Let $(M,g)$ be a compact Riemannian manifold. I'm supposed to show that for any fixed $p\in M$, the exponential map $\text{exp}_p:T_pM \rightarrow M$ is defined on all of $T_p$.
So far I have:
- By Hopf Rinow, since $M$ is complete as a metric space, it is also geodesically complete.
- the domain of $\text{exp}_p$ is star-shaped around zero, so I find a ball $B(\epsilon,p) \subset T_pM$ lying in that domain. Can I expand that domain via Hopf Rinow to get the entire space?
Any help appreciated, thanks!