Let ${M}$ be a complete Riemannian manifold. In the previous post, we saw that the condition that ${M}$ have nonpositive sectional curvature had a reformulation in terms of the “spreading” of geodesics: that is, nearby geodesics in nonpositive curvature spread at least as much as they would in euclidean space. One consequence of this philosophy was the Cartan-Hadamard theorem: the universal cover of ${M}$ is diffeomorphic to ${\mathbb{R}^n}$. In fact, for a ${M}$ simply connected (and complete of nonpositive curvature), the exponential map

$\displaystyle \exp_p : T_p M \rightarrow M$

is actually a diffeomorphism.

This suggests that without the simple connectivity assumption, the “only reason” for two geodesics to meet is the fundamental group. In particular, geodesics have no conjugate points, meaning that the exponential map is always nonsingular. Moreover, again by passing to the universal cover, it follows that for any ${p \in M}$ and ${\alpha \in \pi_1 (M)}$ (relative to the basepoint ${p}$), there is a unique geodesic loop at ${p}$ representing ${\alpha}$. In this post, I’d like to discuss some of the topological consequences of having a metric of nonpositive sectional curvature. Most of this material is from Do Carmo’s Riemannian Geometry.  (more…)