Laplacian algebras, manifold submetries and the Inverse Invariant Theory Problem

Abstract

Manifold submetries of the round sphere are a class of partitions of the round sphere that generalizes both singular Riemannian foliations, and the orbit decompositions by the orthogonal representations of compact groups. We exhibit a one-to-one correspondence between such manifold submetries and maximal Laplacian algebras, thus solving the Inverse Invariant Theory problem for this class of partitions. Moreover, a solution to the analogous problem is provided for two smaller classes, namely orthogonal representations of finite groups, and transnormal systems with closed leaves.

Appendices

Appendix A. Lagrangian families of Jacobi fields

The goal is this section is to recall some results regarding Lagrangian families of Jacobi fields, and important results by Wilking and Lytchak. For a deeper introduction on this topics, we refer the reader to [Wil07, Lyt09] and Chapter 4 of [Rad].

Let I be an interval of any type (it can be a half line or the whole real line as well). Consider a vector bundle $\pi :E\rightarrow I$ together with a smoothly-varying inner product $\langle \,,\, \rangle$ on each fiber, a covariant derivative $D:\Gamma (E)\rightarrow \Gamma (E)$ compatible with the inner product (we will write $X':=D(X)$ for a section $X\in \Gamma (E)$), and a symmetric endomorphism $R\in {\text {Sym}}^2(E)$ called curvature operator. Clearly, given a Riemannian manifold (M, g) and a geodesic $\gamma :I\rightarrow M$, then $E=\gamma '^\perp$ automatically comes equipped with $\langle \,,\, \rangle _t=g_{\gamma (t)}$, $D=\nabla _{\gamma '}$, and $R(t)=R^M(\cdot , \gamma '(t))\gamma '(t)$ where $\nabla$ and $R^M$ denote the Levi Civita connection and the Riemann curvature tensor of g, respectively.

Since I is contractible, E is trivial and thus it can be identified, via parallel transport, to $V\times I\rightarrow I$ for some Euclidean vector space $(V,\langle \,,\, \rangle )$. Via this identification, R becomes a function $R:I\rightarrow {\text {Sym}}^2(V)$.

With this setup, we can define the space of (R-)Jacobi fields as the set of sections

$$\begin{aligned} {\mathcal {J}}=\{J:I\rightarrow V\mid J''(t)+R(t)J(t)=0\quad \forall t\in I\}. \end{aligned}$$

This space has dimension $2\dim V$, isomorphic to $V\oplus V$ via the map $J\mapsto (J(0),J'(0))$. It is easy to see that for $J_1,J_2\in {\mathcal {J}}$ the function $\omega (J_1,J_2)=\langle J_1(t),J_2'(t) \rangle -\langle J_1'(t),J_2(t) \rangle$ is in fact constant, and defines a symplectic product on ${\mathcal {J}}$.

A subspace $W\subset {\mathcal {J}}$ is called isotropic if $\omega |_W=0$. Equivalently, W is isotropic if $\langle J_1'(t),J_2(t) \rangle =\langle J_1(t), J_2'(t) \rangle$ for any $J_1,J_2\in W$. The maximal dimension of an isotropic space is $\dim V$. An isotropic subspace of maximal dimension is called a Lagrangian subspace.

Given a subspace $W\subset {\mathcal {J}}$, define $W_t=\{J\in W\mid J(t)=0\}$ and $W(t)=\{J(t)\mid J\in W\}$. One fundamental property of isotropic subspaces is the following:

Proposition 38

([Lyt09], Lemma 2.2) An isotropic space W of Jacobi fields satisfies $\dim W(t)=\dim W$ for all but discretely values of t.

It follows from the proposition above that the focal function$f_W(t):=\dim (W_t)$ equals zero for all but discretely many values of $t\in I$. Thus, it makes sense to define, for every compact interval $[a,b]\subset I$, the index of W over [a, b] by

$$\begin{aligned} {\text {ind}}_IW=\sum _{t\in [a,b]}f_W(t). \end{aligned}$$

The index satisfies the following semi-continuity property, cf. [Lyt09]:

Proposition 39

Let $R_n:I\rightarrow {\text {Sym}}^2(V)$ be a sequence of families of symmetric endomorphisms converging in the $C^0$ topology to R. Let $W_n$ be isotropic subspaces of $R_n$-Jacobi fields that converge to an isotropic subspace W of R-Jacobi fields. Let $[a, b]\subseteq I$ be a compact interval and assume that $f_{W_n}(a) = f_{W}(a)$ and $f_{W_n}(b) = f_W (b)$, for all n large enough.

Then ${\text {ind}}_{[a,b]}W\ge {\text {ind}}_{[a,b]}W_n$ for all n large enough. If all $W_n$ are Lagrangians then this inequality becomes an equality.

1.1 A.1. Transverse Jacobi equation.

Let $E\simeq V\times I\rightarrow I$ be a vector bundle with $R\in {\text {Sym}}^2(V)$, and $\Lambda$ be a Lagrangian family of R-Jacobi fields, and let W be a subspace of $\Lambda$. Then W is isotropic by default, and by [Wil07] the subspaces

$$\begin{aligned} {\tilde{W}}(t)=\{J(t)\mid J\in W\}\oplus \{J'(t)\mid J\in W_t\}\subset E_t \end{aligned}$$

define a smooth vector bundle $E_W:=\coprod _{t\in I}{\tilde{W}}(t)\rightarrow I$. The quotient $H:=E/E_W$ comes equipped with:

• A Euclidean product $\langle [v_1],[v_2] \rangle :=\langle pr_{E_W^\perp }(v_1),pr_{E_W^\perp }(v_2) \rangle$, where $pr_{E_W^\perp }E\rightarrow E_W^\perp$ denotes the orthogonal projection onto $E_W^\perp$.

• A covariant derivative $D^H([X(t)])=[D(pr_{E_W^\perp }X(t))]$.

• A vector bundle map $A: E_W\rightarrow H$ given by $A(v)=[J'(t)]$, where $J\in W$ is such that $J(t)=v$.

• A symmetric endomorphism $R^H\in {\text {Sym}}^2(H)$ given by

  $$\begin{aligned} R^H_t([v])=[R_t(pr_{E_W^\perp }(v))+3AA^*[v]], \end{aligned}$$

  where $A^*:H\rightarrow E_W$ is the adjoint of A.

Proposition 40

(Transverse Jacobi equation). The projection $E\rightarrow H$ sends the Jacobi fields in $\Lambda$ to an isotropic subspace of $R^H$-Jacobi fields in H, which is isomorphic to $\Lambda /W$ as a vector space.

Because of the proposition above, we can identify the quotient $\Lambda /W$ with the corresponding isotropic space of $R^H$-Jacobi fields. Furthermore, by Lemma 3.1 of [Lyt09], for every $t\in I$ one has

$$\begin{aligned} f_\Lambda (t)=f_W(t)+f_{\Lambda /W}(t) \end{aligned}$$

(4)

and in particular, for every compact subinterval $[a,b]\subset I$,

$$\begin{aligned} {\text {ind}}_{[a,b]}\Lambda ={\text {ind}}_{[a,b]}W+{\text {ind}}_{[a,b]}{\Lambda /W}. \end{aligned}$$

(5)

Example 41

Let $\pi :M\rightarrow B$ be a Riemannian submersion, $\gamma :I\rightarrow M$ a horizontal geodesic, let $\gamma _*=\pi (\gamma )$, and let $E=(\gamma '^\perp )$ be the vector bundle along I. Letting W be the (isotropic) space of Jacobi fields along $\gamma$ such that $\pi _*J\equiv 0$, it follows by the O’Neill’s formulas that $H=E/E_W$ can be canonically identified with $(\gamma _*')^\perp$, in such a way that $R^H(v)=R^B(v, \gamma _*'(t))\gamma _*'(t)$ where $R^B$ denotes the Riemann curvature tensor of B.

Furthermore, letting $\Lambda \supseteq W$ denote the (Lagrangian) subspace of Jacobi fields J along $\gamma$, obtained as variation of horizontal geodesics, and such that $\pi _*(J(0))=0$, then $\Lambda /W$ corresponds to the Lagrangian space of Jacobi fields $J_*(t)$ along $\gamma _*$, such that $J_*(0)=0$. In particular, in this case $f_{\Lambda /W}(t)$ counts the conjugate points of $\gamma _*(0)$ along $\gamma _*$.

Appendix B. Manifold submetries

As mentioned in Sect. 2.1, the definitions of singular Riemannian foliation and manifold submetries are very close. The two key features which characterize singular Riemannian foliations are:

• (1) The leaves are connected.

• (2) There is a family of smooth vector fields which span the tangent spaces to the leaves at all points.

A lot of literature has focused mainly on singular Riemannian foliations, and uses the presence of smooth vector fields in several crucial places. The goal of this section is then to re-develop most of the basic results to the case of manifold submetries.

In this whole section, we will assume $\sigma :M\rightarrow X$ is a $C^2$ -manifold submetry unless states otherwise.

1.1 B.1. Homothetic Transformation Lemma, and stratification.

Let $\sigma :M\rightarrow X$ be a manifold submetry. Since leaves are equidistant, it follows from the first variation formula for the length function that every geodesic starting perpendicular to a leaf, stays perpendicular to all the leaves it meets. Such geodesics are called horizontal geodesics.

The first, fundamental result is the following (cf. [Mol88, Lemma 6.2] for transnormal systems):

Lemma 42

(Homothetic Transformation Lemma). Let $\sigma :M\rightarrow X$ be a manifold submetry, L a fiber of $\sigma$, $P\subset L$ a relatively compact open subset of L (called a plaque), and let $\epsilon >0$ be small enough that for every $v\in \nu ^{<\epsilon }P=\{v\in \nu P\mid \Vert v\Vert <\epsilon \}$, the geodesic $\gamma _v(t)=\exp (tv)$ minimizes the distance between $\gamma _v(1)$ and P. Then for any $\rho _1, \rho _2<\epsilon$ with $\rho _2=\lambda \rho _1$, the map

$$\begin{aligned} h_\lambda : \exp (\nu ^{\rho _1}P)\rightarrow \exp (\nu ^{\rho _2}P), \qquad h_\lambda (\exp v):=\exp (\lambda v) \end{aligned}$$

sends fibers of $\sigma$ into other fibers.

Proof

Let $q=\exp v$, $q'=\exp v' \in \nu ^{\rho _1} P$ be points such that $\sigma (q)=\sigma (q')=q_*$, and let $\sigma (P)=p_*$. By construction, the geodesics $\gamma _v(t)=\exp (tv)$ and $\gamma _{v'}(t)=\exp (tv')$ are projected to distance minimizing geodesics from $p_*$ to a bit past $q_*$. Since there is no bifurcation of geodesics in Alexandrov spaces, it follows that $\sigma (\gamma _v(t))=\sigma (\gamma _{v'}(t))=:\gamma _*(t)$ and therefore $h_\lambda (q)=\gamma _v(\lambda )$ and $h_\lambda (q')=\gamma _{v'}(\lambda )$ both project to $\gamma _*(\lambda )$. $\square$

For any integer r, define $\Sigma ^r\subset M$ the union of $\sigma$-fibers of dimension r. Any point $p\in M$ belongs to some stratum $\Sigma ^r$, and we define the stratum through p, and denote it by $\Sigma _p$ the union of connected components of $\Sigma ^r$ containing the (possibly disconnected) fiber through p. As a direct application of the Homothetic Transformation Lemma, one has

Proposition 43

(cf. [Mol88], Proposition 6.3). Given a manifold submetry $M\rightarrow X$, for every point $p\in M$ the stratum $\Sigma _p$ is a (possibly non-complete) smooth submanifold of M. Furthermore, for any relatively compact open subset $P\subset L$ of the leaf through p, there is an $\epsilon$ such that every horizontal geodesic from p initially tangent to $\Sigma _p$ stays in $\Sigma _p$ at least up to distance $\epsilon$.

Remark 44

It is important to notice that, in particular, if $\Sigma _p$ is disconnected, then different components will still have the same dimension.

Lemma 45

Let $\sigma :M\rightarrow X$, L, P, and $\epsilon$ as above. Consider the closest-point map $f:\exp \nu ^{<\epsilon }P\rightarrow P$. Given a $\sigma$-fiber $L'$ intersecting $\exp \nu ^\epsilon P$, let $P':=L'\cap \exp \nu ^{<\epsilon }P$ and $f'$ be the restriction of f to $P'$. Then:

• 1. The differential $d_qf'$ is surjective.

• 2. For any $p\in P$ and $x\in \nu ^{<\epsilon }_pP$, the fiber $L'$ through $q:=\exp x$ is transverse to the slice $D_p:=\exp \nu ^{<\epsilon }_pP$ at q.

• 3. The function $M\rightarrow {\mathbb {Z}}$, $p\mapsto \dim (L_p)$, is lower semicontinuous.

Proof

• (1) Let $\gamma (t)=\exp tx$. For any vector $v\in T_qP'$, let $J_v(t)$ the Jacobi field defined by $J_v(t)=(h_{t})_*v$. By the Homothetic Transformation Lemma, $J_v(t)$ is tangent to the $\sigma$-fibers for all $t\in [0,1]$. In particular, $J_v(0)\in T_pP$. Let $W=\{J_v\mid v\in T_qP'\}$. Notice that W is contained in the Lagrangian family $\Lambda _L$ consisting of Jacobi fields generated by variations of normal geodesics through L (cf. Appendix A). In particular, W is isotropic and any $J_v\in W$ vanishing at 0 satisfies $J_v'(0)\perp T_pL$. We can also embed W in the Lagrangian space $\Lambda _{L'}$ of Jacobi fields generated by variations of horizontal geodesics through $L'$. Letting $\Lambda _0=\{J\in \Lambda _{L'}\mid J(1)=0,\,J'(1)\perp T_qP'\}$, we have $\Lambda _{L'}=\Lambda _0\oplus W$. By Sect. 10 we get

  $$\begin{aligned} \gamma '(t)^{\perp }=\{J(t)\mid J\in \Lambda _{L'}\}\oplus \{J'(t)\mid J\in \Lambda _{L'}, J(t)=0\}. \end{aligned}$$

  In particular, every $w\in T_pP$ can be written as

  $$\begin{aligned} w=J_{u}(0)+J_{v}'(0)+J_3(0)+J_4'(0), \end{aligned}$$

  (6)

  where $J_{u}, J_v\in W$, $J_3,J_4\in \Lambda _0$, and $J_v(0)=J_4(0)=0$. Notice that:

  • $J_4=0$ because otherwise p and q would be conjugate points.

  • By the discussion above, $J_u(0)\in T_pP$ and $J_v'(0)\in \nu _pP$.

  • Taking the projection of Eq. 6 onto $\nu _pP$ and using the previous points, we get

    $$\begin{aligned} 0=J_v'(0)+pr_{\nu _pP}J_3(0). \end{aligned}$$

    However, by the definition of Lagrangian space of Jacobi fields,

    $$\begin{aligned} -\Vert J_v'(0)\Vert ^2=\langle J_v'(0), pr_{\nu _pP}J_3(0) \rangle =\langle J_v'(0), J_3(0) \rangle =\langle J_v(0), J_3'(0) \rangle =0 \end{aligned}$$

    and thus $J_v'(0)=0$ and $J_3(0)\in T_pP$.

  • $J_3=0$, because otherwise q would be a focal point for q, which is not possible because $\gamma$ keeps minimizing past q.

  Therefore, it must be $w=J_u(0)$. Notice however that $J_u(0)=d_qf'(u)$ and therefore $d_qf':T_qP'\rightarrow T_pP$ is surjective.

• (2) Since the kernel of $d_qf:T_qM\rightarrow T_pL$ is $T_qD_p$ and $d_qf$ is surjective by the previous point, the result follows.

• (3) It is enough to prove that for every $p\in M$ there is a neighborhood U around p such that $\dim L_q\ge \dim L_p$ for every $q\in U$. This is exactly what point (1) shows. $\square$

Remark 46

• (1) In the case of singular Riemannian foliations, the semicontinuity of the dimension of leaves follows immediately from the existence of smooth vector fields spanning the foliation.

• (2) Lemma 45 shows that for every $r_0$, the union $\bigcup _{r\ge r_0}\Sigma ^{r}$ is open. In particular, the regular part, consisting of fibers of maximal dimension, is open in M.

1.2 B.2. Generic strata.

In this section we assume that $\sigma :M\rightarrow X$ is a smooth manifold submetry with connected fibers, and let $M^{(2)}$ be the union of the strata $\Sigma _p$ of codimension $\le 2$ (see Sect. 10). The main result of this section will be to show that the fibers of $\sigma$ form a full singular Riemannian foliation on $M^{(2)}$ (see Definition 48 below).

Lemma 47

There are no strata of codimension 1. Moreover, let $\sigma :M\rightarrow X$ a manifold submetry, and $\Sigma _p$ be a stratum of codimension 2. Let U be a relatively compact neighborhood of p in $\Sigma _p$, and $\epsilon$ small enough that all normal geodesics from U minimize the distance from $\Sigma _p$ up to time $\epsilon$. Let $B_\epsilon (U)=\exp \nu ^{<\epsilon }(U)$. Then for any $q=\exp _{p'}v\in B_\epsilon (U){\setminus } U$, $v\in \nu _{p'}^{<\epsilon }(U)$, the $\sigma$-fiber through q is given by $S_d(L_{p'})\cap S_d(U)$ where $d=dist(q,U)$ and $S_d(L_{p'})$ (resp. $S_d(U)$) denotes the boundary of the tube of distance d around $L_{p'}$ (resp. around U).

Proof

First of all notice that $q\notin \Sigma _p$ and, by Lemma 45 and the Homothetic Transformation Lemma, $\dim (L_q)>\dim L_{p'}$. By definition of $\epsilon$, it follows $d={\text {dist}}(q,U)={\text {dist}}(q, L_{p'})={\text {dist}}(q,p')$. Notice furthermore that $S_d(L_{p'})\cap S_d(U)=\exp \nu ^dU\big |_{L_{p'}}$ is a manifold and, since U has codimension 2 in M by assumption, one has

$$\begin{aligned} \dim S_d(L_{p'})\cap S_d(U)=\dim \exp \nu ^dU\big |_{L_{p'}}= \dim \nu ^dU\big |_{L_{p'}}= \dim L_{p'}+1. \end{aligned}$$

By equidistance of the $\sigma$ fibers, the fiber $L_q$ through q must lie in $S_d(L_{p'})$. Moreover since ${\text {dist}}(\cdot ,U)=\inf _{r\in U}{\text {dist}}(\cdot ,L_{r})$, it follows that the distance from U is constant along the $\sigma$-fibers in $B_\epsilon (U)$, and thus $L_q$ must lie in $S_d(U)$ as well. Therefore, $L_q\cap B_\epsilon (U)$ is contained in $S_d(L_{p'})\cap S_d(U)$. On the other hand, it is easy to see that $\dim (S_d(L_{p'})\cap S_d(U))=\dim L_p+{\text {codim}} \Sigma _p-1$. Thus, if $\Sigma _p$ was a stratum of codimension 1, then $\dim L_q\le \dim L_p$ which would give a contradiction, hence there are no strata of codimension 1. Now letting $\Sigma _p$ be a stratum of codimension 2, this would imply that $\dim (L_q\cap B_{\epsilon }(U))\le \dim (S_d(L_{p'})\cap S_d(U))=\dim L_{p'}+1$ and the only possibility is that the inequality is in fact an equality, in which case $L_q\cap B_{\epsilon }(U)=S_d(L_{p'})\cap S_d(U)$. $\square$

Recall Definition 3 for the notion of singular Riemannian foliation. We now define the concept of full singular Riemannian foliation (see [Lyt10]):

Definition 48

A singular Riemannian foliation $(M,\mathcal {F})$ is called full if for every point $p\in M$ there exists an $\epsilon$, such that the normal exponential map $\exp :\nu ^{<\epsilon }L\rightarrow M$ from the leaf L through p is well defined.

We can now prove the following:

Proposition 49

Let $\sigma :M\rightarrow X$ be a manifold submetry with connected fibers. Then the partition $(M^{(2)},\mathcal {F})$ of $M^{(2)}$ into the fibers of $\sigma |_{M^{(2)}}$ is a full singular Riemannian foliation.

Proof

Since the fibers of $\sigma$ are connected by assumption, the only thing to prove is that every vector v tangent to a $\sigma$-fiber, can be locally extended to a vector field everywhere tangent to the $\sigma$-fibers. Once this is proved, the foliation is automatically full since every leaf L of $\mathcal {F}$ is compact.

Fix $p\in M^{(2)}$, let L be the $\sigma$-fiber through p and $\Sigma _p$ the stratum through p, and fix $v\in T_pL$. Clearly if the codimension of $\Sigma _p$ is zero, then $\sigma$ is a Riemannian submersion around L and it is straightforward to produce a local vector field V everywhere tangent to the $\sigma$-fibers extending v. Furthermore, by Lemma 47$\Sigma _p$ cannot have codimension 1, which only leaves the case of $\Sigma _p$ having codimension 2. In this case, $\sigma |_{\Sigma _p}$ is still a Riemannian submersion, and any vector $v\in T_pL$ can be extended to a vector field $V_1$ in $\Sigma _p$, tangent to the $\sigma$-fibers. Take a neighborhood U of p in $\Sigma _p$, and let $\epsilon$ small enough, as in Lemma 47. We can extend $V_1$ to a vector field V in $B_\epsilon (U)$ as follows: first take any extension $V_2$ of $V_1$ to $B_{\epsilon }(U)$. Secondly, define the linearization of$V_2$alongU as

$$\begin{aligned} V_2^{\ell }=\lim _{\lambda \rightarrow 0}(h_{\lambda })_*^{-1}(V_2\circ h_{\lambda }), \end{aligned}$$

where $h_\lambda :B_\epsilon (U)\rightarrow B_{\lambda \epsilon }(U)$ denotes the homothetic transformation $\exp v\mapsto \exp \lambda v$, $v\in \nu ^{<\epsilon }U$. By the properties of linearized vector fields (cf. [MR19b], Proposition 13) $V_2^\ell$ is still smooth and it projects to $V_1$ via the closest-point-map projection $\pi :B_\epsilon (U)\rightarrow U$. In particular, letting $K\subset T B_\epsilon (U)$ be the smooth distribution given by $\ker (\pi _*)$, the projection $V=pr_{K^\perp }V_2^\ell$ is the unique vector field perpendicular to the $\pi$-fibers which projects to $V_1$ via $\pi$ (that is, V the horizontal extension of $V_1$ with respect to the submersion $\pi :B_{\epsilon }(U)\rightarrow U$). It then follows that the flows $\Phi ^t_V$, $\Phi ^t_{V_1}$ satisfy $\pi \circ \Phi _{V}^t=\Phi _{V_1}^t\circ \pi$. The flow lines of V (which stay at a constant distance from U by the first variation of length) are thus also equidistant to $L'$ thus V is tangent to the intersections $S_d(U)\cap S_d(L')$. These, by Lemma 47, coincide with the leaves $L_q\cap B_{\epsilon }(U)$, $q\in B_\epsilon (U)$.

Summing up V is a local vector field, everywhere tangent to the leaves, which coincides with the vector v at p. Since v was arbitrary, $(M^{(2)},\mathcal {F})$ is a (full) singular Riemannian foliation. $\square$