The Higher Rank Rigidity Theorem for Manifolds With No Focal Points

TL;DR

This paper generalizes the Higher Rank Rigidity Theorem to manifolds with no focal points, showing such manifolds are symmetric spaces under certain conditions, and extends the invariance of rank to this broader class.

Contribution

It extends the Higher Rank Rigidity Theorem to manifolds with no focal points and proves rank invariance for their fundamental groups.

Findings

Manifolds with no focal points satisfying certain conditions are symmetric spaces.

Rank is an invariant of the fundamental group for manifolds with no focal points.

Generalization of rigidity results to broader class of manifolds.

Abstract

We say that a Riemannian manifold M has rank at least k if every geodesic in M admits at least k parallel Jacobi fields. The Rank Rigidity Theorem of Ballmann and Burns-Spatzier, later generalized by Eberlein-Heber, states that a complete, irreducible, simply connected Riemannian manifold M of rank at least 2 (the higher rank assumption), whose isometry group satisfies the condition that the recurrent vectors are dense in the unit tangent bundle SM (the duality condition), is a symmetric space of noncompact type. This includes, for example, higher rank M which admit a finite volume quotient. We adapt the method of Ballmann and Eberlein-Heber to prove a generalization of this theorem where the manifold M is assumed only to have no focal points. We then use this theorem to generalize to no focal points a result of Ballmann-Eberlein stating that for compact manifolds of nonpositive curvature, rank is an invariant of the fundamental group.