Using symmetry arguments only, we show that every spacetime with mirror-symmetric spatial sections is necessarily conformally flat. The general form of the Ricci tensor of such spacetimes is also determined. 1. Introduction. It is well known that the curvature tensor of any four-dimensional differentiable manifold has only 20 algebraically independent components. Ten out of these 20 components ...

We study Ricci solitons in Lorentzian α-Sasakian manifolds. It is shown that a symmetric parallel second order covariant tensor in a Lorentzian α-Sasakian manifold is a constant multiple of the metric tensor. Using this it is shown that if LV g + 2S is parallel, V is a given vector field then (g, V ) is Ricci soliton. Further, by virtue of this result Ricci solitons for (2n + 1)-dimensional Lor...

We study a well-known scalar quantity in Riemannian geometry, the Ricci scalar, in the context of diffusion tensor imaging (DTI), which is an emerging non-invasive medical imaging modality. We derive a physical interpretation for the Ricci scalar and explore experimentally its significance in DTI. We also extend the definition of the Ricci scalar to the case of high angular resolution diffusion...

Our goal in this paper is to obtain further information about the curvature of gradient shrinking Ricci solitons. This is important for a better understanding and ultimately for the classification of these manifolds. The classification of gradient shrinkers is known in dimensions 2 and 3, and assuming locally conformally flatness, in all dimensions n ≥ 4 (see [14, 13, 6, 15, 20, 12, 2]). Many o...

We obtain the evolution equations for the Riemann tensor, the Ricci tensor and the scalar curvature induced by the mean curvature flow. The evolution for the scalar curvature is similar to the Ricci flow, however, negative, rather than positive, curvature is preserved. Our results are valid in any dimension.

We prove that for a solution (M, g(t)), t ∈ [0, T ), where T < ∞, to the Ricci flow on a complete non-compact Riemannian manifold with the Ricci curvature tensor uniformly bounded by some constant C on M × [0, T ), the curvature tensor stays uniformly bounded on M × [0, T ).

Searching for the dynamical foundations of Havrda-Charvát/Daróczy/ Cressie-Read/Tsallis non-additive entropy, we come across a covariant quantity called, alternatively, a generalized Ricci curvature, an N -Ricci curvature or a Bakry-Émery-Ricci curvature in the configuration/phase space of a system. We explore some of the implications of this tensor and its associated curvature and present a co...

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...