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 no solution exists then there is a non-trivial solution of another non-linear limit equation on $1$-forms, which is known as the vacuum Einstein constraint equation.
Abstract: Let $(M,g)$ be a compact Riemannian manifold on which a trace-free and divergence-free $\sigma \in W^{1,p}$ and a positive function $\tau \in W^{1,p}$, $p > n$, are fixed. In this paper, we study the vacuum Einstein constraint equations using the well known conformal method with data $\sigma$ and $\tau$. We show that if no solution exists then there is a non-trivial solution of another non-linear limit equation on $1$-forms. This last equation can be shown to be without solutions no solution in many situations. As a corollary, we get existence of solutions of the vacuum Einstein constraint equation under explicit assumptions which in particular hold on a dense set of metrics $g$ for the $C^0$-topology.
58 citations
TL;DR: In this article, the authors introduced the r-th anisotropic mean curvature for compact hypersurfaces in ℓ n+1, which is a generalization of the usual R-th mean curvatures.
Abstract: Given a positive function F on S
n
which satisfies a convexity condition, we introduce the r-th anisotropic mean curvature M
r
for hypersurfaces in ℝ
n+1 which is a generalization of the usual r-th mean curvature H
r
. We get integral formulas of Minkowski type for compact hypersurfaces in R
n+1. We give some new characterizations of the Wulff shape by the use of our integral formulas of Minkowski type, in case F = 1 which reduces to some well-known results.
55 citations
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TL;DR: In this paper, it was shown that any compact non-trivial almost Ricci soliton with constant scalar curvature is isometric to a Euclidean sphere and the vector field decomposes as the sum of a Killing vector field and the gradient of a suitable function.
Abstract: The aim of this note is to prove that any compact non-trivial almost Ricci soliton $\big(M^n,\,g,\,X,\,\lambda\big)$ with constant scalar curvature is isometric to a Euclidean sphere $\Bbb{S}^{n}$. As a consequence we obtain that every compact non-trivial almost Ricci soliton with constant scalar curvature is gradient. Moreover, the vector field $X$ decomposes as the sum of a Killing vector field $Y$ and the gradient of a suitable function.
55 citations
TL;DR: In this paper, it was shown that if a 3D Kenmotsu metric is a Ricci soliton, then it is of constant curvature −1 and the soliton is expanding.
Abstract: We show that, if a 3-dimensional Kenmotsu metric is a Ricci soliton then it is of constant curvature −1 and the soliton is expanding.
52 citations
TL;DR: In this article, the authors give a short Lie-derivative theoretic proof of the following recent result of Barros et al. that a compact almost Ricci soliton with constant scalar curvature is gradient, and isometric to a Euclidean sphere.
Abstract: We give a short Lie-derivative theoretic proof of the following recent result of Barros et al. “A compact non-trivial almost Ricci soliton with constant scalar curvature is gradient, and isometric to a Euclidean sphere”. Next, we obtain the result: a complete almost Ricci soliton whose metric \(g\) is \(K\)-contact and flow vector field \(X\) is contact, becomes a Ricci soliton with constant scalar curvature. In particular, for \(X\) strict, \(g\) becomes compact Sasakian Einstein.
52 citations