Focal rigidity of flat tori

Anais da Academia Brasileira de Ciências

Ferry Kwakkel²  Marco Martens¹  Mauricio Peixoto² 

[1] ,Instituto de Matemática Pura e AplicadaRio de Janeiro RJ ,Brasil

关键词: Riemannian manifolds;    focal decomposition;    rigidity;    variedade Riemanniana;    decomposição focal;    rigidez;   

DOI  :  10.1590/S0001-37652011005000037

来源: SciELO

【 摘 要 】

Given a closed Riemannian manifold (M, g), i.e. compact and boundaryless, there is a partition of its tangent bundle TM = ∪iΣi called the focal decomposition of TM. The sets Σi are closely associated to focusing of geodesics of (M, g), i.e. to the situation where there are exactly i geodesic arcs of the same length joining points p and q in M. In this note, we study the topological structure of the focal decomposition of a closed Riemannian manifold and its relation with the metric structure of the manifold. Our main result is that flat n-tori, n > 2, are focally rigid in the sense that if two flat tori are focally equivalent then the tori are isometric up to rescaling. The case n = 2 was considered before by F. Kwakkel.

【 授权许可】

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