Cartan–Hadamard theorem

Remarks

The idea of the proof: if $K \le 0$, then for a Jacobi field $J$ with $J(0) = 0$ the function $\abs{J(t)}^2$ is convex. So a nonzero such $J$ never vanishes again, and there are no conjugate points. Hence $\exp_p$ is a local diffeomorphism. Pull the metric back to $T_p M$; the lines through $0$ are geodesics defined for all time, so the pulled-back metric is complete by the Hopf–Rinow theorem. A local isometry from a complete connected Riemannian manifold onto a connected one is a covering map.

Example: for a flat torus $\R^n / \Z^n$, the map $\exp_p$ is the quotient map $\R^n \to \R^n / \Z^n$ followed by a translation. For a compact hyperbolic surface, $\exp_p$ is the universal covering by the hyperbolic plane.

A consequence is that all higher homotopy groups of $M$ vanish. The sign condition is needed: on the round sphere $S^n$, the map $\exp_p$ sends the whole sphere of radius $\pi$ in $T_p S^n$ to the antipode of $p$. So it is not a covering. Completeness is needed too: the punctured plane $\R^2 \setminus \set{0}$ is flat, but $\exp_p$ is not defined on all of $T_p M$.

Going deeper

For a student

The theorem shows that curvature restricts topology: a compact manifold with $K \le 0$ is aspherical, its universal cover is $\R^n$, and its topology is governed by its fundamental group. For example, no sphere $S^n$ with $n \ge 2$ carries a metric of nonpositive curvature. It is the starting point of the study of manifolds of negative curvature.