On projective and affine equivalence of sub-Riemannian metrics

Abstract

Consider a smooth connected manifold M equipped with a bracket generating distribution D. Two sub-Riemannian metrics on (M, D) are said to be projectively (resp. affinely) equivalent if they have the same geodesics up to reparameterization (resp. up to affine reparameterization). A sub-Riemannian metric g is called rigid (resp. conformally rigid) with respect to projective/affine equivalence, if any sub-Riemannian metric which is projectively/affinely equivalent to g is constantly proportional to g (resp. conformal to g). In the Riemannian case the local classification of projectively (resp. affinely) equivalent metrics was done in the classical work (Levi-Civita in Ann Mat Ser 2a 24:255–300, 1896; resp. Eisenhart in Trans Am Math Soc 25(2):297–306, 1923). In particular, a Riemannian metric which is not rigid with respect to one of the above equivalences satisfies the following two special properties: its geodesic flow possesses a collection of nontrivial integrals of special type and the metric induces certain canonical product structure on the ambient manifold. The only proper sub-Riemannian cases to which these classification results were extended so far are sub-Riemannian metrics on contact and quasi-contact distributions (Zelenko in J Math Sci (NY) 135(4):3168–3194, 2006). The general goal is to extend these results to arbitrary sub-Riemannian manifolds. In this article we establish two types of results toward this goal: if a sub-Riemannian metric is not conformally rigid with respect to the projective equivalence, then, first, its flow of normal extremals has at least one nontrivial integral quadratic on the fibers of the cotangent bundle and, second, the nilpotent approximation of the underlying distribution at any point admits a product structure. As a consequence we obtain two types of genericity results for rigidity: first, we show that a generic sub-Riemannian metric on a fixed pair (M, D) is conformally rigid with respect to projective equivalence. Second, we prove that, except for special pairs (m, n), for a generic distribution D of rank m on an n-dimensional manifold, every sub-Riemannian metric on D is conformally rigid with respect to the projective equivalence. For the affine equivalence in both genericity results conformal rigidity can be replaced by usual rigidity.

Appendices

Appendix A. Proof of Proposition 5.5 on projective equivalence of Levi-Civita pairs

Let $(g_1,g_2)$ be a Levi-Civita pair on a distribution D of a manifold M, and fix a point $q_0 \in M$. In local coordinates, the metrics $g_1, g_2$ take the form (5.2) and the distribution D is the product distribution $D=D_1 \times \cdots \times D_N$ on ${\mathbb {R}}^n= {\mathbb {R}}^{n_1}\times \cdots \times {\mathbb {R}}^{n_N}$ defined by (5.1).

Let us construct a frame adapted to $(g_1, g_2)$. For any integer $1\le \ell \le N$, we choose vector fields $Y_1^\ell , \dots , Y^\ell _{k_\ell }$, where $k_\ell = \dim D_\ell$, of the form $Y^\ell _i = \sum _{j=1}^{n_\ell } a^\ell _{ij}(x^\ell )\partial _{x_j^\ell }$ such that $\{ Y_1^\ell , \dots , Y^\ell _{k_\ell }\}$ is a frame of $D_\ell$ and is orthonormal with respect to $\bar{g}_\ell$. We complete $\{ Y_1^\ell , \dots , Y^\ell _{k_\ell }\}$ into a frame adapted to the flag $D_\ell \subset D_\ell ^2 \subset \dots \subset T{\mathbb {R}}^{n_\ell }$ by adding vector fields $X_{k_\ell +1}^\ell , \dots , X^\ell _{n_\ell }$ of the form $[Y^\ell _{i_1}, \dots , [Y_{i_{k-1}}^\ell , Y^\ell _{i_k}]]$. Moreover, setting $X^\ell _i = \frac{1}{\sqrt{\gamma _\ell }}Y^\ell _i$ for $i=1, \dots , k_\ell$, we obtain a $g_1$-orthonormal frame $\{ X_1^\ell , \dots , X^\ell _{k_\ell }\}$ of $D_\ell$.

Grouping all together, we have obtained a frame $\{X^\ell _i, \ 1\le \ell \le N, \ i=1, \dots , k_\ell \}$ of D which is $g_1$-orthonormal and $g_2$-orthogonal, and a frame $\{X^\ell _i, \ 1\le \ell \le N, \ i=1, \dots , n_\ell \}$ of $T{\mathbb {R}}^n$ which is adapted to the pair $(g_1, g_2)$. To simplify the notations we denote by $\{X_1,\dots ,X_m\}$ the frame of D and by $\{X_1,\dots ,X_n\}$ the frame of $T{\mathbb {R}}^n$. For $i=1,\dots ,n$, we denote by $\ell (i)$ the integer in $\{ 1, \dots , N\}$ such that $X_i$ is of the form $X^{\ell (i)}_j$.

The special form of the constructed adapted frame and the form of (5.3) imply the following properties of the structure coefficients $c^k_{ij}$:

• if $\ell (i) \ne \ell (j)$, then $c_{ij}^k=0$ if $k \ne i$ or j; moreover,

  $$\begin{aligned} c^j_{ij} =\left\{ \begin{array}{ll} \frac{\alpha ^2_{\ell (j)} X_i(\alpha ^2_{\ell (i)}) }{4 \alpha _{\ell (i)}^2 \left( \alpha _{\ell (j)}^2 - \alpha _{\ell (i)}^2 \right) }, &{} \quad \hbox {if } j\le m; \\ 0, &{} \quad \hbox {if } j>m; \end{array} \right. \end{aligned}$$

  (A.1)

• if $\ell (i)= \ell (j) \le \ell (k)$, then $c_{ij}^k=0$.

Notice also that we can obtain the following relationship from (5.3),

$$\begin{aligned} X_i(\alpha _{\ell (j)}^2) = \frac{\alpha ^2_{\ell (j)}}{2 \alpha _{\ell (i)}^2} X_i(\alpha _{\ell (i)}^2), \qquad \text {if } {\ell (i)} \ne {\ell (j)}. \end{aligned}$$

(A.2)

These formulas permit us to simplify the equations (3.6) and (3.7) which characterize an orbital diffeomorphism. To simplify (3.6), we have to compute $R_j$. For this, we first show that the first divisibility condition holds for our choice of adapted frame (it results directly from the use of (A.1) and (A.2) in the computation of $\vec {h}_1({\mathcal {P}})$). Then we use the following formula (see [23, Lemma 3]),

$$\begin{aligned} \begin{aligned} R_j =&\sum _{i=1}^{m} (1-\delta _{ij})\left( (\alpha _j^2 - \alpha _i^2)c^i_{ji} - \frac{X_j(\alpha _i^2)}{2}\right) u^2_i + \sum _{i=1}^{m}(1-\delta _{ij})\frac{\alpha _i^2}{2 \alpha _j^2}X_i\left( \frac{\alpha _j^4}{\alpha _i^2}\right) u_i u_j\\&+ \sum _{i=1}^{m} \sum _{k=1}^{m} (1-\delta _{ik})(\alpha _j^2 - \alpha _k^2)c^k_{ji}u_i u_k + \alpha _j^2 \sum _{i=1}^{m}\sum _{k=m+1}^{n}c^k_{ji}u_i u_k. \end{aligned} \end{aligned}$$

We substitute the structure coefficients by the expressions shown above and use the property of functions $\beta _\ell ({\bar{x}}_\ell )$ to be constant if $x_\ell$ is of dimension more then one. We get $R_j = \alpha _j^2 \sum _{i=1}^{m} \sum _{k = m+1}^{n} c^k_{ij} u_i u_k.$ We finally obtain a simplified form of (3.6),

$$\begin{aligned} \sum _{k=m+1}^{n}q_{jk}\Phi _k = \frac{\alpha _j^2}{\alpha } \sum _{k=m+1}^{n} q_{jk} u_k \qquad 1 \le j \le m. \end{aligned}$$

To simplify (3.7), it is sufficient to notice that $X^s_i(\alpha ^2_i) = 0$ if $\left| I_s \right| > 1$. Setting $\Phi _i = \frac{\alpha _i^2 u_i}{\alpha }$ for $i = 1 , \dots , m$ as in (3.5), we obtain

$$\begin{aligned} \vec {h}_1 (\Phi _s) = \sum _{k=1}^{n} q_{sk} \Phi _k. \end{aligned}$$

To summarize, there exists an orbital diffeomorphism between $\vec {h}_1$ and $\vec {h}_2$ if the following equations have a solution:

$$\begin{aligned}&\sum _{k=m+1}^{n}q_{jk}\Phi _k = \frac{\alpha _j^2}{\alpha } \sum _{k=m+1}^{n} q_{jk} u_k \qquad 1 \le j \le m,\\&\vec {h}_1(\Phi _s) = \sum _{k=1}^{n}q_{sk} \Phi _k \qquad m+1 \le s \le n. \end{aligned}$$

It appears that $\Phi _k = \frac{\alpha _k^2u_k}{\alpha }$, $k= m+1, \dots , n$, obviously satisfy this system. Thus $\vec {h}_1$ and $\vec {h}_2$ are orbitally diffeomorphic and, by Proposition 3.3, $g_1, g_2$ are projectively equivalent.

In the case of a pair with constant coefficients, all $\alpha _i$ are constant and thus $\vec {h}_1(\alpha ^2) = 0$. Applying again Proposition 3.3, we deduce that the metrics of a Levi-Civita pair with constant coefficients are affinely equivalent. Conversely, if the metrics of a Levi-Civita pair are affinely equivalent, then by Proposition 4.7 all factors $\alpha _i$ are constant, which implies that all $\beta _i$ are constant. Thus the pair has constant coefficients. This ends the proof of Proposition 5.5.

Appendix B. Proof of Proposition 4.6 on quadratic first-integrals

Proposition 4.6 is the generalization to sub-Riemannian metrics of a result stated for Riemannian metrics in [19], namely Corollary 3 of Theorem 1. It is then sufficient to show the following result, which is the exact generalization to the sub-Riemannian case of that Theorem 1 (in the case of polynomials of degree $d=2$).

Proposition B.1

Let D be a Lie-bracket generating distribution on an open ball $B \subset {\mathbb {R}}^n$ and g be a smooth metric on D. Then, for any $\varepsilon >0$ there exists a metric $\widetilde{g}$ on D which is $\varepsilon$-close to g in the $C^\infty$-topology, and $\varepsilon '>0$ such that for any $C^2$ metric $g'$ on D which is $\varepsilon '$-close to $\widetilde{g}$ in the $C^2$-topology, the normal extremal flow of $g'$ does not admit a non-trivial quadratic first-integral (non-trivial means non proportional to the Hamiltonian $h_{g'}$ associated with $g'$).

Note that we work on an open subset of ${\mathbb {R}}^n$ and not in a general manifold since, as noticed in [19], it is sufficient to prove the result locally. Thus we identify $T^*B$ to $B \times {\mathbb {R}}^n$ and we write a covector $\lambda \in T^*B$ as a pair (x, p), where $x=\pi (\lambda )$.

The proof of Theorem 1 in [19] goes as follows. Choose k sets of N points² in B, $S_\ell =\{x_{\ell ,1}, \dots , x_{\ell ,N}\}$, $\ell =1, \dots , k$, where $N=n(n+1)/2$ and k is an integer larger than 4. Then consider the initial covectors associated with all the geodesics joining the points in different sets. The existence of a quadratic first-integral implies strong constraints on these covectors. If the points are in “general” position, small and localized perturbations of the metric along the geodesics make these constraints incompatible, which prevents the existence of a quadratic first-integral.

This argument is very general, it is not specific to Riemannian geometry. It only requires the following assumption on the kN points:

(H.1) :

no three of the points $x_{1,1}, \dots , x_{k,N}$ lie on one normal geodesic;

(H.2) :

for every sets $S_i \ne S_j$ and every point $x \in S_i$, there exists a 2-decisive set (see below) $p_1,\dots ,p_N \in T_x^*B \simeq {\mathbb {R}}^n$ such that

$$\begin{aligned} S_j=\{ \pi \circ e^{\vec {h}_g}(x,p_1),\dots , \pi \circ e^{\vec {h}_g}(x,p_N) \}; \end{aligned}$$

(H.3) :

for every pair of sets $S_i \ne S_j$ and every pair of points $x \in S_i$ and $y \in S_j$, let $p \in T_{x}^*B \simeq {\mathbb {R}}^n$ be the covector such that $y = \pi \circ e^{\vec {h}_g}(x,p)$; then perturbations $\widetilde{g}$ of the metric g localized near one point of the geodesic $\pi \circ e^{t\vec {h}_g}(x,p)$, $t\in (0,1)$, generate a neighborhood of $e^{\vec {h}_{g}}(x,p)$ in $T^*B$, i.e. the map

$$\begin{aligned} \widetilde{g} \mapsto e^{\vec {h}_{\widetilde{g}}}(x,p) \end{aligned}$$

is a submersion at $\widetilde{g}=g$.

As a consequence, if any sub-Riemannian metric g admits kN points satisfying (H.1)–(H.3), then Proposition B.1 can be proved in the same way as [19, Theorem 1]. Thus we are reduced to proving the existence of such sets of points.

Remark B.2

A set of $N=n(n+1)/2$ vectors of ${\mathbb {R}}^n$ is called 2-decisive if the values of any quadratic polynomial on this set determine the polynomial. Clearly, the set of 2-decisive sets is open and dense in the set of N-tuples of vectors of ${\mathbb {R}}^n$.

Let us first study the perturbation property of (H.3). We denote by ${\mathcal {G}}$ the set of sub-Riemannian $C^2$ metrics on D. Locally ${\mathcal {G}}$ can be identified with an open subset of the Banach space ${\mathcal {S}}$ of $C^2$ maps from B to the set of symmetric $(m \times m)$ matrices.

Lemma B.3

Let g be a sub-Riemannian metric and $\lambda _0 \in T^*B$ be an ample covector with respect to g. Then the map

$$\begin{aligned} \psi : \widetilde{g} \in {\mathcal {G}} \mapsto e^{\vec {h}_{\widetilde{g}}}(\lambda _0) \in T^*B \end{aligned}$$

is a submersion at $\widetilde{g}=g$.

Proof

From standard results on the dependance of differential equations with respect to a parameter, the differential of $\psi$ at g can be written as

$$\begin{aligned} D_g \psi : \widetilde{g} \in {\mathcal {S}} \mapsto {e_*^{\vec {h}_g}} \int _0^1 {e_*^{-s \vec {h}_g}} \left( \frac{\partial \vec {h}_g (\lambda (s))}{\partial g} (g) \cdot \widetilde{g}\right) ds, \end{aligned}$$

where $\lambda (s) = e^{s \vec {h}_g} \lambda _0$, $s \in [0,1]$. Now, we can easily verify that, for a given $\lambda \in T^*B$, the image of the partial differential of $\vec {h}_g$ with respect to g is

$$\begin{aligned} \mathrm {Im} \left( \frac{\partial \vec {h}_g (\lambda )}{\partial g} (g)\right) = \left\{ v \in T_{\lambda }(T^*B) \ : \ \pi _* v \in D \right\} = J_\lambda ^{(1)}, \end{aligned}$$

where the last equality comes from Lemma 2.8. As a consequence, the image of the linear map $D_g \psi$ satisfies

$$\begin{aligned} \mathrm {Im} D_g \psi = {e_*^{\vec {h}_g}} \int _0^1 {e_*^{-s \vec {h}_g}} J_{\lambda (s)}^{(1)} ds, \end{aligned}$$

and $\psi$ is a submersion at g if and only if

$$\begin{aligned} \mathrm {span} \left\{ {e_*^{-s \vec {h}_g}} J_{\lambda (s)}^{(1)} \ : \ s \in [0,1] \right\} = T_{\lambda _0}(T^*B). \end{aligned}$$

(B.1)

Assume by contradiction that (B.1) does not hold. Then there exists $p \in T^*_{\lambda _0}(T^*B)$ such that $\langle p, {e_*^{-s \vec {h}_g}} J_{\lambda (s)}^{(1)} \rangle = 0$ for all $s \in [0,1]$. Note that $J_{\lambda _0}(s) \subset {e_*^{-s \vec {h}_g}} J_{\lambda (s)}^{(1)}$ (see Definition 2.5). Hence, for all smooth curve $l(\cdot )$ such that $l(s) \in J_{\lambda _0}(s)$ for all $s \in [0,1]$, we have $\langle p, l(s) \rangle \equiv 0$. Taking the derivatives with respect to s at 0, we get

$$\begin{aligned} \langle p, \frac{d^j}{dt^j}l(0) \rangle = 0 \quad \hbox {for all integer } j. \end{aligned}$$

From Definition 2.2 this implies $\langle p, J_{\lambda _0}^{(k)} \rangle = 0$ for any integer k, which contradicts the fact that $\lambda _0$ is ample. Thus (B.1) holds and $\psi$ is a submersion at g. $\square$

As a direct consequence of this lemma, if (H.2) is satisfied with ample covectors $p_i$, then (H.3) is satisfied as well.

Let x be a point in B and $\exp _x$ be the exponential mapping at x, $\exp _x : p \in T^*_xB \rightarrow \pi \circ e^{\vec {h}_g} (x, p) \in B$. Since conjugate times are isolated from 0 along a geodesic which is ample at $t=0$ (see for instance [1, Cor. 8.47]), for any ample covector p the map $\exp _x$ is locally open near tp for t small enough. Let us denote by ${\mathcal {A}}_x$ the set of N-tuples of ample covectors $(p_1,\dots ,p_N)$ in $(T_x^*B)^N$ which are 2-decisive, and set

$$\begin{aligned} S(x) = \{ (\exp _x(p_1),\dots , \exp _x(p_N) ) \in B^N \ : \ (p_1,\dots ,p_N) \in {\mathcal {A}}_x \}. \end{aligned}$$

By Remark B.2 and Theorem 2.10, the set ${\mathcal {A}}_x$ is open and dense in $(T_x^*B)^N$. It results then from the local openness of the exponential map that S(x) has a non empty interior with $(x, \dots ,x)\in \overline{\mathrm {int}S(x)}$.

We are now in a position to give the construction of sets of N points $S_\ell$, $\ell =1, \dots , k$, satisfying (H.1)–(H.3). The properties above ensure that we can choose $S_1=\{x_{1,1}, \dots , x_{1,N}\} \in B^N$ such that no three points are aligned and such that the intersection

$$\begin{aligned} \bigcap _{i=1}^N \mathrm {int} S(x_{1,i}) \end{aligned}$$

is non empty. We then choose $S_2=\{x_{2,1}, \dots , x_{2,N}\}$ in this intersection such that no three points in $S_1 \cup S_2$ are aligned and such that the intersection of all sets $\mathrm {int} S(x_{1,i}) \cap \mathrm {int} S(x_{2,i})$ is non empty. Iterating this construction we obtain k sets of N points satisfying (H.1)–(H.3). This together with the argument in [19] shows Proposition B.1 and then Proposition 4.6.