In this paper, we construct smooth forward Ricci flow evolutions of singular initial metrics resulting from rotationally symmetric neckpinches on Sn+1, without performing an intervening surgery. In the restrictive context of rotational symmetry, this construction gives evidence in favor of Perelman’s hope for a “canonically defined Ricci flow through singularities”.

The object of the present paper is to study three-dimensional Lorentzian -Sasakian manifolds which are Ricci-semisymmetry, locally symmetric and have -parallel Ricci tensor. An example of a three-dimensional Lorentzian -Sasakian manifold is given which verifies all the Theorems.

We exhibit Walker manifolds of signature (2, 2) with various commutativity properties for the Ricci operator, the skew-symmetric curvature operator, and the Jacobi operator. If the Walker metric is a Riemannian extension of an underlying affine structure A, these properties are related to the Ricci tensor of A.

The field equations associated with the Born-Infeld-Einstein action are derived using the Palatini variational technique. In this approach the metric and connection are varied independently and the Ricci tensor is generally not symmetric. For sufficiently small curvatures the resulting field equations can be divided into two sets. One set, involving the antisymmetric part of the Ricci tensor Rμ...

A Riemannian manifold is called harmonic if its volume density function expressed in polar coordinates centered at any point of the manifold is radial. Flat and rank-one symmetric spaces are harmonic. The converse (the Lichnerowicz Conjecture) is true for manifolds of nonnegative scalar curvature and for some other classes of manifolds, but is not true in general: there exists a family of homog...

We exhibit several families of Jacobi–Videv pseudo-Riemannian manifolds which are not Einstein. We also exhibit Jacobi–Videv algebraic curvature tensors where the Ricci operator defines an almost complex structure.

The object of the present paper is to study weakly projective symmetric manifolds and its decomposability with several non-trivial examples. Among others it is shown that in a decomposable weakly projective symmetric manifold both the decompositions are weakly Ricci symmetric.