The Rank Rigidity Theorem for Manifolds with No Focal Points.

[摘要] 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 G satisfies the condition that the G-recurrent vectors are dense in SM 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.

[发布机构] University of Michigan

[效力级别] No Focal Points

[关键词] Rigidity;No Focal Points;Higher Rank;Duality Condition;Riemannian Manifolds;MSC - 53C24;Mathematics;Science;Mathematics