I'm trying understand the proof of the following theorem:

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

Proof. Since $M$ is complete $\exp_p: T_pM \longrightarrow M$ is defined for all $p \in M$ and is surjective. By lemma $3.2$, $\exp_p$ is a local diffeomorphism. This allows us to introduce a Riemannian metric on $T_pM$ in such a way that $\exp_p$ is a local isometry. Such metric is complete because the geodesics of $T_pM$ passing through the origin are straight lines (Cf. Theorem $2.8$, $(a) \Longrightarrow (d)$). From lemma $3.3$, $\exp_p$ is a covering map. Since $M$ is simply connected, $\exp_p$ is a diffeomorphism. $\square$

My doubt is why $M$ simply connected and $\exp_p$ a covering map imply that $\exp_p$ is a diffeomorphism? I don't have much background in covering maps (indeed, I never did a course in convering maps, a few that I know it is reading some references, but it doesn't clear for me how to prove that $\exp_p$ is a diffeomorphism).

Thanks in advance!


I think that I understood why $\exp_p$ is a diffeomorphism. I put my attempt below.

Using the isomorphism of $\mathbb{R}^n$ in $T_pM$ ($n = \dim T_pM$), we can show that T_pM is arcwise connected, i. e., given two points in $M$, exist two curves joining this points and are homotopic between them.

The following result is in Differential Geometry of Curves and Surfaces by Manfredo P. do Carmo:

$\textbf{COROLLARY.}$ Let $\pi: \tilde{B} \longrightarrow B$ a covering map, $\tilde{B}$ arcwise connected and $B$ is simply connected. Then $\pi$ is a homeomorphism.

By the previous corollary, $\exp_p$ is a homeomorphism.

The fact that $\exp_p$ is a diffeomorphism follows because $\exp_p$ is a local diffeomorphism, has inverse and the inverse it is differentiable since differentiability is a local property and $\exp_p$ is a local diffeomorphism.


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