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 article, the existence of a new connection in a Riemannian manifold is proved and the curvature tensor of the new connection is also found, where the connection reduces to several symmetric, semi-symmetric and quarter symmetric connections; even some of them are not introduced so far.
Abstract: In a Riemannian manifold, the existence of a new connection is proved. In particular cases, this connection reduces to several symmetric, semi-symmetric and quarter-symmetric connections; even some of them are not introduced so far. We also find formula for curvature tensor of this new connection.
50 citations
Book•
14 Oct 2015
TL;DR: In this article, the authors studied the moduli space of all Riemannian metrics with positive scalar curvature and sectional curvature, and applied the Kreck-Stolz invariant and eta invariant to the case of dimensions 2 and 3.
Abstract: Part I: Positive scalar curvature.- The (moduli) space of all Riemannian metrics.- Clifford algebras and spin.- Dirac operators and index theorems.- Early results on the space of positive scalar curvature metrics.- Kreck-Stolz invariants.- Applications of Kreck-Stolz invariants.- The eta invariant and applications.- The case of dimensions 2 and 3.- The observer moduli space and applications.- Other topological structures.- Negative scalar and Ricci curvature.- Part II: Sectional curvature.- Moduli spaces of compact manifolds with positive or non-negative sectional curvature.- Moduli spaces of compact manifolds with negative and non-positive sectional curvature.- Moduli spaces of non-compact manifolds with non-negative sectional curvature.- Positive pinching and the Klingenberg-Sakai conjecture.
50 citations
TL;DR: In this paper, necessary and sufficient conditions for warped product manifolds (M,g) of dimension 4, with 1-dimensional base, and in particular for generalized Robertson-Walker spacetimes, to satisfy some generalized Einstein metric condition were given.
Abstract: We give necessary and sufficient conditions for warped product manifolds (M,g), of dimension \geqslant 4, with 1-dimensional base, and in particular, for generalized Robertson--Walker spacetimes, to satisfy some generalized Einstein metric condition. Namely, the difference tensor R . C - C . R, formed from the curvature tensor R and the Weyl conformal curvature tensor C, is expressed by the Tachibana tensor Q(S,R) formed from the Ricci tensor S and R. We also construct suitable examples of such manifolds. They are quasi-Einstein, i.e. at every point of M rank (S - a g) \leqslant 1, for some a \in R, or non-quasi-Einstein.
50 citations
TL;DR: In this article, the determinantal form of the Born-Infeld gravity action is analyzed in terms of the metric fluctuations around constant curvature backgrounds, and it is shown that the unitarity of the theory depends only on finite number of terms built from the powers of the curvature tensor.
Abstract: We develop techniques of analyzing the unitarity of general Born-Infeld gravity actions in $D$-dimensional spacetimes. The determinantal form of the action allows us to find a compact expression quadratic in the metric fluctuations around constant curvature backgrounds. This is highly nontrivial since for the Born-Infeld actions, in principle, infinitely many terms in the curvature expansion should contribute to the quadratic action in the metric fluctuations around constant curvature backgrounds, which would render the unitarity analysis intractable. Moreover in even dimensions, unitarity of the theory depends only on finite number of terms built from the powers of the curvature tensor. We apply our techniques to some four-dimensional examples.
50 citations
TL;DR: In this paper, it was shown that a Ricci simple manifold with vanishing divergence of the conformal curvature tensor admits a proper concircular vector field and it is necessarily a generalized Robertson-Walker space-time.
Abstract: A generalized Robertson–Walker (GRW) space-time is the generalization of the classical Robertson–Walker space-time. In the present paper, we show that a Ricci simple manifold with vanishing divergence of the conformal curvature tensor admits a proper concircular vector field and it is necessarily a GRW space-time. Further, we show that a stiff matter perfect fluid space-time or a mass-less scalar field with time-like gradient and with divergence-free Weyl tensor are GRW space-times.
50 citations