Focal rigidity of flat tori

Abstract

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.Key words: Riemannian manifolds, focal decomposition, rigidity. DEFINITIONS AND STATEMENT OF RESULTSIn general, topological characteristics of a Riemannian manifold do not determine its geometry, that is, its metric structure. However, starting in the 1960s, examples have been discovered for which such characteristics do determine the geometry. The manifolds can not be deformed without changing the characteristic. One speaks of rigidity. The prototype rigidity result is due to Mostow (Mostow 1968). MOSTOW'S RIGIDITY THEOREM. Two compact hyperbolic n-manifolds, n ≥ 3, with isomorphic fundamental groups are isometric.Given an analytic manifold M, we denote R ω (M) the class of analytic Riemannian structures on M.This class of structures is dense in the class R ∞ (M) of smooth Riemannian structures in the C ∞ strong Whitney topology, see (Hirsch 1976). Let (M, g) and ( M, g) be closed (compact and boundaryless) analytic manifolds. Two Riemannian manifolds are said to be isometric up to a rescaling if, up to a constant rescaling of the metric g, the manifolds (M, g) and ( M, g) are isometric.

Focal rigidity of hyperbolic surfaces

“…That is, we consider the global counterpart to focal stability. It has been shown in [8] that flat ntori, with n ≥ 2, are focally rigid, in the sense that global topological deformations of the focal decomposition are not possible without essentially changing the metric. Two surfaces M 1 and M 2 are commensurable, if their uniformizing surface groups Γ 1 and Γ 2 are commensurable, that is, if Γ 1 ∩ Γ 2 has finite index in both Γ 1 and Γ 2 .…”

“…Focal spectrum of hyperbolic surfaces. Let us first pose a problem regarding the focal spectrum of a closed hyperbolic surface, see also [8] for the case of flat tori. An analogue of the length spectrum is given by the focal spectrum defined in terms of the focal decomposition.…”