Introduction
Riemannian geometry was first put forward in generality by Bernhard Riemann in the nineteenth century. It deals with a broad range of geometries whose metric properties vary from point to point, as well as two standard types of Non-Euclidean geometry and hyperbolic geometry, as well as Euclidean geometry itself.
Any 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
- Riemannian manifold
- Levi-Civita connection
- Curvature
- Curvature tensor
- List of differential geometry topics
- Glossary of Riemannian and metric geometry
Read more about this topic: Riemannian Geometry
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