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Ricci decomposition

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


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Journal ArticleDOI
18 Oct 2007
TL;DR: In this article, it was shown that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite.
Abstract: We show that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite. In particular, it follows that complete shrinking Ricci solitons and complete smooth metric measure spaces with a positive lower bound on the Bakry-Emery tensor have finite fundamental group. The method of proof is to generalize arguments of Garcia-Rio and Fernandez-Lopez in the compact case.

86 citations

Journal ArticleDOI
TL;DR: Abbena et al. as discussed by the authors completely classified three-dimensional homogeneous Lorentzian manifolds equipped with Einstein-like metrics, and showed that the Ricci tensor of (M, g) being cyclic-parallel is related to natural reductivity (respectively, symmetry) of (m, g).
Abstract: We completely classify three-dimensional homogeneous Lorentzian manifolds, equipped with Einstein-like metrics. Similarly to the Riemannian case (E. Abbena et al., Simon Stevin Quart J Pure Appl Math 66:173–182, 1992), if (M, g) is a three-dimensional homogeneous Lorentzian manifold, the Ricci tensor of (M, g) being cyclic-parallel (respectively, a Codazzi tensor) is related to natural reductivity (respectively, symmetry) of (M, g). However, some exceptional examples arise.

85 citations

Journal ArticleDOI
TL;DR: In this paper, the representations of the Riemann and the Weyl tensors through covariant derivatives of third-order potentials are examined in detail, and the possibility of introducing gauges on the potentials is reexamined in connection with the previous result.
Abstract: The representations of the Riemann and the Weyl tensors of a four-dimensional Riemannian manifold through covariant derivatives of third-order potentials are examined in detail. The Weyl tensor always admits a completely general representation whereas the Riemann tensor does not. Nevertheless there exists a class of Riemannian manifolds whose Riemann tensors may be calculated in terms of potentials; in this connection, specific examples are exhibited explicitly. The possibility of introducing gauges on the potentials is reexamined in connection with the previous result. New properties of the representations are also discussed.

84 citations

Journal ArticleDOI
TL;DR: In this article, the Ricci flow on compact four-manifolds with positive isotropic curvature and no essential incompressible space form was studied, and a long-time existence result was established with surgery on four-dimensional manifolds.
Abstract: In this paper we study the Ricci flow on compact four-manifolds with positive isotropic curvature and with no essential incompressible space form. We establish a long-time existence result of the Ricci flow with surgery on four-dimensional manifolds. As a consequence, we obtain a complete proof to the main theorem of Hamilton. During the proof we have actually provided, up to slight modifications, all necessary details for the part from Section 1 to Section 5 of Perelman’s second paper on the Ricci flow to approach the Poincare conjecture.

83 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202326
202241
20211
20203
20192
201810