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Let $M$ be a compact smooth manifold and let $G$ be a connected compact Lie group acting on $M$. According to an old theorem of Mostow and Palais, there exists a $G$-equivariant embedding of $M$ into some finite dimensional orthogonal representation.

Question: suppose now that $M$ is a manifold with corners. Does the conclusion still hold?

I can't imagine how such a statement could fail to generalize, but I'm hoping that there is some trick which would avoid having to rewrite the proof. (Or perhaps a reference?)

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    $\begingroup$ Option 1: State without proof the correct claim that Palais' argument goes through without change. Option 2: Argue that there is an embedded copy of $M$ contained within its interior (with some careful application of the appropriate kind of collar neighborhoods in the context of corners) and apply Mostow-Palais to the interior of M; get the results for M for free. $\endgroup$
    – mme
    Commented Nov 18, 2022 at 15:58
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    $\begingroup$ @mme Thanks! Concerning Option 2, I guess you mean that there should be some sort of equivariant collar neighborhood theorem for manifolds with corners? I agree that this should be true. $\endgroup$ Commented Nov 18, 2022 at 16:49
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    $\begingroup$ Yes, this is what I mean. You would do this re-embedding locally inductively on the strata. Since M is compact you could even do it one chart at a time. At this point you are almost just rewriting Palais' proof. Option 1 is probably what I would do. :) $\endgroup$
    – mme
    Commented Nov 18, 2022 at 17:00
  • $\begingroup$ @mme Thanks a lot -- sounds reasonable. $\endgroup$ Commented Nov 19, 2022 at 1:07

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