%0 Journal Article
%T Metric Ricci Curvature for Manifolds
%A David Xianfeng Gu
%A Emil Saucan
%J Geometry
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/694169
%X We introduce a metric notion of Ricci curvature for manifolds and study its convergence properties. We also prove a fitting version of the Bonnet-Myers theorem, for surfaces as well as for a large class of higher dimensional manifolds. 1. Introduction Recent years have seen a great ¡°revival¡± of Ricci curvature, mainly due to not only Perelman¡¯s celebrated work on the Ricci flow and the Poincare conjecture [1, 2], but also to its extension to a far larger class of geometric objects than merely smooth 3-manifolds (see [3] and the bibliography therein). In consequence, Ricci curvature has become an object of interest and study in graphics and imaging. The approaches range from the implementations of the combinatorial Ricci curvature of Chow and Luo [4]¡ªsee, for example, [5], through¡ªthe classical approximation methods of smooth differential operators [6, 7] to discrete, purely combinatorial methods [8, 9]. We have addressed the problem of Ricci curvature of surfaces and higher dimensional (piecewise flat) manifolds, from a metric point of fview, both as a tool in studying the combinatorial Ricci flow on surfaces [10, 11] and, in a more general context, in the approximation in secant of curvature measures of manifolds and their applications [12, 13]. Computational applications aside, these and related problems¡ªsee [14, 15]¡ªmake the study of a robust notion of Ricci curvature for spaces a subject of thriving interest in the geometry and topology of (mainly 3-dimensional) manifolds. This paper represents a continuation of both papers above. In all fairness, other definitions of Ricci curvature for -dimensional manifolds have been considered previously. The main contribution, from our point of view, is that of Stone [16, 17] that we will discuss in detail and adapt here. More recent contributions are due to Alsing et al. [18] Glickenstein [19] and Trout [20]. However, our approach is quite different and owns nothing to the mentioned works. The only one of the quoted papers that has a similar starting point with ours¡ªnamely, Stone¡¯s articles¡ªis [20]. Moreover he¡ªas we do (and as Stone originally did) seeks a discrete version of the Bonnet-Myers theorem. However, these facts represent the only common ground: his approach is (as Stone¡¯s was) purely combinatorial whereas ours is metric; his methods are also combinatorial in nature (even the Morse function employed is combinatorial) while, in contrast, our basic tool of study is Wald¡¯s metric curvature. We should also mention the fact that the convergence and related results for the combinatorial Ricci curvature
%U http://www.hindawi.com/journals/geometry/2013/694169/