Mathematical Sciences: Geometry and Topology of Riemannian Manifolds

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9626375 Grove The project deals with global problems in Riemannian geometry. The investigator intends to study approximations of Riemannian manifolds by singular spaces such as Alexandrov spaces and finite metric spaces; to study stability and inverse stability problems for Riemannian manifolds; to investigate the relationship between curvature and symmetry of Riemannian manifolds via the geometry of orbits; to find examples of manifolds with large isometry groups and with positive curvature. Riemannian manifolds are an abstraction of curved spaces possessing a distance function. Unlike curves and surfaces in 3-space these manifolds do not necessarily sit inside an ambient space, hence they are abstract spaces. The proposed research aims to understand various global aspects of these manifolds, in particular, the investigator is interested in approximations of Riemannian manifolds by finite metric spaces and in symmetry properties. Finite metric spaces are closely related to graphs so that the project may impact on discrete applications of Riemannian geometry.