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Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an inner product on the tangent space at each point that varies smoothly from point to point). This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.

Elliptic geometry is also sometimes called "Riemannian geometry".

Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "Ueber die Hypothesen, welche der Geometrie zu Grunde liegen" ("On the Hypotheses on which Geometry is Based").[1] It is a very broad and abstract generalization of the differential geometry of surfaces in R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. It enabled the formulation of Einstein's general theory of relativity, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology.

Metric tensor

Riemannian manifold

Levi-Civita connection

Curvature

Riemann curvature tensor

List of differential geometry topics

Glossary of Riemannian and metric geometry

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.[2][3]


The following articles provide some useful introductory material:

Shape of the universe

Introduction to the mathematics of general relativity

Normal coordinates

Systolic geometry

in Einstein–Cartan theory (motivation)

Riemann–Cartan geometry

Riemann's minimal surface

Reilly formula

(2000), Riemannian Geometry During the Second Half of the Twentieth Century, University Lecture Series, vol. 17, Rhode Island: American Mathematical Society, ISBN 0-8218-2052-4. (Provides a historical review and survey, including hundreds of references.)

Berger, Marcel

; Ebin, David G. (2008), Comparison theorems in Riemannian geometry, Providence, RI: AMS Chelsea Publishing; Revised reprint of the 1975 original.

Cheeger, Jeff

Gallot, Sylvestre; ; Lafontaine, Jacques (2004), Riemannian geometry, Universitext (3rd ed.), Berlin: Springer-Verlag.

Hulin, Dominique

Jost, Jürgen (2002), Riemannian Geometry and Geometric Analysis, Berlin: Springer-Verlag,  3-540-42627-2.

ISBN

Petersen, Peter (2006), Riemannian Geometry, Berlin: Springer-Verlag,  0-387-98212-4

ISBN

by V. A. Toponogov at the Encyclopedia of Mathematics

Riemannian geometry

, "Riemannian Geometry", MathWorld

Weisstein, Eric W.