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 paper, the authors studied properties of quasi-Einstein almost hyperbolic Hermitian manifold with quasi-constant curvature and showed that the curvature of such a manifold can be characterized by a Gaussian distribution.
Abstract: In 1954 almost hyperbolic Hermitian manifold introduced by P. Libermann were classified for the first time in 1988 by C. L. Bejan. Recently in 1998 C. L. Bejan and L. Ornea constructed an example of an almost hyperbolic Hermitian manifold. Object of present paper is to study properties of quasi-Einstein almost hyperbolic Hermitian manifold with quasi-constant curvature.
2 citations
TL;DR: In this article, a theorem due to Lichnerowicz which establishes a lower bound on the lowest nonzero eigenvalue of the Laplacian acting on functions on a compact, closed manifold is reviewed.
Abstract: A theorem due to Lichnerowicz which establishes a lower bound on the lowest nonzero eigenvalue of the Laplacian acting on functions on a compact, closed manifold is reviewed. It is shown how this theorem can be extended to the case of a manifold with nonempty boundary. Lower bounds for different boundary conditions, analogous to the empty boundary case, are formulated and some novel proofs are presented.
2 citations
Journal Article•
TL;DR: In this article, it was shown that a decomposable Riemannian manifold is nearly quasi-Einstein if and only if both the decompositions of the manifold are Einstein.
Abstract: The object of the present paper is to study nearly quasi-Einstein manifold. Also we have studied decomposable Riemannian manifold and it is shown that a decomposable Riemannian manifold is nearly quasi-Einstein if and only if both the decompositions are Einstein.
2 citations
Posted Content•
TL;DR: In this paper, it was shown that there exists a universal constant $C, depending only on positive integers $n\geq 3$ and $p\leq n-1$, such that if $M^n$ is a compact free boundary submanifold of dimension $n$ immersed in the Euclidean unit ball whose size of the traceless second fundamental form is less than $C, then the $p$th cohomology group of $M$ vanishes.
Abstract: In this paper, we prove that there exists a universal constant $C$, depending only on positive integers $n\geq 3$ and $p\leq n-1$, such that if $M^n$ is a compact free boundary submanifold of dimension $n$ immersed in the Euclidean unit ball $\mathbb{B}^{n+k}$ whose size of the traceless second fundamental form is less than $C$, then the $p$th cohomology group of $M^n$ vanishes. Also, employing a different technique, we obtain a rigidity result for compact free boundary surfaces minimally immersed in the unit ball $\mathbb{B}^{2+k}$.
2 citations
Posted Content•
TL;DR: In this paper, the Hodge-de-Rham Laplacian that acts on conformal and projective Killing one-forms of a compact Riemannian manifold is considered.
Abstract: In the present paper, we consider the Hodge-de Rham Laplacian that acts on conformal Killing and projective Killing one-forms of a compact Riemannian manifold.
2 citations