Differential geometry of singular spaces
Final Report Abstract
In this project we investigate differential geometric properties of spaces with conical and cuspidal singularities. The first part of the investigation clarifies the intrinsic structures on the rescaled cotangent bundle on manifolds with boundary which arises from resolving such singularities by blow-up. We identify a number of intrinsically defined differential operators on these bundles, defined at the boundary. These allow us to identify natural classes of degenerate metrics on manifolds with boundary generalizing those arising from pull-back of metrics on the singular space that are restrictions of smooth metrics on an ambient manifold in which the singular space is embedded. In the second part we investigate the geodesic flow for such metrics, which allows us to draw conclusions on the existence and properties of an exponential map based at the singularity. We also find the precise asymptotic properties of the Levi-Civitá connection and Riemann curvature tensor for these metrics. We also consider the geodesic flow on generalized conic metrics and show that it behaves similarly to the geodesic flow for cusp metrics.
