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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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Journal ArticleDOI
TL;DR: In this paper, Bergery and Ikemakhenceforth gave a classification of the holonomy algebras for Lorentzian manifolds M of dimension ⩽11.
Abstract: The holonomy algebra g of an indecomposable Lorentzian ( n + 2 ) -dimensional manifold M is a weakly-irreducible subalgebra of the Lorentzian algebra so 1 , n + 1 . L. Berard Bergery and A. Ikemakhen divided weakly-irreducible not irreducible subalgebras into 4 types and associated with each such subalgebra g a subalgebra h ⊂ so n of the orthogonal Lie algebra. We give a description of the spaces R ( g ) of the curvature tensors for algebras of each type in terms of the space P ( h ) of h -valued 1-forms on R n that satisfy the Bianchi identity and reduce the classification of the holonomy algebras of Lorentzian manifolds to the classification of irreducible subalgebras h of so ( n ) with L ( P ( h ) ) = h . We prove that for n ⩽ 9 any such subalgebra h is the holonomy algebra of a Riemannian manifold. This gives a classification of the holonomy algebras for Lorentzian manifolds M of dimension ⩽11.

35 citations

Journal ArticleDOI
TL;DR: In this paper, a complete local classification of curvature homogeneous manifolds is presented, and the techniques presented in this paper can be applied to obtain a complete (local) classification of these manifolds, and to construct a number of new examples of such manifolds.

35 citations

Journal ArticleDOI
01 Jan 1976
TL;DR: In this paper, it was shown that the Hopf conjecture does not hold for a tensor having the symmetries of a curvature tensor and having positive sectional curvatures and negative Gauss-Bonnet integrand.
Abstract: An example is given, in dimension six, of a curvature tensor having positive sectional curvatures and negative Gauss-Bonnet integrand. A large class of questions in differential geometry involves the relationship between the topology and the geometry of a compact Riemannian manifold. One of these is the Hopf conjecture: If, in even dimensions, the sectional curvatures of such a manifold are positive, then so is the Euler number. The Hopf conjecture is known to be true in dimensions two and four by the following argument (Milnor, unpublished; [2]). One first writes down the Gauss-Bonnet formula, which, in every even dimension, equates the Euler number of the manifold to a certain integral over the manifold, where the integrand involves only the curvature tensor, and that only algebraically. One then shows (in dimensions two and four) that, at each point, positivity of the sectional curvatures implies positivity of this integrand. Most attempts to prove the full Hopf conjecture have been attempts to generalize this argument [1], [3], [4], [5], [6]. Thus, there arises the following, purely algebraic, question: Over a vector space of any even dimension, does a tensor having the symmetries of a curvature tensor and having positive sectional curvatures necessarily have positive Gauss-Bonnet integrand? We here answer this question in the negative. Fix a real, six-dimensional vector space V. A wedge denotes the wedge product, and a star a Hodge star operator.2 Denote by V2 the vector space of second-rank, antisymmetric tensors over V, by V2 its dual (the space of 2-forms over V), and by V22 the vector space of symmetric linear mappings from V2 to V2. We shall make use of the following fact: For any element A of V2, (1) ((A A A)* A (A A A)*)* 9(A A A A A)*A. For A any element of V2, denote by TA the element of V22 with action Received by the editors September 3, 1974. AMS (MOS) subject classifications (1970). Primary 53B20. i Supported in part by the National Science Foundation under contract GP-34721Xi, and by the Sloan Foundation. 2 Our conventions for the star operation are these: For any form A, A* = A; for B a 2-form and C a 4-form, B(C*) = C(B*) = (B A C)* = (B* A C*)*. Note that we introduce no metric on V. ? American Mathematical Society 1976 267 This content downloaded from 157.55.39.17 on Wed, 31 Aug 2016 04:16:18 UTC All use subject to http://about.jstor.org/terms

35 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that the kth-order Riemann-Lovelock tensor is determined by its traces in dimensions 2k ⩽ D < 4k.
Abstract: In order to study the properties of Lovelock gravity theories in low dimensions, we define the kth-order Riemann–Lovelock tensor as a certain quantity having a total 4k-indices, which is kth order in the Riemann curvature tensor and shares its basic algebraic and differential properties. We show that the kth-order Riemann–Lovelock tensor is determined by its traces in dimensions 2k ⩽ D < 4k. In D = 2k + 1 this identity implies that all solutions of pure kth-order Lovelock gravity are ‘Riemann–Lovelock’ flat. It is verified that the static, spherically symmetric solutions of these theories, which are missing solid angle spacetimes, indeed satisfy this flatness property. This generalizes results from Einstein gravity in D = 3, which corresponds to the k = 1 case. We speculate about some possible further consequences of Riemann–Lovelock curvature.

35 citations


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