$ with multiplicities 2, $n-2$, $n-2$, 1, respectively. The principal curvature spaces $T_0$, $T_1$, $T_{-1}$, $T_{2\tanh (2r)}$ satisfy

$$\begin{aligned} T_0&= {\mathcal {C}}\ominus {\mathcal {Q}},\\ T_{2\tanh (2r)}&= \mathbb {R}J\zeta _r = {\mathcal {C}}^\perp ,\\ T_1 \oplus T_{-1}&= {\mathcal {Q}} \text{ and } JT_1 = T_{-1}. \end{aligned}$$

Moreover, the shape operator $A^\alpha$ and the structure tensor field $\phi$ of $M^{2n-1}_\alpha$ satisfy

$$\begin{aligned} A^\alpha \phi + \phi A^\alpha = 0. \end{aligned}$$

The principal curvature spaces $T_{2\tanh (2r)}$, $T_0$, $T_1$ and $T_{-1}$ of the shape operator $A^\alpha$ and the eigenspaces $E_0$, $E_{-1}$ and $E_{-4}$ of the normal Jacobi operator ${\mathcal {K}}^\alpha = {\mathcal {K}}_{\zeta _r}$ satisfy

$$\begin{aligned} T_{2\tanh (2r)} = E_{-4},\,\ T_0 = E_0 ,\,\ T_{-1} \oplus T_1 = E_{-1}. \end{aligned}$$

It follows that $A^\alpha$ and ${\mathcal {K}}^\alpha$ are simultaneously diagonalizable and hence $A^\alpha {\mathcal {K}}^\alpha = {\mathcal {K}}^\alpha A^\alpha$. This implies:

Corollary 7.2

The equidistant real hypersurface $M^{2n-1}_\alpha$, $0< \alpha < 2$, to the minimal homogeneous Hopf hypersurface $M^{2n-1}_0$ in $({Q^n}^*,g)$ is curvature-adapted.

By construction, the integral curves of the Reeb vector field on $M^{2n-1}_\alpha$ are congruent to an equidistant curve at distance $r = \frac{1}{2}\tanh ^{-1}(\frac{\alpha }{2})$ to a geodesic in a complex hyperbolic line $\mathbb {C}H^1(-4)$. Such an equidistant curve has constant geodesic curvature $2\tanh (2r)$. As in previous cases, this leads to the geometric interpretation of $M^{2n-1}_\alpha$ given by attaching copies of the homogeneous complex hypersurface $P^{n-1} \cong \mathbb {C}H^{n-1}(-4)$ to such an equidistant curve to a geodesic in the boundary component $B_1 \cong \mathbb {C}H^1(-4)$. Equivalently, $M^{2n-1}_\alpha$ is the canonical extension of an equidistant curve at distance $r = \frac{1}{2}\tanh ^{-1}(\frac{\alpha }{2})$ to a geodesic in the boundary component $B_1 \cong \mathbb {C}H^1(-4)$.

8 The homogeneous Hopf hypersurface of horocyclic type

In this section we discuss the canonical extension of a horocycle in the boundary component $B_1 \cong \mathbb {C}H^1(-4)$, which leads to the homogeneous real hypersurface $M^{2n-1}_2$ in Theorem 1.1. We first define the solvable subalgebra

$$\begin{aligned} {\mathfrak {h}}_1 = {\mathfrak {a}}_1 \oplus {\mathfrak {n}}= \left\{ \begin{pmatrix} 0 &{} x+y &{} a &{} x-y &{} v_1 &{} \cdots &{} v_{n-2} \\ -x-y &{} 0 &{} x+y &{} a &{} w_1 &{} \cdots &{} w_{n-2} \\ a &{} x+y &{} 0 &{} x-y &{} v_1 &{} \cdots &{} v_{n-2} \\ x-y &{} a &{} -x+y &{} 0 &{} w_1 &{} \cdots &{} w_{n-2} \\ v_1 &{} w_1 &{} -v_1 &{} -w_1 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{n-2} &{} w_{n-2} &{} -v_{n-2} &{} -w_{n-2} &{} 0 &{} \cdots &{} 0 \end{pmatrix}: \begin{array}{l} a,x,y \in \mathbb {R},\\ v,w \in \mathbb {R}^{n-2} \end{array} \right\} \end{aligned}$$

of ${\mathfrak {a}}\oplus {\mathfrak {n}}$. Recall that ${\mathfrak {a}}^1 \oplus {\mathfrak {g}}_{\alpha _1} = \mathbb {R}H_{\alpha _1} \oplus {\mathfrak {g}}_{\alpha _1}$ generates the boundary component $B_1 \cong \mathbb {C}H^1(-4)$. The orbit containing o of the 1-dimensional Lie group generated by ${\mathfrak {g}}_{\alpha _1}$ is a horocycle in the boundary component $B_1$. Since the tangent vectors of $B_1$ are ${\mathfrak {A}}$-isotropic, the horocycle is ${\mathfrak {A}}$-isotropic. The canonical extension of this cohomogeneity one action on $B_1$ is the cohomogeneity one action on ${Q^n}^*$ by the subgroup $H_1$ of AN with Lie algebra ${\mathfrak {h}}_1$. Let $\hat{M}^{2n-1}_2 = H_1 \cdot o \cong H_1$ be the orbit of the $H_1$-action on $(AN,\langle \cdot , \cdot \rangle )$ containing o and $M^{2n-1}_2 = H_1 \cdot o$ be the orbit of the $H_1$-action on $({Q^n}^*,g)$ containing o.

The normal space $\nu _o\hat{M}^{2n-1}_2$ of $\hat{M}^{2n-1}_2$ at o is

$$\begin{aligned} \nu _o\hat{M}^{2n-1}_2 = {\mathfrak {a}}^1 = \mathbb {R}H_{\alpha _1}. \end{aligned}$$

Since $\langle H_{\alpha _1}, H_{\alpha _1} \rangle = 4$, the vector $\hat{\zeta }= \frac{1}{2}H_{\alpha _1} \in {\mathfrak {a}}$ is a unit normal vector of $\hat{M}^{2n-1}_2$ at o. Let $\hat{A}$ be the shape operator of $\hat{M}^{2n-1}_2$ in $(AN,\langle \cdot , \cdot \rangle )$ with respect to $\hat{\zeta }$. As in previous sections, we can show that the shape operator $\hat{A}$ of $\hat{M}^{2n-1}_2$ is given by

$$\begin{aligned} \hat{A} \hat{X} = [\zeta ,\hat{X}]_{{\mathfrak {h}}_1} \end{aligned}$$

for all $\hat{X} \in {\mathfrak {h}}_1$, where

$$\begin{aligned} \zeta = \frac{1}{2}(\hat{\zeta }- \theta (\hat{\zeta })) = \hat{\zeta }= \frac{1}{2}H_{\alpha _1} \end{aligned}$$

and $[\, \cdot \ ]_{{\mathfrak {h}}_1}$ is the orthogonal projection onto ${\mathfrak {h}}_1$.

For

$$\begin{aligned} \hat{X} = \begin{pmatrix} 0 &{} x+y &{} a &{} x-y &{} v_1 &{} \cdots &{} v_{n-2} \\ -x-y &{} 0 &{} x+y &{} a &{} w_1 &{} \cdots &{} w_{n-2} \\ a &{} x+y &{} 0 &{} x-y &{} v_1 &{} \cdots &{} v_{n-2} \\ x-y &{} a &{} -x+y &{} 0 &{} w_1 &{} \cdots &{} w_{n-2} \\ v_1 &{} w_1 &{} -v_1 &{} -w_1 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{n-2} &{} w_{n-2} &{} -v_{n-2} &{} -w_{n-2} &{} 0 &{} \cdots &{} 0 \end{pmatrix} \in {\mathfrak {h}}_1 \end{aligned}$$

we then compute

$$\begin{aligned}{}[\zeta ,\hat{X}] = \begin{pmatrix} 0 &{} 2x &{} 0 &{} 2x &{} v_1 &{} \cdots &{} v_{n-2} \\ -2x &{} 0 &{} 2x &{} 0 &{} -w_1 &{} \cdots &{} -w_{n-2} \\ 0 &{} 2x &{} 0 &{} 2x &{} v_1 &{} \cdots &{} v_{n-2}\\ 2x &{} 0 &{} -2x &{} 0 &{} -w_1 &{} \cdots &{} -w_{n-2} \\ v_1 &{} -w_1 &{} -v_1 &{} w_1 &{} 0 &{} \cdots &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{n-2} &{} -w_{n-2} &{} -v_{n-2} &{} w_{n-2} &{} 0 &{} \cdots &{} 0 \end{pmatrix}. \end{aligned}$$

Since the last matrix is in ${\mathfrak {h}}_1$, the orthogonal projection of $[\zeta ,\hat{X}]$ onto ${\mathfrak {h}}_1$ is $[\zeta ,\hat{X}]$. We conclude that the shape operator $\hat{A}$ of $\hat{M}^{2n-1}_2$ in $(AN,\langle \cdot , \cdot \rangle )$ is given by

$$\begin{aligned} \hat{A} \hat{X} = \begin{pmatrix} 0 &{} 2x &{} 0 &{} 2x &{} v_1 &{} \cdots &{} v_{n-2} \\ -2x &{} 0 &{} 2x &{} 0 &{} -w_1 &{} \cdots &{} -w_{n-2} \\ 0 &{} 2x &{} 0 &{} 2x &{} v_1 &{} \cdots &{} v_{n-2}\\ 2x &{} 0 &{} -2x &{} 0 &{} -w_1 &{} \cdots &{} -w_{n-2} \\ v_1 &{} -w_1 &{} -v_1 &{} w_1 &{} 0 &{} \cdots &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{n-2} &{} -w_{n-2} &{} -v_{n-2} &{} w_{n-2} &{} 0 &{} \cdots &{} 0 \end{pmatrix} \end{aligned}$$

with

$$\begin{aligned} \hat{X} = \begin{pmatrix} 0 &{} x+y &{} a &{} x-y &{} v_1 &{} \cdots &{} v_{n-2} \\ -x-y &{} 0 &{} x+y &{} a &{} w_1 &{} \cdots &{} w_{n-2} \\ a &{} x+y &{} 0 &{} x-y &{} v_1 &{} \cdots &{} v_{n-2} \\ x-y &{} a &{} -x+y &{} 0 &{} w_1 &{} \cdots &{} w_{n-2} \\ v_1 &{} w_1 &{} -v_1 &{} -w_1 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{n-2} &{} w_{n-2} &{} -v_{n-2} &{} -w_{n-2} &{} 0 &{} \cdots &{} 0 \end{pmatrix} \in {\mathfrak {h}}_1. \end{aligned}$$

From this we deduce that the principal curvatures of $\hat{M}^{2n-1}_2$ are 2, 0, 1, $-1$ with corresponding multiplicities 1, 2, $n-2$, $n-2$, respectively. The corresponding principal curvature spaces are

$$\begin{aligned} \hat{T}_2 = {\mathfrak {g}}_{\alpha _1}\,\ \hat{T}_0 = {\mathfrak {a}}_1 \oplus {\mathfrak {g}}_{\alpha _1\oplus 2\alpha _2} \,\ \hat{T}_1 = {\mathfrak {g}}_{\alpha _1+\alpha _2}\,\ \hat{T}_{-1} = {\mathfrak {g}}_{\alpha _2}. \end{aligned}$$

All of the above calculations are with respect to the metric $\langle \cdot , \cdot \rangle$ on AN. We now switch to the Riemannian metric g on ${Q^n}^*$ and the Cartan decomposition ${\mathfrak {g}}= {\mathfrak {k}}\oplus {\mathfrak {p}}$. Recall that, by construction, $(AN,\langle \cdot , \cdot \rangle )$ and $({Q^n}^*,g)$ are isometric and the metrics are related by

$$\begin{aligned} \langle H_1 + \hat{X}_1, H_2 + \hat{X}_2 \rangle = g(H_1,H_2) + g(X_1,X_2) \end{aligned}$$

with $H_1,H_2 \in {\mathfrak {a}}$ and $\hat{X}_1,\hat{X}_2 \in {\mathfrak {n}}$.

Let A be the shape operator of $M^{2n-1}_2$ in $({Q^n}^*,g)$ with respect to $\zeta$. The above calculations then imply

$$\begin{aligned} A X = \begin{pmatrix} 0 &{} 0 &{} 0 &{} 2x &{} v_1 &{} \cdots &{} v_{n-2} \\ 0 &{} 0 &{} 2x &{} 0 &{} -w_1 &{} \cdots &{} -w_{n-2} \\ 0 &{} 2x &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0\\ 2x &{} 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ v_1 &{} -w_1 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{n-2} &{} -w_{n-2} &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \end{pmatrix} \end{aligned}$$

with

$$\begin{aligned} X = \begin{pmatrix} 0 &{} 0 &{} a &{} x-y &{} v_1 &{} \cdots &{} v_{n-2} \\ 0 &{} 0 &{} x+y &{} a &{} w_1 &{} \cdots &{} w_{n-2} \\ a &{} x+y &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ x-y &{} a &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ v_1 &{} w_1 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{n-2} &{} w_{n-2} &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 \end{pmatrix} \in T_oM^{2n-1}_2 \subset {\mathfrak {p}}. \end{aligned}$$

From this we deduce the following result.

Theorem 8.1

The homogeneous real hypersurface $M^{2n-1}_2$ in $({Q^n}^*,g)$ has four distinct constant principal curvatures 2, 0, 1, $-1$ with multiplicities 1, 2, $n-2$, $n-2$, respectively. The principal curvature spaces $T_2$, $T_0$, $T_1$ and $T_{-1}$ are

$$\begin{aligned} T_2 = {\mathfrak {p}}_{\alpha _1} = \mathbb {R}J\zeta ,\ T_0 = \mathbb {R}H_{\alpha +2\alpha _2} \oplus {\mathfrak {p}}_{\alpha _1\oplus 2\alpha _2} = {\mathcal {C}}\ominus {\mathcal {Q}},\ T_1 = {\mathfrak {p}}_{\alpha _1+\alpha _2},\ T_{-1} = {\mathfrak {p}}_{\alpha _2}. \end{aligned}$$

In particular, $T_1$ and $T_{-1}$ are mapped into each other by the structure tensor field $\phi$. Moreover, the shape operator A and the structure tensor field $\phi$ of $M^{2n-1}_2$ satisfy

$$\begin{aligned} A \phi + \phi A = 0. \end{aligned}$$

Note that $T_1 \subset V(C_0)$ and $T_{-1} \subset JV(C_0)$.

The eigenspaces $T_2$, $T_0$, $T_1$ and $T_{-1}$ of the shape operator A and the eigenspaces $E_0$, $E_{-1}$ and $E_{-4}$ of the normal Jacobi operator ${\mathcal {K}}= {\mathcal {K}}_\zeta$ satisfy

$$\begin{aligned} T_2 = E_{-4},\ T_0 = E_0,\ T_{-1} \oplus T_1 = E_{-1}. \end{aligned}$$

It follows that A and ${\mathcal {K}}$ are simultaneously diagonalizable and hence $A{\mathcal {K}}= {\mathcal {K}}A$. Thus we have proved the following.

Corollary 8.2

The homogeneous Hopf hypersurface $M^{2n-1}_2$ in $({Q^n}^*,g)$ is curvature-adapted.

By construction, the integral curves of the Reeb vector field $\xi$ are congruent to a horocycle in a complex hyperbolic line $\mathbb {C}H^1(-4)$. Such a horocycle has constant geodesic curvature 2. As in previous cases, this leads to the geometric interpretation of $M^{2n-1}_2$ being obtained by attaching isometric copies of the homogeneous complex hypersurface $P^{n-1} \cong \mathbb {C}H^{n-1}(-4)$ to the horocycle in a suitable way. Equivalently, $M^{2n-1}_2$ is the canonical extension of a horocycle in the boundary component $B_1 \cong \mathbb {C}H^1(-4)$.

9 Curvature

In this section we compute the Ricci tensor $\mathrm{{Ric}}_\alpha$ and the scalar curvature $s_\alpha$ of the homogeneous Hopf hypersurface $M^{2n-1}_\alpha$ in $({Q^n}^*,g)$. Let $R_\alpha$, $\mathrm{{Ric}}_\alpha$, $s_\alpha$ be the Riemannian curvature tensor, Ricci tensor, scalar curvature of $M^{2n-1}_\alpha$, respectively. Let $A_\alpha$ and ${\mathcal {K}}_\alpha$ be the shape operator and normal Jacobi operator of $M^{2n-1}_\alpha$ with respect to the unit normal vector $\zeta _\alpha$, respectively. The Gauss equation tells us that

$$\begin{aligned} g(\bar{R}(X,Y)Z, W) = g(R_\alpha (X, Y)Z,W) - g(A_\alpha Y,Z)g(A_\alpha X,W) + g(A_\alpha X,Z)g(A_\alpha Y,W) \end{aligned}$$

for all $X,Y,Z,W \in {\mathfrak {X}}(M^{2n-1}_\alpha )$. Contracting the Gauss equation gives, after some straightforward computations, the expression

$$\begin{aligned} \mathrm{{Ric}}_\alpha X = -2nX - {\mathcal {K}}_\alpha X + \alpha A_\alpha X - A_\alpha ^2X, \end{aligned}$$

where we used the fact that the Ricci tensor of $({Q^n}^*,g)$ is equal to $-2ng$ and $\mathrm{{tr}}(A_\alpha ) = \alpha$ by Theorem 1.1. Since the unit normal vector $\zeta _\alpha$ of $M^{2n-1}_\alpha$ is ${\mathfrak {A}}$-isotropic, the normal Jacobi operator ${\mathcal {K}}_\alpha$ of $M^{2n-1}_\alpha$ satisfies

$$\begin{aligned} {\mathcal {K}}_\alpha X = {\left\{ \begin{array}{ll} 0 &{}, \text{ if } X \in {\mathcal {C}}\ominus {\mathcal {Q}}= T_0, \\ -X &{}, \text{ if } X \in {\mathcal {Q}}= T_{-1} \oplus T_1,\\ -4X &{}, \text{ if } X \in {\mathcal {C}}^\perp = \mathbb {R}\xi = T_\alpha \end{array}\right. } \end{aligned}$$

by Theorem 1.1 and the description of the Jacobi operator in Sect. 3. It follows that

$$\begin{aligned} \mathrm{{Ric}}_\alpha X = {\left\{ \begin{array}{ll} -2nX &{}, \text{ if } X \in {\mathcal {C}}\ominus {\mathcal {Q}}= T_0, \\ (-2n-\alpha )X &{}, \text{ if } X \in T_{-1}, \\ (-2n+\alpha )X &{}, \text{ if } X \in T_1, \\ (-2n+4)X &{}, \text{ if } X \in {\mathcal {C}}^\perp = \mathbb {R}\xi = T_\alpha . \end{array}\right. } \end{aligned}$$

It follows that the Ricci tensor of $M^{2n-1}_\alpha$ has two (if $\alpha = 0$), three (if $\alpha = 4$) or four (if $\alpha \notin \{0,4\}$) constant eigenvalues. More specifically, for $\alpha = 0$ we obtain

$$\begin{aligned} \mathrm{{Ric}}_0 X = -2n X + 4\eta (X)\xi , \end{aligned}$$

which means that $M^{2n-1}_0$ is pseudo-Einstein (see [12]).

Proposition 9.1

The minimal homogeneous real hypersurface $M^{2n-1}_0$ is a pseudo-Einstein Hopf hypersurface in $({Q^n}^*,g)$. In particular, the Ricci tensor $\mathrm{{Ric}}_0$ of $M^{2n-1}_0$ is $\phi$-invariant, that is, $\mathrm{{Ric}}_0 \circ \phi = \phi \circ \mathrm{{Ric}}_0$.

We also see that

$$\begin{aligned} \mathrm{{Ric}}_\alpha \circ \phi + \phi \circ \mathrm{{Ric}}_\alpha = -4n \phi . \end{aligned}$$

This equation is motivated by Ricci solitons (see [3], Lemma 3.3.11). However, none of the homogeneous Hopf hypersurfaces $M^{2n-1}_\alpha$ is a Ricci soliton.

By contracting the Ricci tensor we see that the scalar curvature of $M^{2n-1}_\alpha$ is independent of $\alpha$.

Proposition 9.2

The scalar curvature $s_\alpha$ of the homogeneous Hopf hypersurface $M^{2n-1}_\alpha$ in $({Q^n}^*,g)$ does not depend on $\alpha$ and satisfies

$$\begin{aligned} s_\alpha = 4-2n(2n-1). \end{aligned}$$