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, the Ricci soliton is shown to be Ricci flat and locally isometric with respect to the Euclidean distance of the potential vector field when the manifold satisfies gradient almost.
Abstract: In the present paper, we initiate the study of $$*$$
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-Ricci soliton within the framework of Kenmotsu manifolds as a characterization of Einstein metrics. Here we display that a Kenmotsu metric as a $$*$$
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-Ricci soliton is Einstein metric if the soliton vector field is contact. Further, we have developed the characterization of the Kenmotsu manifold or the nature of the potential vector field when the manifold satisfies gradient almost $$*$$
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$$\eta $$
-Ricci soliton. Next, we deliberate $$*$$
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$$\eta $$
-Ricci soliton admitting $$(\kappa ,\mu )^\prime $$
-almost Kenmotsu manifold and proved that the manifold is Ricci flat and is locally isometric to $${\mathbb {H}}^{n+1}(-4)\times {\mathbb {R}}^n$$
. Finally we present some examples to decorate the existence of $$*$$
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$$\eta $$
-Ricci soliton, gradient almost $$*$$
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$$\eta $$
-Ricci soliton on Kenmotsu manifold.
8 citations
TL;DR: In this paper, the dimensional reduction of a Weyl space WN of N=4+n dimensions to a principal fiber bundle P(W4,Gn) over a four-dimensional space-time W4 with structural group Gn of dimension n arising from the existence of n conformal Killing vector fields of the original N-metric is studied.
Abstract: The dimensional reduction of a Weyl space WN of N=4+n dimensions to a principal fiber bundle P(W4,Gn) over a four‐dimensional space–time W4 with structural group Gn of dimension n arising from the existence of n conformal Killing vector fields of the original N‐metric is studied. The framework of a Weyl geometry is adopted in order to investigate conformal rescalings of the metric on the bundle P(W4,Gn) obtained. The Weyl symmetry is then, finally, broken again by choosing a particular Weyl gauge in which the internal, i.e., fiber metric, is of constant Cartan–Killing form. This choice of gauge, yielding a Riemannian theory, forces the internal metric to play no dynamical role in the theory, as is usually assumed to be the case in non‐Abelian gauge theories. However, this gauge induces a conformal transformation of the metric in the space–time base of P compared to the space–time metric obtained by the ordinary Kaluza–Klein reduction of a Riemannian space VN. The role of vector torsion in this dimen...
8 citations
TL;DR: In this paper, the curvature operator acting on the space of covariant traceless symmetric 2-tensors has been studied on an n-dimensional compact, orientable, connected Riemannian manifold and it has been shown that the dimension of the vector space of conformally Killing p-forms on such a manifold is not zero.
Abstract: On an n-dimensional compact, orientable, connected Riemannian manifold, we consider the curvature operator acting on the space of covariant traceless symmetric 2-tensors. We prove that, if the curvature operator is negative, then the manifold admits no nonzero conformally Killing p-forms for p = 1, 2, …, n − 1. On the other hand, we prove that the dimension of the vector space of conformally Killing p-forms on an n-dimensional compact simply-connected conformally flat Riemannian manifold (M,g) is not zero.
8 citations
8 citations
TL;DR: The basic assumptions underlying nonrelativistic classical and quantum physics are i) space is absolute, Euclidean and 3-dimensional, ii) time is absolute and flows uniformly and iii) path is a basic concept in mechanics and can be suitably extended to quantum theory as mentioned in this paper.
Abstract: The basic assumptions underlying nonrelativistic classical and quantum physics are i) space is absolute, Euclidean and 3-dimensional, ii) time is absolute and flows uniformly. In classical mechanics one further assumes that the solution of a problem consists in determining positions as a function of time. Thus path is a basic concept in mechanics and can be suitably extended to quantum theory. In the following we deduce all dynamical symmetries from this principle. Newton's equations in generalized co-ordinates are Dvi/dt ~ .F ~, or
8 citations