The Schwarzschild metric, its Reissner-Nordstrom-de Sitter generalizations to higher dimensions, and some further generalizations all share the feature that gtt grr = −1 in Schwarzschild-like coordinates. In this pedagogical note we trace this feature to the condition that the Ricci tensor (and stress-energy tensor in a solution to Einstein’s equation) has vanishing radial null-null component, ...

The object of the present paper is to study Z-symmetric manifold with projective curvature tensor. At first, we case Z-tensor and Ricci tensor being Codazzi type. Next, consider recurrent We also divergence-free Z-tensor. Finally, construct an example

We show that the recently found anti-de Sitter (AdS)-plane and AdS-spherical wave solutions of quadratic curvature gravity also solve the most general higher derivative theory in D dimensions. More generally, we show that the field equations of such theories reduce to an equation linear in the Ricci tensor for Kerr-Schild spacetimes having type-N Weyl and type-N traceless Ricci tensors.

We use the framework used by Bakry and Emery in their work on logarithmic Sobolev inequalities to define a notion of coarse Ricci curvature on smooth metric measure spaces alternative to the notion proposed by Y. Ollivier. This function can be used to recover the Ricci tensor on smooth Riemannian manifolds by the formula Ric(γ′ (0) , γ′ (0)) = 1 2 d ds2 Ric4g (x, γ (s)) for any curve γ(s).

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We show that the splitting feature of the Einstein tensor, as the first term of the Lovelock tensor, into two parts (the Ricci tensor and the term proportional to the curvature scalar) with the trace relation between them is a common feature of any other homogeneous terms in the Lovelock tensor. Motivated by the principle of general invariance, we find that this property can be generalized, wit...

In a Riemannian manifold with smooth positive function that weights the associated Hausdorff measures we study stable sets, i.e., second order minima of weighted perimeter under variations preserving volume. By assuming local convexity boundary and certain behavior Bakry–Émery–Ricci tensor deduce rigidity properties for sets by using deformations constructed from parallel vector fields tangent ...

Abstract In this article, we investigate perfect fluid spacetimes equipped with concircular vector field. At first, in a spacetime admitting field, prove that the velocity field annihilates conformal curvature tensor. addition, dimension 4, show is generalized Robertson–Walker Einstein fibre. It proved if furnished admits second order symmetric parallel tensor P , then either equation of state ...

We try to provide a visual introduction to some objects used in Riemannian geometry: parallel transport, sectional curvature, Ricci curvature, Bianchi identities... We then explain some of the strategies used to define analogues of curvature in non-smooth or discrete spaces, beginning with Alexandrov curvature and δ-hyperbolic spaces, and insisting on various notions of generalized Ricci curvat...

It is proved that a compact Kähler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kähler-Einstein manifolds. A stronger result is established for the case of Kähler surfaces. Irreducible Kähler manifolds with two distinct constant eigenvalues of the Ricci tensor are shown to exist in various situations: there are homogeneous examples...