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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: Static, spherically symmetric solutions of the field equations for a particular dimensional continuation of general relativity with negative cosmological constant are studied and the causal structure is analyzed and the Penrose diagrams are exhibited.
Abstract: Static, spherically symmetric solutions of the field equations for a particular dimensional continuation of general relativity with a negative cosmological constant are studied. The action is, in odd dimensions, the Chern-Simons form for the anti-de Sitter group and, in even dimensions, the Euler density constructed with the Lorentz part of the anti-de Sitter curvature tensor. Both actions are special cases of the Lovelock action, and they reduce to the Hilbert action (with a negative cosmological constant) in the lower dimensional cases $\mathcal{D}=3$ and $\mathcal{D}=4$. Exact black hole solutions characterized by mass ($M$) and electric charge ($Q$) are found. In odd dimensions a negative cosmological constant is necessary to obtain a black hole, while in even dimensions both asymptotically flat and asymptotically anti-de Sitter black holes exist. The causal structure is analyzed and the Penrose diagrams are exhibited. The curvature tensor is singular at the origin for all dimensions greater than three. In dimensions of the form $\mathcal{D}=4k,4k\ensuremath{-}1$, the number of horizons may be zero, one, or two, depending on the relative values of $M$ and $Q$, while for a negative mass there is no horizon for any real value of $Q$. In the other cases, $\mathcal{D}=4k+1,4k+2$, both naked and dressed singularities with a positive mass exist. As in three dimensions, in all odd dimensions anti-de Sitter space appears as a "bound state" of mass $M=\ensuremath{-}1$, separated from the continuous spectrum ($M\ensuremath{\ge}0$) by a gap of naked curvature singularities. In even dimensions anti-de Sitter space has zero mass. The analysis is Hamiltonian throughout, considerably simplifying the discussion of the boundary terms in the action and the thermodynamics. The Euclidean black hole has the topology ${R}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{\mathcal{D}\ensuremath{-}2}$. Evaluation of the Euclidean action gives explicit expressions for all the relevant thermodynamical parameters of the system. The entropy, defined as a surface term in the action coming from the horizon, is shown to be a monotonically increasing function of the black hole radius, different from the area for $\mathcal{D}g4$.

224 citations

Journal ArticleDOI
TL;DR: The analytical formulas presented here are useful in treating the propagation and transformation of partially coherent anisotropic GSM beams, which include previous results for completely coherent Gaussian beams as special cases.
Abstract: A 4 x 4 complex curvature tensor M>(-1) is introduced to describe partially coherent anisotropic Gaussian-Schell model (GSM) beams. An analytical propagation formula for the cross-spectral density of partially coherent anisotropic GSM beams is derived. The propagation law of M(-1) that is also derived may be called partially coherent tensor ABCD law. The analytical formulas presented here are useful in treating the propagation and transformation of partially coherent anisotropic GSM beams, which include previous results for completely coherent Gaussian beams as special cases.

224 citations

Journal ArticleDOI
TL;DR: A new and efficient algorithm for the decomposition of 3D arbitrary triangle meshes and particularly optimized triangulated CAD meshes based on the curvature tensor field analysis is presented, which decomposes the object into near constant curvature patches and corrects boundaries by suppressing their artefacts or discontinuities.
Abstract: This paper presents a new and efficient algorithm for the decomposition of 3D arbitrary triangle meshes and particularly optimized triangulated CAD meshes. The algorithm is based on the curvature tensor field analysis and presents two distinct complementary steps: a region based segmentation, which is an improvement of that presented by Lavoue et al. [Lavoue G, Dupont F, Baskurt A. Constant curvature region decomposition of 3D-meshes by a mixed approach vertex-triangle, J WSCG 2004;12(2):245-52] and which decomposes the object into near constant curvature patches, and a boundary rectification based on curvature tensor directions, which corrects boundaries by suppressing their artefacts or discontinuities. Experiments conducted on various models including both CAD and natural objects, show satisfactory results. Resulting segmented patches, by virtue of their properties (homogeneous curvature, clean boundaries) are particularly adapted to computer graphics tasks like parametric or subdivision surface fitting in an adaptive compression objective.

219 citations

Journal ArticleDOI
TL;DR: In this paper, the curvature invariance of the Randall-Sundrum (RS) single brane-world solution was analyzed in the direction of the Cauchy horizon.
Abstract: We carefully investigate the gravitational perturbation of the Randall-Sundrum (RS) single brane-world solution [L. Randall and R. Sundrum, Phys. Rev. Lett. $83,$ 4690 (1999)], based on a covariant curvature tensor formalism recently developed by us. Using this curvature formalism, it is known that the ``electric'' part of the five-dimensional Weyl tensor, denoted by ${E}_{\ensuremath{\mu}\ensuremath{ u}},$ gives the leading order correction to the conventional Einstein equations on the brane. We consider the general solution of the perturbation equations for the five-dimensional Weyl tensor caused by the matter fluctuations on the brane. By analyzing its asymptotic behavior in the direction of the fifth dimension, we find the curvature invariant diverges as we approach the Cauchy horizon. However, in the limit of asymptotic future in the vicinity of the Cauchy horizon, the curvature invariant falls off fast enough to render the divergence harmless to the brane world. We also obtain the asymptotic behavior of ${E}_{\ensuremath{\mu}\ensuremath{ u}}$ on the brane at spatial infinity, assuming that the matter perturbation is localized. We find it falls off sufficiently fast and will not affect the conserved quantities at spatial infinity. This indicates strongly that the usual conservation law, such as the ADM energy conservation, holds on the brane as far as asymptotically flat spacetimes are concerned.

218 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that there is no finiteness result for complete Riemannian manifolds with Ricci curvature > 0 in the low-dimensional special cases n = 2 and n = 3, where all notions of curvature coincide.
Abstract: Complete open Riemannian manifolds (Mn, g) with nonnegative sectional curvature are well understood. The basic results are Toponogov's Splitting Theorem and the Soul Theorem [CG1]. The Splitting Theorem has been extended to manifolds of nonnegative Ricci curvature [CG2]. On the other hand, the Soul Theorem does not extend even topologically, according to recent examples in [GM2]. A different method to construct manifolds which carry a metric with Ric > 0, but no metric with nonnegative sectional curvature, has been given by L. Berard Bergery [BB]. This leads to the question (cf. also [Y1]): Is there any finiteness result for complete Riemannian manifolds with Ric > 0 ? The answer is certainly affirmative in the low-dimensional special cases n = 2, where all notions of curvature coincide, and n = 3, where nonnegative Ricci curvature has been studied by means of stable minimal surfaces [MSY, SY]. On the other hand, J. P. Sha and D. G. Yang [ShY] have constructed complete manifolds with strictly positive Ricci curvature in higher dimensions. For example they can choose the underlying space to be R4 x S3 with infinitely many copies of S3 x CP 2 attached to it by surgery; cf. also [ShY 1]. It is therefore clear that any finiteness result for arbitrary dimensions requires additional assumptions. The purpose of this paper is to establish the following main result.

215 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202364
2022152
2021169
2020163
2019174
2018180