[Paper Review] On projective and affine equivalence of sub-Riemannian metrics

Frédéric Jean, Sofya Maslovskaya+1 more|arXiv (Cornell University)|Jan 12, 2018

Geometric Analysis and Curvature Flows21 references3 citations

TL;DR

This paper establishes that non-conformally rigid sub-Riemannian metrics admit at least one nontrivial quadratic integral on the cotangent bundle fibers and induce a product structure in their nilpotent approximation. It proves generic sub-Riemannian metrics are conformally rigid under projective equivalence, and under affine equivalence, they are rigid, extending classical Riemannian results to sub-Riemannian geometry via geometric control and Jacobi curve analysis.

ABSTRACT

Consider a smooth 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 and affinely equivalent metrics is classical (Levi-Civita, Eisenhart). In particular, a Riemannian metric which is not rigid satisfies the following two special properties: its geodesic flow possesses nontrivial integrals and the metric induces certain canonical product structure on the ambient manifold. These classification results were extended to contact and quasi-contact distributions by Zelenko. Our general goal is to extend these results to arbitrary sub-Riemannian manifolds, and we establish two types of results toward this goal: if a sub-Riemannian metric is not projectively conformally rigid, 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: first, we show that a generic sub-Riemannian metric on a fixed pair $(M,D)$ is projectively conformally rigid. Second, we prove that, except for special pairs $(m,n)$, every sub-Riemannian metric on a rank $m$ generic distribution in an $n$-dimensional manifold is projectively conformally rigid. For the affine equivalence in both genericity results conformal rigidity can be replaced by usual rigidity.

Motivation & Objective

• To extend classical Riemannian results on projective and affine equivalence to sub-Riemannian geometry.
• To characterize non-rigid sub-Riemannian metrics by identifying geometric and algebraic obstructions.
• To establish genericity results for conformal and standard rigidity in sub-Riemannian settings.
• To analyze the structure of Jacobi curves and their role in determining equivalence classes of sub-Riemannian metrics.
• To prove that non-conformally rigid metrics must have nontrivial quadratic first integrals and decomposable nilpotent approximations.

Proposed method

• Analyzes geodesics and normal extremals using the Pontryagin Maximum Principle and Jacobi curves.
• Applies orbital diffeomorphism techniques to study projective and affine equivalence of sub-Riemannian metrics.
• Uses the fundamental algebraic system to derive conditions for equivalence and integrability.
• Employs nilpotent approximation to study local structure of distributions and their equivalence classes.
• Applies submersion arguments to show openness and density of ample covectors and 2-decisive sets.
• Utilizes exponential maps and perturbation theory to construct finite sets of points satisfying geometric and algebraic conditions (H.1)–(H.3).

Experimental results

Research questions

• RQ1 Under what conditions are two sub-Riemannian metrics projectively equivalent?
• RQ2 What geometric and algebraic structures must a non-conformally rigid sub-Riemannian metric possess?
• RQ3 How does the existence of quadratic first integrals relate to the structure of the nilpotent approximation?
• RQ4 What is the generic behavior of sub-Riemannian metrics with respect to projective and affine equivalence?
• RQ5 Can the rigidity of sub-Riemannian metrics be characterized in terms of the decomposability of their nilpotent approximations?

Key findings

• A sub-Riemannian metric that is not conformally rigid with respect to projective equivalence admits at least one nontrivial quadratic integral on the fibers of the cotangent bundle.
• The nilpotent approximation of any non-conformally rigid sub-Riemannian metric at any point admits a canonical product structure.
• For a fixed distribution on a manifold, a generic sub-Riemannian metric is conformally rigid under projective equivalence.
• For generic distributions of rank $ m $ on an $ n $-dimensional manifold, excluding special $ (m,n) $ pairs, every sub-Riemannian metric is conformally rigid under projective equivalence.
• Under affine equivalence, conformal rigidity can be strengthened to standard rigidity in the generic settings.
• The existence of $ N = n(n+1)/2 $ 2-decisive points ensures the validity of key propositions on quadratic first integrals and equivalence.