I am working on a problem from Do-Carmo Riemannian Geometry book, I have been working on it all day, I would appreciate any advice or suggestion, the problem says:
Let p be a point in a complete, non-compact riemannian manifold M. Prove there is a geodesic $ \gamma : [0 , \infty] \to M $ with initial value $\ \gamma (0) = p$, having the property that $ \gamma$ is a minimal geodesic between $\gamma(0)$ and $\gamma(s)$ for any $ s \in [0, \infty ]$ .
My ideas: If there did not exist that geodesic, that would mean that for every $v \in TpM$ with $|v|=1$ I would have an correspondence : v $\to$ max{t} such that the geodesic $exp_p(tv)$ minimizes the distance. If that transformation is continuous, It would reach it maximum and being $exp_p$ suprayective (Hopf-Rinow theorem) I could conclude that M is compact, Contradiction! But of course i can not see why that transformation would be continuous.
Another idea would be that at points where the geodesic don't minimize distance, For the Hopf-Rinow theorem, there would be another geodesic which would minimize, therefore that point is a critical point of $dexp_p$ and i get a Jacobi field. But I don't know what else i can do working on this idea.