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 if the metric of a 3-dimensional α-Sasakian manifold is a Ricci soliton, then it is either of constant curvature or of constant scalar curvature.
Abstract: We prove that if the metric of a 3-dimensional α-Sasakian manifold is a Ricci soliton, then it is either of constant curvature or of constant scalar curvature We also establish some properties of the potential vector field U of the Ricci soliton Finally, we give an example of an α-Sasakian 3-metric as a nontrivial Ricci soliton
3 citations
30 Sep 2020
TL;DR: In this paper, it was shown that a compact Ricci almost Ricci soliton is isometric to a sphere provided either it has constant scalar curvature or its associated vector field is conformal.
Abstract: In this paper, we shall give some structure equations for -almost solitons which generalize previous results for Ricci almost solitons. As a consequence of these equations we also derive an integral formula for the compact gradient -almost solitons which enables us to show that a compact nontrivial almost Ricci soliton is isometric to a sphere provided either it has constant scalar curvature or its associated vector field is conformal.
3 citations
TL;DR: In this paper, the Hamiltonian constraint equation of the Lovelock gravity theories was studied in the case of an empty, compact, conformally flat, time-symmetric, and space-like manifold.
Abstract: This paper is devoted to the study of the constraint equations of the Lovelock gravity theories. In the case of an empty, compact, conformally flat, time-symmetric, and space-like manifold, we show that the Hamiltonian constraint equation becomes a generalisation of the $\sigma_k$-Yamabe problem. That is to say, the prescription of a linear combination of the $\sigma_k$-curvatures of the manifold. We search solutions in a conformal class for a compact manifold. Using the existing results on the $\sigma_k$-Yamabe problem, we describe some cases in which they can be extended to this new problem. This requires to study the concavity of some polynomial. We do it in two ways: regarding the concavity of an entire root of this polynomial, which is connected to algebraic properties of the polynomial; and seeking analytically a concavifying function. This gives several cases in which a conformal solution exists. At last we show an implicit function theorem in the case of a manifold with negative scalar curvature, and find a conformal solution when the Lovelock theories are close to General Relativity.
3 citations
3 citations
01 Jan 1996
3 citations