Properties
As a consequence of the Bianchi identities, the Ricci tensor of a Riemannian manifold is symmetric, in the sense that
It thus follows that the Ricci tensor is completely determined by knowing the quantity for all vectors of unit length. This function on the set of unit tangent vectors is often simply called the Ricci curvature, since knowing it is equivalent to knowing the Ricci curvature tensor.
The Ricci curvature is determined by the sectional curvatures of a Riemannian manifold, but generally contains less information. Indeed, if is a vector of unit length on a Riemannian n-manifold, then Ric(ξ,ξ) is precisely (n−1) times the average value of the sectional curvature, taken over all the 2-planes containing . There is an (n−2)-dimensional family of such 2-planes, and so only in dimensions 2 and 3 does the Ricci tensor determine the full curvature tensor. A notable exception is when the manifold is given a priori as a hypersurface of Euclidean space. The second fundamental form, which determines the full curvature via the Gauss–Codazzi equation, is itself determined by the Ricci tensor and the principal directions of the hypersurface are also the eigendirections of the Ricci tensor. The tensor was introduced by Ricci for this reason.
If the Ricci curvature function Ric(ξ,ξ) is constant on the set of unit tangent vectors ξ, the Riemannian manifold is said to have constant Ricci curvature, or to be an Einstein manifold. This happens if and only if the Ricci tensor Ric is a constant multiple of the metric tensor g.
The Ricci curvature is usefully thought of as a multiple of the Laplacian of the metric tensor (Chow & Knopf 2004, Lemma 3.32). Specifically, if xi are harmonic local coordinates, then
where Δ is the Laplace–Beltrami operator regarded here as acting on the functions gij. This fact motivates, for instance, the introduction of the Ricci flow equation as a natural extension of the heat equation for the metric. Alternatively, in a normal coordinate system based at p, at the point p
Read more about this topic: Ricci Curvature
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