Riemann curvature tensor

About: Riemann curvature tensor is a research topic. Over the lifetime, 6248 publications have been published within this topic receiving 138871 citations. The topic is also known as: Riemann–Christoffel tensor.

Conformal deformation of a Riemannian metric to constant scalar curvature

Abstract: A well-known open question in differential geometry is the question of whether a given compact Riemannian manifold is necessarily conformally equivalent to one of constant scalar curvature. This problem is known as the Yamabe problem because it was formulated by Yamabe [8] in 1960, While Yamabe's paper claimed to solve the problem in the affirmative, it was found by N. Trudinger [6] in 1968 that Yamabe's paper was seriously incorrect. Trudinger was able to correct Yamabe's proof in case the scalar curvature is nonpositive. Progress was made on the case of positive scalar curvature by T. Aubin [1] in 1976. Aubin showed that if dim M > 6 and M is not conformally flat, then M can be conformally changed to constant scalar curvature. Up until this time, Aubin's method has given no information on the Yamabe problem in dimensions 3, 4, and 5. Moreover, his method exploits only the local geometry of M in a small neighborhood of a point, and hence could not be used on a conformally flat manifold where the Yamabe problem is clearly a global problem. Recently, a number of geometers have been interested in the conformally flat manifolds of positive scalar curvature where a solution of Yamabe's problem gives a conformally flat metric of constant scalar curvature, a metric of some geometric interest. Note that the class of conformally flat manifolds of positive scalar curvature is closed under the operation of connected sum, and hence contains connected sums of spherical space forms with copies of S X S~. In this paper we introduce a new global idea into the problem and we solve it in the affirmative in all remaining cases; that is, we assert the existence of a positive solution u on M of the equation

Riemannian Manifolds: An Introduction to Curvature

Abstract: What Is Curvature?.- Review of Tensors, Manifolds, and Vector Bundles.- Definitions and Examples of Riemannian Metrics.- Connections.- Riemannian Geodesics.- Geodesics and Distance.- Curvature.- Riemannian Submanifolds.- The Gauss-Bonnet Theorem.- Jacobi Fields.- Curvature and Topology.

On the structure of spaces with Ricci curvature bounded below. I

Abstract: In this paper and in we study the structure of spaces Y which are pointed Gromov Hausdor limits of sequences f M i pi g of complete connected Riemannian manifolds whose Ricci curvatures have a de nite lower bound say RicMn i n In Sections and sometimes in we also assume a lower volume bound Vol B pi v In this case the sequence is said to be non collapsing If limi Vol B pi then the sequence is said to collapse It turns out that a convergent sequence is noncollapsing if and only if the limit has positive n dimensional Hausdor measure In par ticular any convergent sequence is either collapsing or noncollapsing Moreover if the sequence is collapsing it turns out that the Hausdor dimension of the limit is actually n see Sections and Our theorems on the in nitesimal structure of limit spaces have equivalent statements in terms of or implications for the structure on a small but de nite scale of manifolds with RicMn n Al though both contexts are signi cant for the most part it is the limit spaces which are emphasized here Typically the relation between corre sponding statements for manifolds and limit spaces follows directly from the continuity of the geometric quantities in question under Gromov Hausdor limits together with Gromov s compactness theorem Theorems see also Remark are examples of