# Ricci tensor

(redirected from Ricci curvature)
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## Ricci tensor

[′rē‚chē ‚ten·sər]
(mathematics)
Mentioned in ?
References in periodicals archive ?
When the Ricci curvature is non-negative, says Klartag, log-concave measure are obtained when conditioning the Riemannian volume measure with respect to a geodesic foliation that is orthogonal to the level of sets of a Lipschitz function.
1) If the transversal Ricci curvature of F is positive-definite, then any [L.
Key words: Ricci curvature tensors, Einstein Field Equations, Black hole, Vacuum Solutions
Li, Manifolds with Nonnegative Ricci Curvature and Mean Convex Boundary, arXiv: 1204.
where R and S are respectively Riemann and Ricci curvature tensors.
On the other hand, if one chooses to sample the manifold using Ricci curvature, as proposed in , , one needs to encode just one additional, number, namely the Ricci curvature Ric (or, rather, its absolute value).
Chen established a sharp relationship between the Ricci curvature (an intrinsic invariant) and the squared mean curvature (the main extrinsic invariant) for any n-dimensional Riemannian submanifold of a real space form [?
The problem of finding the metric from the Ricci curvature has been studied by DeTurk , ,  and , and from the curvature in general relativity by Hall  and Hall and McIntosh .
Other subjects are local Gromov-Witten invariants of cubic surfaces, flop invariance of the topological vertex, Lagrangian fibrations and theta functions, and ends of metric measure spaces with non-negative Ricci curvature.
Next we observe that the evolution of the connection is given by the first covariant derivative of the Ricci curvature, so that there is a formula of the form
for any X, Y [member of] [GAMMA](TM), where S, Q and [GAMMA](TM) denote the Ricci curvature tensor, the Ricci operator with respect to the metric g and the Lie algebra of all vector fields on [M.
Now, precisely because there is only one purely geometric integrand here, namely the Ricci curvature scalar R (apart from the metric volume term [square root of -g]).

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