---
tag: 428R02EYWS
kind: statement
role: theorem
name: Cartan–Hadamard theorem
msc: [53C20, 53C22]
uses: [7CBT4SM83Y, 1X42JAA349, Y7Z0AGY1A7, 3XZRRE165N, Y1YPE6PM52, QCCD2JNFBT, 9AHMM41PJ3, ZPXE5VSZ35]
about: [1X42JAA349, Y7Z0AGY1A7]
sources:
  - url: https://en.wikipedia.org/wiki/Cartan%E2%80%93Hadamard_theorem
same_as:
  wikidata: Q5047048
  wikipedia: Cartan–Hadamard theorem
---

## Statement

Let $(M, g)$ be a connected complete [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$ whose [[sectional curvature|1X42JAA349]] satisfies $K \le 0$ everywhere.
Let $p \in M$.
Then the [[exponential map|Y7Z0AGY1A7]] $\exp_p \colon T_p M \to M$ is a [[covering map|3XZRRE165N]].
In particular, the [[universal cover|Y1YPE6PM52]] of $M$ is diffeomorphic to $\R^n$, and if $M$ is simply connected, then $M$ is diffeomorphic to $\R^n$.

## Remarks

The idea of the proof: if $K \le 0$, then for a [[Jacobi field|9AHMM41PJ3]] $J$ 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|ZPXE5VSZ35]].
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|QCCD2JNFBT]].
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 $T_p M$.

## Explanations

### L1 human

On a space that curves like a saddle everywhere, geodesics that start at one point spread apart and never meet again.
So straight lines from one point unroll the space onto a flat plane, perhaps wrapping it several times around.
In particular, such a space with no holes is just a stretched copy of $\R^n$.

### L2 human

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.