Resumen

KWAKKEL, Ferry; MARTENS, Marco  y  PEIXOTO, Mauricio. Focal rigidity of flat tori. An. Acad. Bras. Ciênc. [online]. 2011, vol.83, n.4, pp. 1149-1158.  Epub 30-Sep-2011. ISSN 0001-3765.  http://dx.doi.org/10.1590/S0001-37652011005000037.

Given a closed Riemannian manifold (M, g), i.e. compact and boundaryless, there is a partition of its tangent bundle TM = ∪_iΣ_i called the focal decomposition of TM. The sets Σ_i 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.

Palabras llave : Riemannian manifolds; focal decomposition; rigidity.