How to use Klingenberg's Lemma to prove Hadamard's theorem

How can I use Klingenberg's Lemma to construct an alternative proof of Hadamard's theorem?

Let $$M$$ be a complete Riemannian manifold, simply connected, with sectional curvature $$K(p,\sigma) \le 0$$, for all $$p \in M$$ and for all $$\sigma \subset T_p(M)$$. Then $$M$$ is diffeomorphic to $$\mathbb R^n$$, $$n = \dim M$$; more precisely, $$\exp_p : T_p M \to M$$ is a diffeomorphism.

(do Carmo, Riemannian Geometry, p. 149)

Klingenberg's Lemma

Let $$M$$ be a complete Riemannian manifold with sectional curvature $$K \le K_0$$, where $$K_0$$ is a positive constant. Let $$p,q \in M$$ and let $$\gamma_0$$ and $$\gamma_1$$ be two different geodesics joining $$p$$ to $$q$$ with $$\ell(\gamma_0) \le \ell(\gamma_1)$$. Assume that $$\gamma_0$$ is homotopic to $$\gamma_1$$, that is, there exists a continuous family of curves $$\alpha_t$$, $$t \in [0,1]$$ such that $$\alpha_0=\gamma_0$$ and $$\alpha_1=\gamma_1$$. Then there exists $$t_0 \in [0,1]$$ such that $$\ell(\gamma_0)+\ell(\alpha_{t_0}) \ge \frac{2\pi}{\sqrt{K_0}}$$.

(do Carmo, Riemannian Geometry, pp, 235-236)

Attempted proof:

Since $$M$$ is complete, by the Hopf-Rinow Theorem (Theorem 2.8 of Chapter 7 in do Carmo), there exists a geodesic $$\gamma$$ joining $$p,q \in M$$. Since $$M$$ is simply connected, the geodesic is unique. Because if two such geodesics exist, then they would be homotopic, and Klingenberg's lemma would assert that there exist a sequence of curves of lengths $$\ge \pi\sqrt n$$ in this homotopy.

Question:

I am not sure how the fact that the geodesic being unique implies that $$\exp_p : T_p M \to M$$ is a diffeomorphism. However, if this implication is true, then I think that should complete my proof. At the very least least, I am aware that, since $$M$$ is (geodesically) complete, $$\exp_p$$ is defined for all $$\sigma \in T_p(M)$$.

• Your statement of Klingenberg's lemma is messed up -- the hypothesis should be $K\le K_0$, not $K\ge K_0$. And more importantly, you've left out half of the hypothesis and all of the conclusion. May 16 '17 at 1:17

Surjectiveness : For any $q\in M$, there is a minimizing geodesic from $p$ to $q$, i.e., $c(t)=\exp_p\ tv,\ |v|=1$. Hence $c({\rm dist}\ (p,q))=q$ so that the map $\exp_p$ is a surjective map.
Injectiveness : If $\exp_p\ v=\exp_p\ w=q\ \ast$, then we have two geodesics $c_1(t)=\exp_p\ tv,\ c_2(t)=\exp_p\ tw$ from $p$ to $q$. By uniqueness, $v=Cw$ for some $C$. And $\ast$ implies that $C=1$.