Cartan-Hadamard Theorem

The Cartan-Hadamard theorem describes the global topology and geometry of complete Riemannian manifolds with non-positive sectional curvature, showing they are diffeomorphic to Euclidean space.

Statement

Theorem8.7Cartan-Hadamard theorem

Let $(M^n, g)$ be a complete, simply connected Riemannian manifold with non-positive sectional curvature ( $K \leq 0$ ). Then:

The exponential map $\exp_p: T_pM \to M$ is a diffeomorphism for every $p \in M$ .
In particular, $M$ is diffeomorphic to $\mathbb{R}^n$ .
Any two points are connected by a unique geodesic.

Proof

Step 1: No conjugate points. If $K \leq 0$ , the Jacobi equation $J'' + R(J, \gamma')\gamma' = 0$ has the form $J'' = -R(J, \gamma')\gamma'$ where the "potential" $-R(\cdot, \gamma')\gamma'$ is a positive semidefinite operator (since $K \leq 0$ ). A Sturm comparison argument shows that the norm $|J(t)|$ is convex: $\frac{d^2}{dt^2}|J|^2 \geq 2|J'|^2 \geq 0$ (using $K \leq 0$ ).

If $J(0) = 0$ and $J'(0) \neq 0$ , then $|J(t)| > 0$ for all $t > 0$ (since $|J|^2$ is convex, starts at 0, and has positive initial derivative). Therefore there are no conjugate points along any geodesic.

Step 2: $\exp_p$ is a local diffeomorphism. The differential $(d\exp_p)_v$ maps Jacobi fields along the radial geodesic $t \mapsto \exp_p(tv/|v|)$ . Since there are no conjugate points, $(d\exp_p)_v$ is non-singular for all $v \in T_pM$ . So $\exp_p$ is a local diffeomorphism.

Step 3: $\exp_p$ is a covering map. Since $M$ is complete, $\exp_p$ is defined on all of $T_pM$ (Hopf-Rinow). Being a local diffeomorphism from a complete space, it is a covering map (by the path-lifting property).

Step 4: $\exp_p$ is a diffeomorphism. Since $M$ is simply connected, any covering map to $M$ must be trivial. Therefore $\exp_p$ is a diffeomorphism. $\blacksquare$

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Consequences

ExampleHadamard manifolds

A complete, simply connected manifold with $K \leq 0$ is called a Hadamard manifold. Examples:

$\mathbb{R}^n$ ( $K = 0$ ): The trivial case.
$\mathbb{H}^n$ ( $K = -1$ ): Hyperbolic space, diffeomorphic to $\mathbb{R}^n$ .
Symmetric spaces of non-compact type: $\mathrm{SL}(n, \mathbb{R})/SO(n)$ , $Sp(2n, \mathbb{R})/U(n)$ , etc.
Universal covers: The universal cover of any compact manifold with $K \leq 0$ .

RemarkNon-simply-connected case

Without simple connectivity, the conclusion changes: a complete manifold with $K \leq 0$ has universal cover diffeomorphic to $\mathbb{R}^n$ , so it is a $K(\pi, 1)$ space (Eilenberg-MacLane space). In particular, $\pi_k(M) = 0$ for $k \geq 2$ , and the topology is entirely determined by $\pi_1(M)$ .

RemarkComparison with positive curvature

The Cartan-Hadamard theorem contrasts sharply with the positive curvature case (Bonnet-Myers): positive Ricci curvature implies compactness and finite fundamental group, while non-positive sectional curvature implies the manifold is topologically "as simple as Euclidean space" (if simply connected). This is a manifestation of the general principle that positive curvature causes geodesics to converge (compactness) while negative curvature causes them to diverge (non-compactness, uniqueness).