Topic
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.
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TL;DR: In this paper, it was shown that for every n ≥ 4 there exists a closed n-dimensional manifold V which carries a Riemannian metric with negative sectional curvature, but admits no metric with constant curvature K≡−1.
Abstract: We show in this paper that for everyn≧4 there exists a closedn-dimensional manifoldV which carries a Riemannian metric with negative sectional curvatureK but which admits no metric with constant curvatureK≡−1. We also estimate the (pinching) constantsH for which our manifoldsV admit metrics with −1≧K≧−H.
230 citations
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TL;DR: In this article, the cosmological constant and the gravitational constant of topological black holes are reduced to two independent parameters, i.e., cosmologically constant and gravitational constant.
Abstract: We investigate topological black holes in a special class of Lovelock gravity. In odd dimensions, the action is the Chern-Simons form for the anti--de Sitter group. In even dimensions, it is the Euler density constructed with the Lorentz part of the anti--de Sitter curvature tensor. The Lovelock coefficients are reduced to two independent parameters: the cosmological constant and gravitational constant. The event horizons of these topological black holes may have constant positive, zero, or negative curvature. Their thermodynamics is analyzed and electrically charged topological black holes are also considered. We emphasize the differences due to the different curvatures of event horizons.
230 citations
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TL;DR: In this article, the authors propose a method to solve the problem of the problem: without abstracts, without abstractions, without Abstracts. (Without Abstract) (without Abstract)
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225 citations
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TL;DR: In this paper, the existence of a foliation by surfaces of constant mean curvature in a Lorentzian manifold has been studied, and it has been shown that the foliation can be found on surfaces of prescribed mean curvatures.
Abstract: We consider surfaces of prescribed mean curvature in a Lorentzian manifold and show the existence of a foliation by surfaces of constant mean curvature.
225 citations
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TL;DR: In this article, a 3-dimensional topological sigma-model, whose target space is a hyper-Kahler manifold X, is studied and a Feynman diagram calculation of its partition function demonstrates that it is a finite type invariant of 3-manifolds which is similar in structure to those appearing in the perturbative calculation of the Chern-Simons partition function.
Abstract: We study a 3-dimensional topological sigma-model, whose target space is a hyper-Kahler manifold X. A Feynman diagram calculation of its partition function demonstrates that it is a finite type invariant of 3-manifolds which is similar in structure to those appearing in the perturbative calculation of the Chern-Simons partition function. The sigma-model suggests a new system of weights for finite type invariants of 3-manifolds, described by trivalent graphs. The Riemann curvature of X plays the role of Lie algebra structure constants in Chern-Simons theory, and the Bianchi identity plays the role of the Jacobi identity in guaranteeing the so-called IHX relation among the weights. We argue that, for special choices of X, the partition function of the sigma-model yields the Casson-Walker invariant and its generalizations. We also derive Walker's surgery formula from the SL(2, Z) action on the finite-dimensional Hilbert space obtained by quantizing the sigma-model on a two-dimensional torus.
225 citations