Riemannian geometry was first put forward in generality by Bernhard Riemann in the 19th century. It deals with a broad range of geometries whose metric properties vary from point to point, including the standard types of non-Euclidean 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. Other generalizations of Riemannian geometry include Finsler geometry.

There exists a close analogy of differential geometry with the mathematical structure of defects in regular crystals. Dislocations and disclinations produce torsions and curvature.

The following articles provide some useful introductory material:

[Metric tensor](https://en.wikipedia.org/wiki/Metric_tensor)
[Riemannian manifold](https://en.wikipedia.org/wiki/Riemannian_manifold)
[Levi-Civita connection](https://en.wikipedia.org/wiki/Levi-Civita_connection)
[Curvature](https://en.wikipedia.org/wiki/Curvature)
[Riemann curvature tensor](https://en.wikipedia.org/wiki/Riemann_curvature_tensor)
[List of differential geometry topics](https://en.wikipedia.org/wiki/List_of_differential_geometry_topics)
[Glossary of Riemannian and metric geometry](https://en.wikipedia.org/wiki/Glossary_of_Riemannian_and_metric_geometry)
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