Let $\mathcal{M}$ be a Riemannian submanifold and $\Sigma \subset \mathcal{M}$ be a $C^1$ submanifold. Near $\Sigma$, let $d(x)=\pm d(x, \Sigma)$ be the signed distance function to $\Sigma$. Then what can we say about the smoothness of $d$ near $\Sigma$? If $\Sigma$ is a $C^{k,\alpha}$ submanifold, is $d$ also $C^{k,\alpha}$? In Euclidean space, we know that if $\Sigma$ is $C^k$,$k \ge 2$. Then $d$ is also $C^k$ near $\Sigma$. Is this correct in Riemannian manifolds?
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1$\begingroup$ In Chapter 6 of Lee's Riemannian Manifolds book, he proves that $d$ is $C^\infty$ near $\Sigma$ under the assumption that $\Sigma$ is $C^\infty$. (Theorem 6.40) Perhaps the proof can be generalised. $\endgroup$– Nate RiverCommented Dec 30, 2023 at 9:35
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$\begingroup$ If $d$ is $C^1$ in a neighborhood of $\Sigma$, then normal curvatures of $\Sigma$ are bounded. Therefore $\Sigma$ should be at least $C^2$. $\endgroup$– Anton PetruninCommented Jun 12 at 14:33
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