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Let $(M,g)$ be a complete smooth Riemannian manifold. Assume that for two points $p$ and $q$ in $M$, there is a unique minimizing geodesic connecting them. Denote the distance function to $p$ by $d_p(x)$. Then is $d_p(x)$ differentiable at $q$? If this is true, then how to prove it? Furthermore, is $d_p(x)$ smooth at $q$?

Uncertain Wang
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    This has been asked before on MO: https://mathoverflow.net/questions/21295/smoothness-of-distance-function-in-riemannian-manifolds – Jack D'Aurizio Dec 03 '17 at 02:55

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