Focal rigidity of flat tori

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