Topic
Curvature of Riemannian manifolds
About: Curvature of Riemannian manifolds is a research topic. Over the lifetime, 2933 publications have been published within this topic receiving 118023 citations.
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Book•
11 Jul 2011
TL;DR: In this article, the authors introduce Semi-Riemannian and Lorenz geometries for manifold theory, including Lie groups and Covering Manifolds, as well as the Calculus of Variations.
Abstract: Manifold Theory. Tensors. Semi-Riemannian Manifolds. Semi-Riemannian Submanifolds. Riemannian and Lorenz Geometry. Special Relativity. Constructions. Symmetry and Constant Curvature. Isometries. Calculus of Variations. Homogeneous and Symmetric Spaces. General Relativity. Cosmology. Schwarzschild Geometry. Causality in Lorentz Manifolds. Fundamental Groups and Covering Manifolds. Lie Groups. Newtonian Gravitation.
3,593 citations
Posted Content•
TL;DR: In this article, a monotonic expression for Ricci flow, valid in all dimensions and without curvature assumptions, is presented, interpreted as an entropy for a certain canonical ensemble.
Abstract: We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric applications are given. In particular, (1) Ricci flow, considered on the space of riemannian metrics modulo diffeomorphism and scaling, has no nontrivial periodic orbits (that is, other than fixed points); (2) In a region, where singularity is forming in finite time, the injectivity radius is controlled by the curvature; (3) Ricci flow can not quickly turn an almost euclidean region into a very curved one, no matter what happens far away. We also verify several assertions related to Richard Hamilton's program for the proof of Thurston geometrization conjecture for closed three-manifolds, and give a sketch of an eclectic proof of this conjecture, making use of earlier results on collapsing with local lower curvature bound.
3,091 citations
TL;DR: In this paper, the Ricci form of some Kahler metric is shown to be closed and its cohomology class must represent the first Chern class of M. This conjecture of Calabi can be reduced to a problem in non-linear partial differential equation.
Abstract: Therefore a necessary condition for a (1,l) form ( G I a ' r r ) I,,, Rlr dz' A d? to be the Ricci form of some Kahler metric is that it must be closed and its cohomology class must represent the first Chern class of M. More than twenty years ago, E. Calabi [3] conjectured that the above necessary condition is in fact sufficient. This conjecture of Calabi can be reduced to a problem in non-linear partial differential equation.
2,903 citations
2,205 citations
Book•
01 Jan 1984
TL;DR: The Dirichlet Heat Kernel for Regular Domains as mentioned in this paper is a heat kernel for non-compact manifolds that is based on the Laplacian on forms (LFP).
Abstract: Preface. The Laplacian. The Basic Examples. Curvature. Isoperimetric Inequalities. Eigenvalues and Kinematic Measure. The Heat Kernel for Compact Manifolds. The Dirichlet Heat Kernel for Regular Domains. The Heat Kernel for Noncompact Manifolds. Topological Perturbations with Negligible Effect. Surfaces of Constant Negative Curvature. The Selberg Trace Formula. Miscellanea. Laplacian on Forms. Bibliography. Index.
2,059 citations