Mathematics – Differential Geometry
Scientific paper
2011-04-29
An. Acad. Bras. Ci\^enc. vol.83 no.4, 2011
Mathematics
Differential Geometry
10 pages
Scientific paper
Given a closed Riemannian manifold (M, g), there is a partition \Sigma_i of its tangent bundle TM called the focal decomposition. The sets \Sigma_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 the flat n-tori are focally rigid, in the sense that if two flat tori are focally equivalent, then the tori are isometric up to rescaling.
Kwakkel Ferry
Martens Marco
Peixoto Mauricio
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