Search results
Results from the WOW.Com Content Network
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 Weyl tensor has the same basic symmetries as the Riemann tensor, but its 'analogue' of the Ricci tensor is zero: = = = = The Ricci tensor, the Einstein tensor, and the traceless Ricci tensor are symmetric 2-tensors:
The generator is recognized as a ... is the Riemann curvature tensor, ... this group of symmetries has the expected dimension (as a Lie group). Heuristically, we can ...
In mathematics, a Killing tensor or Killing tensor field is a generalization of a Killing vector, for symmetric tensor fields instead of just vector fields. It is a concept in Riemannian and pseudo-Riemannian geometry, and is mainly used in the theory of general relativity. Killing tensors satisfy an equation similar to Killing's equation for ...
The other two 1-forms in the Cartan structural equations are given by θ 1 = β and θ 2 = γ. The structural equations themselves are just the Maurer–Cartan equations. In other words; The Cartan structural equations for SO(3)/SO(2) reduce to the Maurer–Cartan equations for the left invariant 1-forms on SO(3). Since α is the connection form,
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.
Every compact Riemann surface of genus greater than 1 (with its usual metric of constant curvature −1) is a locally symmetric space but not a symmetric space. Every lens space is locally symmetric but not symmetric, with the exception of L ( 2 , 1 ) {\displaystyle L(2,1)} , which is symmetric.
The Riemann curvature tensor measures precisely the extent to which parallel transporting vectors around a small rectangle is not the identity map. [28] The Riemann curvature tensor is 0 at every point if and only if the manifold is locally isometric to Euclidean space. [29] Fix a connection on .