scispace - formally typeset
Search or ask a question
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.
Citations
More filters
Posted Content
TL;DR: The notion of a closed vector field is introduced and investigated in this article, and various characterizations of such closed vector fields are established in relation to certain special Finsler spaces.
Abstract: The $\pi$-exterior derivative ${\o}d$, which is the Finslerian generalization of the (usual) exterior derivative $d$ of Riemannian geometry, is defined. The notion of a ${\o}d$-closed vector field is introduced and investigated. Various characterizations of ${\o}d$-closed vector fields are established. Some results concerning ${\o}d$-closed vector fields in relation to certain special Finsler spaces are obtained.

1 citations

DOI
01 Jan 2021
TL;DR: In this article, the authors investigated k-almost Yamabe solitons in the setting of threedimensional Kenmotsu manifolds and showed that they can be solved by gradient-k-almost-Yamabe solITons.
Abstract: In this current article, we intend to investigate k-almost Yamabe and gradient k-almost Yamabe solitons inside the setting of threedimensional Kenmotsu manifolds.

1 citations

Posted Content
TL;DR: In this paper, a set of geometric properties and physical applications of a mixed quasi-Einstein spacetime was discussed, where the Ricci soliton structure in a semiconformally flat Ricci pseudosymmetric spacetime is investigated.
Abstract: In the present paper we discuss about a set of geometric properties and physical applications of a mixed quasi-Einstein spacetime$[M(QE)_{4}]$, which is a special type of nearly quasi-Einstein spacetime$[N(QE)_{4}]$ Firstly we consider a nearly quasi-Einstein manifold$[N(QE)_{n}]$ along with a mixed quasi-Einstein manifold$[M(QE)_{n}]$ and study some geometric conditions with respect to different curvature tensors on them Then we consider a mixed quasi-Einstein spacetime and study $W_{2}$-Ricci pseudosymmetry and projective Ricci pseudosymmetry on it Then we study the conditions $H \cdot S=0$ and $\tilde C \cdot S=0$ in an $M(QE)_{4}$ spacetime, where $H, \tilde C$ are the conharmonical curvature tensor and semiconformal curvature tensor respectively Next we consider a semiconformally flat Ricci pseudosymmetric $M(QE)_{4}$ spacetime and find the nature of that spacetime In the next section we study about some applications of a perfect fluid $M(QE)_{4}$ spacetime in the general theory of relativity Followed by we discuss about the Ricci soliton structure in a semiconformally flat $M(QE)_{4}$ spacetime satisfying Einstein field equation without cosmological constant Finally we give a nontrivial example of a $M(QE)_{4}$ spacetime to establish the existence of it
Posted Content
TL;DR: In this article, the authors characterize the Einstein metrics in such broader classes of metrics as almost $\eta$-Ricci solitons and gradient almost ǫ-RICci soliton on Kenmotsu manifolds, and generalize some results of other authors.
Abstract: In this paper we characterize the Einstein metrics in such broader classes of metrics as almost $\eta$-Ricci solitons and $\eta$-Ricci solitons on Kenmotsu manifolds, and generalize some results of other authors. 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.
Posted Content
TL;DR: This paper uses properties of conformal vector fields to find several sufficient conditions on the soliton vector fields of Yamabe solitons under which their metrics are Yamabe metrics.
Abstract: In this paper, we use less topological restrictions and more geometric and analytic conditions to obtain some sufficient conditions on Yamabe solitons such that their metrics are Yamabe metrics, that is, metrics of constant scalar curvature. More precisely, we use properties of conformal vector fields to find several sufficient conditions on the soliton vector fields of Yamabe solitons under which their metrics are of Yamabe metrics.