Differential geometry of singular spaces

点击翻译按钮获取中文摘要

The goal of this project is to study the differential geometry of singular stratified spaces embedded in a smooth Riemannian manifold. The ambient metric induces a Riemannian metric on each stratum of the singular space, and among the objects of interest are geodesics, curvature and distance function on the singular space, especially their asymptotic behavior near the boundary of each stratum. Also, we will define and study the exponential map based at a singular point (i.e. a point in a non-maximal stratum) and the volume asymptotics of small balls centered at singular points. The cases of conical and cuspidal singularities are quite well-understood, and our goal is to extend these results to semi-algebraic surfaces and to spaces with polyhedral singularities.Our approach is based on resolutions of the singularities, combined with modern methods of singular analysis applied to the Hamiltonian system describing the geodesic flow. As shown in preliminary work, this method can be expected to yield full asymptotic information on the desired differential geometric quantities.