Riemannian Geometry - Introduction

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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