Given a real analytic compact manifold $M$ with boundary $\partial M$, suppose that $M$ is embedded in an open analytic manifold $N$ which has the same dimension as $M$. Is the distance function $d(x)$ to the boundary $\partial M$ real analytic in a neighborhood of $\partial M$ in $N$? Or how can we define the distance function to make it analytic in that neighborhood? Is there any literature for that?
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1$\begingroup$ As stated it is false, but for the "wrong reasons". You are basically asking whether the distance function to an analytic submanifold of an analytic manifold is analytic. But this is false even for $\mathbb{R}^k \hookrightarrow \mathbb{R}^N$. Do you want the "signed distance" function instead? In that case you need that $M$ has the same dimension as $N$ to even define it. $\endgroup$– Willie WongCommented Dec 11, 2016 at 23:47
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$\begingroup$ $M$ and $N$ are in the same dimension. $\endgroup$– mathpdeCommented Dec 11, 2016 at 23:50
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1$\begingroup$ @mathpde, edits to your question should be made in the question, not just in the comments. $\endgroup$– LSpiceCommented Dec 12, 2016 at 0:21
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1$\begingroup$ I believe this question arose in an earlier question of yours. However, in that question, you don't need the distance function to be real analytic. You just need the distance function restricted to $M$ to be extended as a real analytic function to a neighborhood of $\partial M$. As Ben McKay indicated, this can be done, but it won't be the distance function outside $M$. It's easy to see that the distance function can't work by choosing $M$ to be a half-line and $\partial M$ the single point on the boundary. $\endgroup$– Deane YangCommented Dec 12, 2016 at 0:40
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1$\begingroup$ The first google entry for "signed distance function" is the wikipedia page en.wikipedia.org/wiki/Signed_distance_function $\endgroup$– Ben McKayCommented Dec 24, 2016 at 22:08
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