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 [41], [43], 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 [2], [3], [4] and [5], and from the curvature in general relativity by Hall [6] and Hall and McIntosh [7].

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]).