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In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field).
The three identities form a complete list of symmetries of the curvature tensor, i.e. given any tensor that satisfies the identities above, one could find a Riemannian manifold with such a curvature tensor at some point. Simple calculations show that such a tensor has / independent components.
The variation formula computations above define the principal symbol of the mapping which sends a pseudo-Riemannian metric to its Riemann tensor, Ricci tensor, or scalar curvature.
For example, for the tangent bundle of a Riemannian manifold, the structure group is O(n) and Ω is a 2-form with values in the Lie algebra of O(n), i.e. the antisymmetric matrices. In this case the form Ω is an alternative description of the curvature tensor, i.e. (,) = (,),
Since any Riemannian metric is parallel with respect to its Levi-Civita connection, this shows that the Riemann tensor of any constant-curvature space is also parallel. The Ricci tensor is then given by Ric = ( n − 1 ) κ g {\displaystyle \operatorname {Ric} =(n-1)\kappa g} and the scalar curvature is n ( n − 1 ) κ . {\displaystyle n(n-1 ...
A Riemannian space form is a Riemannian manifold with constant curvature which is additionally connected and geodesically complete. A Riemannian space form is said to be a spherical space form if the curvature is positive, a Euclidean space form if the curvature is zero, and a hyperbolic space form or hyperbolic manifold if
A four-dimensional closed and oriented manifold supporting any Einstein Riemannian metric must satisfy the Hitchin–Thorpe inequality on its topological data. As particular cases of well-known theorems on Riemannian manifolds of nonnegative Ricci curvature, any manifold with a complete Ricci-flat Riemannian metric must: [12]
Riemannian circle; Riemannian submersion; Riemannian Penrose inequality; Riemannian holonomy; Riemann curvature tensor; Riemannian connection. Riemannian connection on a surface; Riemannian symmetric space; Riemannian volume form; Riemannian bundle metric; List of topics named after Bernhard Riemann; but may also refer to Hugo Riemann: Neo ...