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!

  • 2
    $\begingroup$ Aren't you done? The statement that you want to show is precisely that of geodesically completness. $\endgroup$ – user99914 Jan 16 '18 at 14:14
  • $\begingroup$ we defined geodesically completeness as "every maximal geodesic is defined for all $t \in \mathbb{R}$". but that's obviously the same isn't it? @JohnMa $\endgroup$ – Simonsays Jan 16 '18 at 14:22
  • $\begingroup$ @Simonsays: The difference of what you wrote with geodesic completeness is purely semantical. $\endgroup$ – Moishe Kohan Jan 17 '18 at 18:30

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