Let $(M, g)$ be a complete Riemannian manifold and let $(\tilde{M}, \tilde{g})$ its universal cover. Let $\pi : \tilde{M} \to M$ be the covering map.
Let $\gamma : I \to (M, g)$ be a smooth curve with $I \subset \mathbb{R}$ be an interval. Let $t_0 \in I$ and let $\tilde{x} \in \tilde{M}$ be a point such that $\pi(\tilde x) = \gamma(t_0)$.
Then I want to prove that there exists a unique curve $\tilde{\gamma} : I \to (\tilde M, \tilde g)$ such that $\pi \circ \tilde \gamma = \gamma$ and $\tilde\gamma(t_0) = \tilde x$.
EDIT: I am interested in this question because I'm trying to understand the proof of Theorem69[Structure Theorem for Nonnegative Ricci Curvature] of Section 3.5 of Chapter 9 of Riemannian Geometry book by Petersen. I also can't understand why the geodesics $\gamma_i$ defined in the proof converge to a geodesic $\hat \gamma$. But maybe I will ask this in an other question.
$2^{nd}$ EDIT: I think I have understood why the geodesic $\gamma_i$ converge to $\hat \gamma$. :)