Let $(M,g)$ be a Riemannain manifold and let $p\in M$. Let $\gamma:[0,1] \to M$ be a smooth curve and let $p \notin \gamma([0,1])$. Assume further that for each $t \in [0,1]$ there is a unique (unit speed) geodesic from $p$ to $\gamma(t)$ with initial speed $v(t) \in T_p M$. Is $t \mapsto v(t)$ differentiable? If so, what is the derivative of $v$ (in terms of $\gamma, g, p$)?
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$\begingroup$ In the Euclidean case, wouldn't $v(t)= \gamma(t)-p$ (in the usual coordinates)? In this case the smoothness of $\gamma$ implies the smoothness of $v$. $\endgroup$– MathavCommented Mar 1, 2022 at 15:19
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$\begingroup$ yes, sorry I didn't read the problem correctly. $\endgroup$– Ben McKayCommented Mar 1, 2022 at 15:29
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