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, the Ricci and Weyl tensors on Generalized Robertson-Walker space-times of dimension n ≥ 3 were shown to be a quasi-Einstein manifold.
Abstract: We prove theorems about the Ricci and the Weyl tensors on Generalized Robertson-Walker space-times of dimension n ≥ 3. In particular, we show that the concircular vector introduced by Chen decomposes the Ricci tensor as a perfect fluid term plus a term linear in the contracted Weyl tensor. The Weyl tensor is harmonic if and only if it is annihilated by Chen’s vector, and any of the two conditions is necessary and sufficient for the Generalized Robertson-Walker (GRW) space-time to be a quasi-Einstein (perfect fluid) manifold. Finally, the general structure of the Riemann tensor for Robertson-Walker space-times is given, in terms of Chen’s vector. In n = 4, a GRW space-time with harmonic Weyl tensor is a Robertson-Walker space-time.
58 citations
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TL;DR: In this article, the classifications of Einstein spaces by Schell and Petrov are combined and certain nonlocal results are obtained and it is shown that an Einstein space cannot be type I with a rank four Riemann tensor in a four-dimensional region.
Abstract: The classifications of Einstein spaces by Schell and Petrov are combined and certain nonlocal results are obtained. In particular, we show that an Einstein space cannot be type I with a rank four Riemann tensor in a four‐dimensional region. On using the notion of a perfect or imperfect infinitesimal‐holonomy group, we establish the conditions under which an Einstein space possesses a two‐, four‐, or six‐parameter group. We find that two‐ and four‐parameter groups are associated with special cases of type II null and type III, respectively.
58 citations
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TL;DR: In this article, the authors developed an analogue of S-duality for linearized gravity in (3+1)-dimensions, and showed that strong-weak coupling duality is an exact symmetry and implies small-large duality for the cosmological constant.
58 citations
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TL;DR: In this paper, the existence of three-dimensional Lorentzian manifolds which are curvature homogeneous up to order one but which are not locally homogeneous was investigated, and a complete local classification of these spaces was obtained.
Abstract: In this paper we investigate the existence of three-dimensional Lorentzian manifolds which are curvature homogeneous up to order one but which are not locally homogeneous, and we obtain a complete local classification of these spaces. As a corollary we determine, for each Segre type of the Ricci curvature tensor, the smallest k ∈ N for which curvature homogeneity up to order k guarantees local homogeneity of the three-dimensional manifold under consideration.
58 citations
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TL;DR: In this paper, the contractibility radius of complete manifolds with RicM > 1, KM '-K 2 and volume of M > the volume of the (r e)-ball on the unit m-sphere, m = dim M is estimated.
Abstract: Instead of injectivity radius, the contractibility radius is estimated for a class of complete manifolds such that RicM > 1, KM '-K 2 and the volume of M > the volume of the (r e)-ball on the unit m-sphere, m = dim M. Then for a suitable choice of e = e(m, K) every M belonging to this class is homeomorphic to
58 citations