Differential Geometry

This group gathers theorems from differential geometry, the study of curved surfaces and manifolds using the tools of calculus. It covers the intrinsic curvature results of Gauss and Gauss-Bonnet that measure how a surface bends independently of how it sits in space, the theory of geodesics and comparison theorems, including the Hopf-Rinow, Bonnet-Myers, Cartan-Hadamard, and Rauch comparison theorems, that describe shortest paths on curved spaces, and the Nash embedding theorem showing that every such abstract curved space can be realized concretely inside ordinary Euclidean space.