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Journal ArticleDOI

Integral formulas in riemannian geometry

01 Mar 1973-Bulletin of The London Mathematical Society (Oxford University Press (OUP))-Vol. 5, Iss: 1, pp 124-125
About: This article is published in Bulletin of The London Mathematical Society.The article was published on 1973-03-01. It has received 331 citations till now. The article focuses on the topics: Riemannian geometry & Fundamental theorem of Riemannian geometry.
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TL;DR: In this paper, a review of the ideas in cosmology and the solutions which describe them is presented, including Friedmann/Robertson/Walker and spherically-symmetric ones with expanding, singular hypersurfaces which are spherical in ordinary space.

10 citations

Journal ArticleDOI
11 May 2021
TL;DR: In this paper, the authors characterized the Einstein metrics in such broad classes of metrics as almost $$\eta $$¯¯ -Ricci solitons and almost $€  ¯¯¯¯ -RICci soliton on Kenmotsu manifolds, and generalized some known results.
Abstract: We characterize the Einstein metrics in such broad classes of metrics as almost $$\eta $$ -Ricci solitons and $$\eta $$ -Ricci solitons on Kenmotsu manifolds, and generalize some known results. First, we prove that a Kenmotsu metric as an $$\eta $$ -Ricci soliton is Einstein metric if either it is $$\eta $$ -Einstein or the potential vector field V is an infinitesimal contact transformation or V is collinear to the Reeb vector field. Further, we prove that if a Kenmotsu manifold admits a gradient almost $$\eta $$ -Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then it is an Einstein manifold. Finally, we present new examples of $$\eta $$ -Ricci solitons and gradient $$\eta $$ -Ricci solitons, which illustrate our results.

9 citations

Journal ArticleDOI
TL;DR: For systems of two independent variables, complete integrability in the present sense implies the existence of a Lax pair for the system, for which the theory of the inverse scattering method is applicable as discussed by the authors.

9 citations

Posted Content
TL;DR: In this paper, the authors investigated the nature of the conformal Ricci soliton within the framework of Kenmotsu manifolds and established a relation between the potential vector field and the Reeb vector field.
Abstract: The object of the present paper is to characterize the class of Kenmotsu manifolds which admits conformal $\eta$-Ricci soliton. Here, we have investigated the nature of the conformal $\eta$-Ricci soliton within the framework of Kenmotsu manifolds. It is shown that an $\eta$-Einstein Kenmotsu manifold admitting conformal $\eta$-Ricci soliton is an Einstein one. Moving further, we have considered gradient conformal $\eta$-Ricci soliton on Kenmotsu manifold and established a relation between the potential vector field and the Reeb vector field. Next, it is proved that under certain condition, a conformal $\eta$-Ricci soliton on Kenmotu manifolds under generalized D-conformal deformation remains invariant. Finally, we have constructed an example for the existence of conformal $\eta$-Ricci soliton on Kenmotsu manifold.

9 citations

Journal ArticleDOI
TL;DR: In this article, an n-dimensional, compact, minimal CR-submanifold of CR-dimension n − 1 and a sufficient condition for it to be a tube over a totally geodesic complex subspace was given.
Abstract: We study an n-dimensional, compact, minimal CR-submanifold of CR-dimension n − 1 and give a sufficient condition for the submanifold to be a tube over a totally geodesic complex subspace.

9 citations