Riemannian geometry

Every smooth manifold admits a Riemannian metric, which often helps to solve problems of differential topology.

It also serves as an entry level for the more complicated structure of pseudo-Riemannian manifolds, which (in four dimensions) are the main objects of the theory of general relativity.

There exists a close analogy of differential geometry with the mathematical structure of defects in regular crystals.

[2][3] The following articles provide some useful introductory material: What follows is an incomplete list of the most classical theorems in Riemannian geometry.

In all of the following theorems we assume some local behavior of the space (usually formulated using curvature assumption) to derive some information about the global structure of the space, including either some information on the topological type of the manifold or on the behavior of points at "sufficiently large" distances.