Focal rigidity of flat tori

• Autor(a) principal: Kwakkel,Ferry
• Data de Publicação: 2011
• Outros Autores: Martens,Marco, Peixoto,Mauricio
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Anais da Academia Brasileira de Ciências (Online)
• Texto Completo: http://old.scielo.br/scielo.php?script=sci_arttext&pid=S0001-37652011000400002
• topic: Riemannian manifolds; focal decomposition; rigidity

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.