| Abstract: |
| Sub-Riemannian metrics on a manifold are defined by a distribution (a subbundle of the tangent bundle) together with a Euclidean structure on each fiber. The Riemannian metrics correspond to the case when the distribution is the whole tangent bundle. Two sub-Riemannian metrics are called projectively equivalent if they have the same geodesics up to a reparameterization and affinely equivalent if they have the same geodesics up to affine reparameterization. In the Riemannian case, both equivalence problems are classical: local classifications of projectively and affinely equivalent Riemannian metrics were established by Levi-Civita in 1898 and Eisenhart in 1923, respectively. In particular, a Riemannian metric admitting a nontrivial (i.e. non-constant proportional) affinely equivalent metric must be a product of two Riemannian metrics i.e. separation of variables (the de Rham decomposition) occur, while for the analogous property in the projective equivalence case, a more involved (``twisted) product structure is necessary. The latter is also related to the existence of commuting nontrivial integrals quadratic with respect to velocities for the corresponding geodesic flow. We will describe the recent progress toward the generalization of these classical results to sub-Riemannian metrics. In particular, we will discuss the genericity of metrics that do not admit non-constantly proportional affinely/projectively equivalent metrics on a given distribution (such metrics are called projectively/ affinely rigid) and the genericity of distributions for which all sub-Riemannian metrics are projective/affinely rigid. |
|