Geodesic Rigidity in Compact Nonpositively Curved Manifolds

Abstract

Our goal is to find geometric properties that are shared by homotopically equivalent compact Riemannian manifolds of sectional curvature $K \leqslant 0$. In this paper we consider mainly properties of free homotopy classes of closed curves. Each free homotopy class is represented by at least one smooth periodic geodesic, and the nonpositive curvature condition implies that any two periodic geodesic representatives are connected by a flat totally geodesic homotopy of periodic geodesic representatives. By imposing certain geometric conditions on these periodic geodesic representatives we define and study three types of free homotopy classes: Clifford, bounded and rank $1$. Let $M$, $M\prime$ denote compact Riemannian manifolds with $K \leqslant 0$, and let $\theta :{\pi _1}(M, m) \to {\pi _1}(M\prime , m\prime )$ be an isomorphism. Let $\theta$ also denote the induced bijection on free homotopy classes. Theorem A. The free homotopy class $[\alpha ]$ in $M$ is, respectively, Clifford, bounded or rank $1$ if and only if the class $\theta [\alpha ]$ in $M\prime$ is of the same type. Theorem B. If $M$, $M\prime$ have dimension $3$ and do not have a rank $1$ free homotopy class then they have diffeomorphic finite covers of the form ${S^1} \times {M^2}$. The proofs of Theorems A and B use the fact that $\theta$ is induced by a homotopy equivalence $f:(M, m) \to (M\prime , m\prime )$. Theorem C. The manifold $M$ satisfies the Visibility axiom if and only if $M\prime$ satisfies the Visibility axiom.