As the title suggests, I am trying to prove that a connected, simply connected, complete riemannian manifold that is locally isometric to $\mathbb{R}^n,S^n, H^n$ is isometric to $\mathbb{R}^n,S^n,H^n$.
The idea of the proof I have consists in extending a local isometry (from $M$ to one of those three spaces, which I will call $X$) $\phi$ defined in a geodesic ball centered at $p \in M$ to all the manifold, and this should be done the following way. Consider a point $q$ that is not in the domain of $\phi$. Then consider a geodesic starting from $p$ at speed $v$ and ending in $q$ at the time $t$. I define $\\psi (q)$ as the point where the geodesic in $X$ starting at $\phi(p)$ with speed $d\phi_{p}(v)$. The question is: how do I continue? I have only the "idea" for what comes next, but I can't put it all together. I know it should be all "well-defined" because of the simple connectedness, and that this $\psi$ that extends $\phi$ should be a local isometry, and surjective. From this fact one should conclude that $\psi$ is an isometry, but I really don't know how to put it all together. There should also be some geodesic balls that "cover" the geodesic from $p$ to $q$, but I really don't have a clue about their role in the proof.
I really don't need a perfectly rigorous proof, I'd be happy to just understand what is going on here.
See Theorem 4.1 (in chapter 8) on page 163 of M. Do Carmo "Riemannian Geometry".