Characterization of perfect fluid spacetimes admitting gradient η-Ricci and gradient Einstein solitons
TL;DR: In this article, the properties of perfect fluid spacetimes endowed with the gradient η -Ricci and gradient Einstein solitons were studied, and the authors set the goal to study the properties.
About: This article is published in Journal of Geometry and Physics.The article was published on 2021-04-01. It has received 11 citations till now. The article focuses on the topics: Perfect fluid & Einstein.
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TL;DR: In this paper, it was shown that the Weyl tensor is divergence-free and the potential function of the concircular vector field is pointwise collinear with the velocity vector field of perfect fluid spacetime.
Abstract: This paper deals with the study of perfect fluid spacetimes. It is proven that a perfect fluid spacetime is Ricci recurrent if and only if the velocity vector field of perfect fluid spacetime is parallel and α = β. In addition, in a stiff matter perfect fluid Yang pure space with p + σ ≠ 0, the integral curves generated by the velocity vector field are geodesics. Moreover, it is shown that in a generalized Robertson–Walker perfect fluid spacetime, the Weyl tensor is divergence-free and the gradient of the potential function of the concircular vector field is pointwise collinear with the velocity vector field of perfect fluid spacetime. We also characterize the perfect fluid spacetimes whose Lorentzian metrics are Yamabe and gradient Yamabe solitons, respectively.
22 citations
TL;DR: In this article, the properties of generalized Ricci recurrent perfect fluid spacetimes and the generalized R-RWRW spacetime were studied, and the main goal of this paper is to study the properties.
Abstract: The main goal of this paper is to study the properties of generalized Ricci recurrent perfect fluid spacetimes and the generalized Ricci recurrent (generalized Robertson–Walker (GRW)) spacetimes. I...
6 citations
01 Jan 2021
TL;DR: In this article, the authors obtained some results on almost Einstein solitons with unit geodesic potential vector field and provided necessary and sufficient conditions for the soliton to be trivial.
Abstract: We obtain some results on almost Einstein solitons with unit geodesic potential vector field and provide necessary and sufficient conditions for the soliton to be trivial.
4 citations
TL;DR: In this paper , the authors studied the properties of invariant submanifolds of hyperbolic Sasakian manifolds and proved that a 3D invariant Submanifold of a 5D hyperskakian manifold is geodesic if and only if it is invariant.
Abstract: We set the goal to study the properties of invariant submanifolds of the
hyperbolic Sasakian manifolds. It is proven that a three-dimensional
submanifold of a hyperbolic Sasakian manifold is totally geodesic if and
only if it is invariant. Also, we discuss the properties of
?-Ricci-Bourguignon solitons on invariant submanifolds of the hyperbolic
Sasakian manifolds. Finally, we construct a non-trivial example of a
three-dimensional invariant submanifold of five-dimensional hyperbolic
Sasakian manifold and validate some of our results.
3 citations
TL;DR: In this article, the authors characterized the Lorentzian manifold with a semi-symmetric non-metric connection and proved that it is a GRW space-time.
3 citations
References
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Book•
11 Jul 2011
TL;DR: In this article, the authors introduce Semi-Riemannian and Lorenz geometries for manifold theory, including Lie groups and Covering Manifolds, as well as the Calculus of Variations.
Abstract: Manifold Theory. Tensors. Semi-Riemannian Manifolds. Semi-Riemannian Submanifolds. Riemannian and Lorenz Geometry. Special Relativity. Constructions. Symmetry and Constant Curvature. Isometries. Calculus of Variations. Homogeneous and Symmetric Spaces. General Relativity. Cosmology. Schwarzschild Geometry. Causality in Lorentz Manifolds. Fundamental Groups and Covering Manifolds. Lie Groups. Newtonian Gravitation.
3,593 citations
Posted Content•
TL;DR: In this article, a monotonic expression for Ricci flow, valid in all dimensions and without curvature assumptions, is presented, interpreted as an entropy for a certain canonical ensemble.
Abstract: We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric applications are given. In particular, (1) Ricci flow, considered on the space of riemannian metrics modulo diffeomorphism and scaling, has no nontrivial periodic orbits (that is, other than fixed points); (2) In a region, where singularity is forming in finite time, the injectivity radius is controlled by the curvature; (3) Ricci flow can not quickly turn an almost euclidean region into a very curved one, no matter what happens far away. We also verify several assertions related to Richard Hamilton's program for the proof of Thurston geometrization conjecture for closed three-manifolds, and give a sketch of an eclectic proof of this conjecture, making use of earlier results on collapsing with local lower curvature bound.
3,091 citations
3,014 citations
Posted Content•
TL;DR: In this article, the Ricci flow with surgeries was constructed, and a lower bound on the volume of maximal horns and the smoothness of solutions was established. But this lower bound was later shown to be unjustified and irrelevant for the other conclusions.
Abstract: This is a technical paper, which is a continuation of math.DG/0211159. Here we construct Ricci flow with surgeries and verify most of the assertions, made in section 13 of that e-print: the exceptions are (1) the statement that manifolds that can collapse with local lower bound on sectional curvature are graph manifolds - this is deferred to a separate paper, since the proof has nothing to do with the Ricci flow, and (2) the claim on the lower bound for the volume of maximal horns and the smoothness of solutions from some time on, which turned out to be unjustified and, on the other hand, irrelevant for the other conclusions.
1,200 citations
TL;DR: In this article, the Ricci flow deforms a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures to a constant curvature metric.
Abstract: Let (M,g_0) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. We show that the Ricci flow deforms g_0 to a constant curvature metric. The proof uses the fact, also established in this paper, that positive isotropic curvature is preserved by the Ricci flow in all dimensions. We also rely on earlier work of Hamilton and of Bohm and Wilking.
330 citations