Some characterizations of Lorentzian manifolds
15 Jan 2019-International Journal of Geometric Methods in Modern Physics (World Scientific Publishing Company)-Vol. 16, Iss: 1, pp 1950016
TL;DR: In this paper, a generalization of the Robertson-Walker (RW) spacetime is proposed, and a further generalisation of the RW spacetime, the twisted spacetime (SST), is introduced.
Abstract: Generalized Robertson–Walker (GRW) spacetime is the generalization of the Robertson–Walker (RW) spacetime and a further generalization of GRW spacetime is the twisted spacetime. In this paper, we g...
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TL;DR: In this paper, it was proved that a Lorentzian manifold endowed with a semi-symmetric metric connection is a GRW spacetime and characterized the Ricci semisymmetric manifold.
Abstract: We set a type of semi-symmetric metric connection on the Lorentzian manifolds. It is proved that a Lorentzian manifold endowed with a semi-symmetric metric $$\rho $$
-connection is a GRW spacetime. We also characterize the Ricci semisymmetric Lorentzian manifold and study the solution of Eisenhart problem of finding the second order parallel (skew-)symmetric tensor on Lorentzian manifolds. Finally, we address physical interpretation of some geometric results of our paper.
28 citations
TL;DR: In this article , the main object of the paper is to characterize the perfect fluid spacetimes if its metrics are Ricci solitons, gradient Ricci, gradient Schouten, gradient A-Einstein and gradient A.
Abstract: The main object of this paper is to characterize the perfect fluid spacetimes if its metrics are Ricci solitons, gradient Ricci solitons, gradient A-Einstein solitons and gradient Schouten solitons.
2 citations
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TL;DR: In this article, a geometrical approach yields a local isometry between a semi-Riemannian manifold and a Robertson-Walker space of the same dimension, curvature and metric tensor sign.
Abstract: In the context of mathematical cosmology, the study of necessary and sufficient conditions for a semi-Riemannian manifold to be a (generalised) Robertson-Walker space-time is important. In particular, it is a requirement for the development of initial data to reproduce or approximate the standard cosmological model. Usually these conditions involve the Einstein field equations, which change if one considers alternative theories of gravity or if the coupling matter fields change. Therefore, the derivation of conditions which do not depend on the field equations is an advantage. In this work we present a geometric derivation of such a condition. We require the existence of a unit vector field to distinguish at each point of space two (non-equal) sectional curvatures. This is equivalent for the Riemann tensor to adopt a specific form. Our geometrical approach yields a local isometry between the space and a Robertson-Walker space of the same dimension, curvature and metric tensor sign (the dimension of the largest subspace on which the metric tensor is negative definite). Remarkably, if the space is simply-connected, the isometry is global. Our result generalises the theorem that spaces of the same curvature, dimension and metric tensor sign must be locally isometric to a class of spaces that have non-constant curvature. Because we do not make any assumptions regarding field equations, matter fields or metric tensor sign, one can readily use this result to study cosmological models within alternative theories of gravity or with different matter fields.
1 citations
Additional excerpts
...In particular, the research of [2, 6, 7] focus on the characterisation of GRW and RW spaces via conditions on the curvature tensor, more specifically the Ricci and Weyl ones....
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TL;DR: In this article , a geometrical approach yields a local isometry between a semi-Riemannian manifold and a Robertson-Walker space of the same dimension, curvature and metric tensor sign.
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1,042 citations
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24 May 2004TL;DR: The Ricci flow of special geometries Special and limit solutions Short time existence Maximum principles The Ricci Flow on surfaces Three-manifolds of positive Ricci curvature Derivative estimates Singularities and the limits of their dilations Type I singularities as discussed by the authors.
Abstract: The Ricci flow of special geometries Special and limit solutions Short time existence Maximum principles The Ricci flow on surfaces Three-manifolds of positive Ricci curvature Derivative estimates Singularities and the limits of their dilations Type I singularities The Ricci calculus Some results in comparison geometry Bibliography Index.
715 citations
01 Jan 1940
273 citations
TL;DR: In this paper, a new technique is introduced in order to solve the following question: when is a complete spacelike hypersurface of constant mean curvature in a generalized Robertson-Walker spacetime totally umbilical and a slice?
Abstract: A new technique is introduced in order to solve the following question:When is a complete spacelike hypersurface of constant mean curvature in a generalized Robertson-Walker spacetime totally umbilical and a slice? (Generalized Robertson-Walker spacetimes extend classical Robertson-Walker ones to include the cases in which the fiber has not constant sectional curvature.) First, we determine when this hypersurface must be compact. Then, all these compact hypersurfaces in (necessarily spatially closed) spacetimes are shown to be totally umbilical and, except in very exceptional cases, slices. This leads to proof of a new Bernstein-type result. The power of the introduced tools is also shown by reproving and extending several known results.
262 citations
Book•
23 Mar 2011
TL;DR: Pseudo-Riemannian Manifolds Pseudo Riemannians Warped Products and Twisted Products Robertson-Walker Spacetimes Hodge Theory and Elliptic Differential Operators Submanifolds of Finite Type Total Mean Curvature as discussed by the authors.
Abstract: Pseudo-Riemannian Manifolds Pseudo-Riemannian Submanifolds Warped Products and Twisted Products Robertson-Walker Spacetimes Hodge Theory and Elliptic Differential Operators Submanifolds of Finite Type Total Mean Curvature δ-Invariants, Inequalities Submanifolds of Kaehler Manifolds Submanifolds of Para-Kaehler Manifolds Pseudo-Riemannian Submersions Affine Hypersurface Contact Geometry Applications of δ-Invariants Applications to Relativity.
249 citations