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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.
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.
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]
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.
Berger spheres, constructed as left-invariant metrics on the special unitary group SU(2), are among the simplest examples of the collapsing phenomena, in which a simply-connected Riemannian manifold can have small volume without having large curvature. [38] They also give an example of a Riemannian metric which has constant scalar curvature but ...
Curvature of general surfaces was first studied by Euler. In 1760 [4] he proved a formula for the curvature of a plane section of a surface and in 1771 [5] he considered surfaces represented in a parametric form. Monge laid down the foundations of their theory in his classical memoir L'application de l'analyse à la géometrie which appeared in ...
Here is the Ricci curvature tensor and represents the Lie derivative. If there exists a function f : M → R {\displaystyle f:M\rightarrow \mathbb {R} } such that V = ∇ f {\displaystyle V=\nabla f} we call ( M , g ) {\displaystyle (M,g)} a gradient Ricci soliton and the soliton equation becomes
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 ...