Ricci decomposition

About: Ricci decomposition is a research topic. Over the lifetime, 1972 publications have been published within this topic receiving 45295 citations.

The entropy formula for linear heat equation

Abstract: We derive the entropy formula for the linear heat equation on general Riemannian manifolds and prove that it is monotone non-increasing on manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman ’s recent results on volume non-collapsing for Ricci flow on compact manifolds. We also prove that if the entropy for the heat kernel achieves its maximum value zero at some positive time, on any complete Riamannian manifold with nonnegative Ricci curvature, if and only if the manifold is isometric to the Euclidean space.

The Classification of Spaces Defining Gravitational Fields

Abstract: In this paper written in 1954 Alexei Petrov describes his famous classification of spaces according to the algebraical structure of the curvature tensor, that determines the classes of the gravitational fields permitted therein. Now this classification of spaces (and, respectively, of the gravitational fields) is known as Petrov’s classification. This paper was originally published, in Russian, in Scientific Transactions of Kazan State University: Petrov A. Z. Klassifikazija prostranstv, opredelajuschikh polja tjagotenia. Uchenye Zapiski Kazanskogo Gosudarstvennogo Universiteta, 1954, vol. 114, book 8, pages 55–69. Translated from Russian in 2008 by Vladimir Yershov, England–Pulkovo. In this paper, the detailed proof of results obtained and published by the author earlier in 1951 [1]. Namely, it is shown that by examining the algebraic structure of the curvature tensor V4 one can establish a classification of the gravitational fields defined by this tensor and given in the form ds = gij dx dx , (1) with the fundamental tensor satisfying the field equations

Isotropic singularities in cosmological models

Abstract: Motivated by the ideas of quiescent cosmology and Penrose's Weyl tensor hypothesis concerning the 'big bang', the authors give a geometric (and hence coordinate-independent) definition of the concept of 'isotropic singularity' in a spacetime. The definition generalises previous work on 'quasi-isotropic' and 'Friedman-like' singularities. They discuss simple consequences of the definition. In particular it is shown that an isotropic singularity is a scalar polynomial curvature singularity at which the Weyl tensor is dominated by the Ricci tensor. Finally they impose the Einstein field equations with irrotational perfect fluid source. This enables them to give a detailed description of the geometric structure of an isotropic singularity.

Ricci curvature of finite Markov chains via convexity of the entropy

Abstract: We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy. Using this notion of Ricci curvature we prove discrete analogues of fundamental results by Bakry--Emery and Otto--Villani. Furthermore we show that Ricci curvature bounds are preserved under tensorisation. As a special case we obtain the sharp Ricci curvature lower bound for the discrete hypercube.