Question: Given a geodesically complete regular surface $S\in\mathbb{R}^3$, $p\in S$, there is a well-defined exponential map $\exp_{p}:T_p S\rightarrow S$ which is a local diffeomorphism at $0\in T_p S$. It seems that $\exp_p$ must be continuous on $T_p S$, but I cannot prove it. I look at my textbook and see that the author uses a theorem about local solutions of ODE to prove that $\exp_{p}$ is locally smooth and hence locally continuous at $0$. So I don't see how to prove the global continuity of $\exp_{p}$. Could anyone suggest a way to verify the continuity?
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2The fact that $S$ is complete means that the solutions of the geodesic equation exist and are smooth for all time. To show $\exp_p$ is smooth, one must show that the solutions of the geodesic equation also depend smoothly on the initial velocity. – Kajelad Aug 09 '20 at 17:41