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Broadly, one could analogize the role of the Ricci curvature in Riemannian geometry to that of the Laplacian in the analysis of functions; in this analogy, the Riemann curvature tensor, of which the Ricci curvature is a natural by-product, would correspond to the full matrix of second derivatives of a function.
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:
With this convention, the Ricci tensor is a (0,2)-tensor field defined by R jk =g il R ijkl and the scalar curvature is defined by R=g jk R jk. (Note that this is the less common sign convention for the Ricci tensor; it is more standard to define it by contracting either the first and third or the second and fourth indices, which yields a Ricci ...
This is the classical method used by Ricci and Levi-Civita to obtain an expression for the Riemann curvature tensor. [7] This identity can be generalized to get the commutators for two covariant derivatives of arbitrary tensors as follows [ 8 ]
Given a Riemannian metric g, the scalar curvature Scal is defined as the trace of the Ricci curvature tensor with respect to the metric: [1] = . The scalar curvature cannot be computed directly from the Ricci curvature since the latter is a (0,2)-tensor field; the metric must be used to raise an index to obtain a (1,1)-tensor field in order to take the trace.
In general relativity and tensor calculus, the contracted Bianchi identities are: [1] = where is the Ricci tensor, the scalar curvature, and indicates covariant differentiation.
The Einstein tensor is a tensor of order 2 defined over pseudo-Riemannian manifolds.In index-free notation it is defined as =, where is the Ricci tensor, is the metric tensor and is the scalar curvature, which is computed as the trace of the Ricci tensor by = .
The Ricci curvature tensor plays a defining role in the theory of Einstein manifolds, which has applications to the study of gravity. A (pseudo-)Riemannian metric is called an Einstein metric if Einstein's equation = for some constant