Curvature Tensor

In differential geometry, the term curvature tensor may refer to:

  • the Riemann curvature tensor of a Riemannian manifold — see also Curvature of Riemannian manifolds;
  • the curvature of an affine connection or covariant derivative (on tensors);
  • the curvature form of an Ehresmann connection: see Ehresmann connection, connection (principal bundle) or connection (vector bundle).

Other articles related to "tensors, tensor, curvature tensor, curvature":

Ricci Decomposition - Mathematical Definition
... is the decomposition of the space of all tensors having the symmetries of the Riemann tensor into its irreducible representations for the action of the orthogonal group (Besse 1987, Chapter 1, §G) ... V be an n-dimensional vector space, equipped with a metric tensor (of possibly mixed signature) ... cotangent space at a point, so that a curvature tensor R (with all indices lowered) is an element of the tensor product V⊗V⊗V⊗V ...
Curvature Of Riemannian Manifolds - Ways To Express The Curvature of A Riemannian Manifold - The Riemann Curvature Tensor - Symmetries and Identities
... The curvature tensor has the following symmetries The last identity was discovered by Ricci, but is often called the first Bianchi identity, just because it looks similar to the Bianchi identity below ... All three together should be named pseudo-orthogonal curvature structure ... They give rise to a tensor only by identifications with objects of the tensor algebra - but likewise there are identifications with concepts in the Clifford-algebra ...
Theoretical Motivation For General Relativity - Geodesic Equation For Circular Orbits - Ricci Curvature Tensor and Trace
... The Ricci curvature tensor is a special curvature tensor given by the contraction ... The trace of the Ricci tensor, called the scalar curvature, is ...