Geometry Seminar

Date: October 2, 2020

Time: 4:00PM - 5:00PM

Location: zoom

Speaker: David, Sykes, TAMU

Title: Local equivalence problems for 2-nondegenerate, hypersurface-type CR geometry studied via dynamical Legendrian contact structures.

Abstract: The local differential geometry of Levi-nondegenerate CR structures is well understood due in large part to classical results of Cartan, Tanaka, Chern, and Moser, and yet comparatively little is known about other CR structures. There is a natural association between 2-nondegenerate, hypersurface-type CR structures – which are the main focus of this talk – and dynamical Legendrian contact structures, and, moreover, there is a broad class of these CR structures that can be uniquely recovered from their associated dynamical Legendrian contact structure. For these recoverable structures, we construct canonical absolute parallelisms on fiber bundles defined over a manifold with the given CR structure. The construction can be applied to discern local equivalence between CR structures. Other applications that will be discussed include upper bounds for the dimension of a CR manifold’s symmetry group and a characterization of local invariants of certain homogeneous CR manifolds. The latter application, coupled with results by Curtis Porter and Igor Zelenko, enables us to classify the local geometry of homogeneous, 2-nondegenerate, hypersurface-type CR manifolds in low dimensions.