Promotion Talk by Igor Zelenko
Date: September 2, 2022
Time: 4:00PM - 5:00PM
Location: Bloc 117
Description: Title: Projective and affine equivalence of sub-Riemannian metrics
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. The talk is based on joint works with Frederic Jean, Sofya
Maslovskaya (my former Ph.D. student), Zaifeng Lin (my current Ph.D.
student), and Andrew Castillo (my former Master's student).