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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.[1][2]