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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.
Notions of Ricci curvature on discrete manifolds have been defined on graphs and networks, where they quantify local divergence properties of edges. Ollivier's Ricci curvature is defined using optimal transport theory. [4] A different (and earlier) notion, Forman's Ricci curvature, is based on topological arguments. [5]
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.
Although individually, the Weyl tensor and Ricci tensor do not in general determine the full curvature tensor, the Riemann curvature tensor can be decomposed into a Weyl part and a Ricci part. This decomposition is known as the Ricci decomposition, and plays an important role in the conformal geometry of Riemannian manifolds.
This formula is often called the Ricci identity. [6] 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 ]
This implies that the Ricci curvature is given by R jk = (n – 1)κg jk and the scalar curvature is n(n – 1)κ, where n is the dimension of the manifold. In particular, every Riemannian manifold of constant curvature is an Einstein manifold, thereby having constant scalar curvature.
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 ...
Here R denotes scalar curvature. This is called the normalized Ricci flow equation. Thus, with an explicitly defined change of scale Ψ and a reparametrization of the parameter values, a Ricci flow can be converted into a normalized Ricci flow. The converse also holds, by reversing the above calculations.