Intégrales de courbure sur des variétés feuilletées

Brito, Fabiano G. B.; Langevin, Rémi; Rosenberg, Harold

doi:10.4310/jdg/1214435986

Cited by 32 publications

(30 citation statements)

References 3 publications

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“…In the early 80's there was obtained a notable result by Brito, Langevin and Rosenberg [11]. The authors considered codimension-one foliations of a closed space form M p+1 (κ).…”

Section: Introductionmentioning

confidence: 99%

Generalized Newton transformation and its applications to extrinsic geometry

Andrzejewski

Kozłowski

Niedziałomski

2016

Asian Journal of Mathematics

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Abstract. In this article we introduce a generalization of the Newton transformation to the case of a system of endomorphisms. We show that it can be used in the context of extrinsic geometry of foliations and distributions yielding new integral formulas containing generalized extrinsic curvatures. IntroductionAnalyzing the study of Riemannian geometry we see that its basic concepts are related with some operators, such as shape, Ricci, Schouten operator, etc. and functions constructed of them, such as mean curvature, scalar curvature, Gauss-Kronecker curvature, etc. The most natural and useful functions are the ones derived from algebraic invariants of these operators e.g. by taking trace, determinant and in general the r-th symmetric functions σ r . However, the case r > 1 is strongly nonlinear and therefore more complicated. The powerful tool to deal with this problem is the Newton transformation T r of an endomorphism A (strictly related with the Newton's identities) which, in a sense, enables a linearization of σ r , (r + 1)σ r+1 = tr (AT r ).Although this operator appeared in geometry many years ago (see, e.g., [21,29]), there is a continues increase of applications of this operator in different areas of geometry in the last years (see, among others, [1,2,3,8,10,17,18,23,24,25,28]).All these results cause a natural question, what happen if we have a family of operators i.e. how to define the Newton transformation for a family of endomorphisms. A partial answer to this question can be found in the literature (operator T r and the scalar S r for even r [5,15]), nevertheless, we expect that this case is much more subtle. This is because in the case of family of operators we should obtain more natural functions as in the case of one and consequently more information about geometry. In order to do this, for any multi-index u 2000 Mathematics Subject Classification. 53C12; 53C65.

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“…In the early 80's there was obtained a notable result by Brito, Langevin and Rosenberg [11]. The authors considered codimension-one foliations of a closed space form M p+1 (κ).…”

Section: Introductionmentioning

confidence: 99%

Generalized Newton transformation and its applications to extrinsic geometry

Andrzejewski

Kozłowski

Niedziałomski

2016

Asian Journal of Mathematics

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“…Later on, Asimov [4] and Brito et al [6] have shown that integrals of mean curvatures (of arbitrary higher order k) of codimension-one foliations F of closed manifolds M of constant curvature c depend only on k, c, volume and dimension of M, not on F . Next, one of this formulae has been extended to foliations of arbitrary Riemannian manifolds:…”

Section: Introductionmentioning

confidence: 99%

“…(4)] and has been applied by several authors in different contexts (see [5,7,14,17,18] etc.). Recently, first Rovenski and the second author [15] on symmetric spaces and then Andrzejewski [2] (see also [3]) on arbitrary Riemannian manifolds have found series of integral formulae for codimension-one foliations, formulae which extend (1) and all the equalities proved in [4] and [6]. For more about integral formulae for foliations, we refer to [16].…”

Section: Introductionmentioning

confidence: 99%

New integral formulae for two complementary orthogonal distributions on Riemannian manifolds

Lużyńczyk

Walczak

2015

Ann Glob Anal Geom

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We derive and apply a new integral formula for a closed Riemannian manifold equipped with a pair of complementary orthogonal distributions (plane fields). The integrand depends on the second fundamental forms and integrability tensors of the distributions, their covariant derivatives, and of the Ricci curvature of the ambient manifold. Also, we discuss some applications of this formula and of another formula of this sort, the one obtained earlier by the second author, and show that both formulae may hold when the distributions are defined only outside a "reasonable" closed subset of the manifold under consideration.

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“…Em [25], os autores conjecturam que, Seja M n+1 (c) uma variedade fechada, munida de uma métrica com curvatura seccional constante c, e seja F uma folheação de codimensão um e transversalmente orientada de M . Brito et al, em [10], mostraram que as integrais em M (c) das funções simétricas dos autovalores da segunda forma fundamental das folhas, não dependem da folheação,…”

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“…Todos os casos conhecidos, [1], [10], [42] e as referências neles contidas, as fórmulas integrais são independentes da folheação, mas ainda dependem da geometria do ambiente folheado.…”

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Aspectos topológicos na teoria geométrica de folheações

Gonçalves¹

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Intégrales de courbure sur des variétés feuilletées

Cited by 32 publications

References 3 publications

Generalized Newton transformation and its applications to extrinsic geometry

Generalized Newton transformation and its applications to extrinsic geometry

New integral formulae for two complementary orthogonal distributions on Riemannian manifolds

Aspectos topológicos na teoria geométrica de folheações

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