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 article, it was shown that the Ricci soliton of a three-dimensional (3D)-Einstein almost-Einstein soliton is a Ricci manifold of constant sectional curvature.
Abstract: Let the metric $g$ of a three-dimensional $\eta$-Einstein almost Kenmotsu manifold $M$ be a Ricci soliton, we prove that $M$ is a Kenmotsu manifold of constant sectional curvature $-1$ and the soliton is expanding.
33 citations
01 Jan 2003
TL;DR: In this article, the existence of a 4-dimensional Lorentz manifold with Ricci tensors of type (0, 2) is established, and it is shown that such a manifold represents a perfect fluid space-time in cosmology.
Abstract: Recently, Prof. M. C. Chaki introduced the notion of a quasi Einstein manifold [1], denoted by (QE)n, whose Ricci tensor S of type (0,2) is not identically zero and satisfies the condition: S(X,Y) = a g(X,Y) + b A(X)A(Y) where a, b are scalars of which b>0 and A is a non-zero 1-form such that g(X,U) = A(X) for all vector fields X, U being a unit vector field. If the existence of a 4-dimensional Lorentz manifold is established whose Ricci tensor is of the form given above, then it is found that such a space-time represents a perfect fluid space-time in cosmology. Investigations by Karcher [2] and others have revealed that a conformally flat perfect fluid space-time has the geometric structure of quasi-constant curvature. It is found that a manifold of quasi-constant curvature is a natural sub-class of quasi Einstein manifold. Investigations on quasi Einstein manifolds help us to have a deeper understanding of the global character of the universe [3] including the topology. Consequently, we can study the nature of the singularities defined from a differential geometric standpoint. In a subsequent paper [4], Prof. Chaki introduced the generalized quasi Einstein manifolds denoted by G(QE)n. Chen and Yano [5] had introduced the notion of a manifold of quasi-constant curvature denoted by (QC)n. a generalization of a manifold of quasi-constant curvature, called a manifold of generalized quasi-constant curvature, denoted by G(QC)n, has been done by Prof. Chaki [4]. This is necessary for the study of G(QE)n. It is found that every G(QC)n (n > 3) is a G(QE)n, while every G(QC)n (n >3) is a conformally flat G(QE)n. The importance of a G(QE)n lies in the fact that such a 4- dimensional semi-Riemannian manifold is relevant to the study of a general relativistic fluid space-time admitting heat flux [6]. The global properties of such a space-time is under investigation. Study of space-times admitting fluid viscosity and electromagnetic fields require further generalization of the Ricci tensor and is under process.
30 citations
TL;DR: In this paper, it was shown that if an almost co-Kahler manifold of dimension greater than three satisfying rigid motions of the Minkowski 2-space can be shown to have a rigid motion.
Abstract: In this paper, we prove that if an almost co-Kahler manifold of dimension greater than three satisfying of rigid motions of the Minkowski 2-space.
30 citations
TL;DR: Weakly cyclic Ricci symmetric manifolds as mentioned in this paper are a type of non-flat Riemannian manifolds, and their properties are studied in detail in this paper.
Abstract: We introduce a type of non-flat Riemannian manifolds called weakly cyclic Ricci symmetric manifolds and study their geometric properties. The existence of such manifolds is shown by several non-trivial examples.
30 citations
TL;DR: In this article, some characterizations of rank-one symmetric Riemannian manifolds by the existence of nontrivial solutions to certain partial differential equations on RiemANNIAN manifolds are surveyed.
Abstract: Some characterizations of certain rank-one symmetric Riemannian manifolds by the existence of nontrivial solutions to certain partial differential equations on Riemannian manifolds are surveyed.
29 citations