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 authors studied smooth submanifolds immersed in a k-step Carnot group of homogeneous dimension Q and proved an isoperimetric inequality for the case of a $C^2$-smooth compact hypersurface S with -or without - boundary $\partial S$; S and S$ are endowed with their homogeneous measures, actually equivalent to the intrinsic (Q-1)-dimensional and (Q 2)-dimensional Hausdorff measures with respect to some homogeneous metric on G.
Abstract: In this paper we shall study smooth submanifolds immersed in a k-step Carnot group G of homogeneous dimension Q Among other results, we shall prove an isoperimetric inequality for the case of a $C^2$-smooth compact hypersurface S with - or without - boundary $\partial S$; S and $\partial S$ are endowed with their homogeneous measures, actually equivalent to the intrinsic (Q-1)-dimensional and (Q-2)-dimensional Hausdorff measures with respect to some homogeneous metric $\varrho$ on G; see Section 5 This generalizes a classical inequality, involving the mean curvature of the hypersurface, proven by Michael and Simon [63] and, independently by Allard [1] In particular, from this result one may deduce some related Sobolev-type inequalities; see Section 7 The strategy of the proof is inspired by the classical one In particular, we shall begin by proving some linear isoperimetric inequalities Once this is proven, one can deduce a local monotonicity formula and then conclude the proof by a covering argument We stress however that there are many differences, due to our different geometric setting Some of the tools which have been developed ad hoc in this paper are, in order, a ``blow-up'' theorem, which also holds for characteristic points, and a smooth Coarea Formula for the HS-gradient; see Section 3 and Section 4 Other tools are the horizontal integration by parts formula and the 1st variation of the H-perimeter already developed in [68], [69], and here generalized to hypersurfaces having non-empty characteristic set Some natural applications of these results are in the study of minimal and constant horizontal mean curvature hypersurfaces Moreover we shall prove some purely horizontal, local and global Poincare-type inequalities as well as some related facts and consequences; see Section 4 and Section 5
5 citations
TL;DR: In this paper, the double inner product of two miscellaneous tensors of rank 2 in a Riemannian space is derived and the corresponding determinantal representation as well as the general representation of outer inverses in the Riemanian space are derived.
Abstract: Starting from a known determinantal representation of outer inverses, we derive their determinantal representation in terms of the inner product in the Euclidean space. We define the double inner product of two miscellaneous tensors of rank 2 in a Riemannian space. The corresponding determinantal representation as well as the general representation of outer inverses in the Riemannian space are derived. A non-zero {2}-inverse X of a given tensor A obeying ρ(X) = s with 1 ≤ s ≤ r = ρ(A) is expressed in terms of the double inner product involving compound tensors with minors of order s, extracted from A and appropriate tensors.
5 citations
TL;DR: The purpose of this paper is to study n-dimensional compact CR-submanifolds of complex hyperbolic space CH(n) using LaSalle's inequality.
Abstract: The purpose of this paper is to study n-dimensional compact CR-submanifolds of complex hyperbolic space CH(n
5 citations
4 citations
16 Dec 2002
TL;DR: In this article, the inverse boundary value problem for nonlinear elliptic equations has been solved for Riemannian metric tensors by the restriction of the tensor £ X (g) - (e σ ⊇ g. (e- σ X)) g to the gradient field l, where £ X is the Lie derivative of the metric tensor g under the vector field X and a = logdet(g).
Abstract: Let g be a C 2,α Riemannian metric defined on a bounded domain Ω ⊂ R 2 with C 3,α boundary and let X be a C 2,α vector field on Ω satisfying X|∂ Ω = 0. We show that if I is a gradient field of a solution u to the equation Δ g u - (⊇ g σ, ⊇ g u) g = 0 on Ω, then both inner products (l; X) g and (l⊥, X) g are uniquely determined by the restriction of the tensor £ X (g) - (e σ ⊇ g . (e- σ X)) g to the gradient field l, where £ X (g) is the Lie derivative of the metric tensor g under the vector field X and a = logdet(g). This work solves a problem related to an inverse boundary value problem for nonlinear elliptic equations.
4 citations