I read the theorem 1.4.7. in "Riemannian Geometry and Geometric Analysis" (2005) that states:
$\textbf{Theorem 1.4.7.}$ Let $M$ be a compact Riemannian manifold. Then for any $p \in M$, the exponential map $\exp_p$ is defined on all of $T_pM$, and any geodesic may be extended indefinitely in each direction.
$\textbf{My question:}$ Let the sphere $\mathbb{S}^2$ to be the compact Riemannian manifold, then can we deduce the fact from the theorem that: you have a geodesic as the trajectory for a particle on the surface of the sphere, if the sphere is expanded indefinitely, the particle will still reach point B due to that $\exp_p$ is defined on all $T_p\mathbb{S}^2$.