Hilbert's fifth problem and proper actions of Lie groups
Soren Illman
(
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Suppose $G$ is a locally euclidean group and $M$
is a locally euclidean space, and let
\begin{equation}
\Phi : G \times M \longrightarrow M
\end{equation}
be a continuous action of $G$ on $M$. In his fifth problem Hilbert
asks if one then can choose the local coordinates in $G$ and $M$ so
that $\Phi$ is real analytic.
When $G = M$ and
\begin{equation}
\Phi : G \times G \longrightarrow G
\end{equation}
is the multiplication in the group $G$ the answer to Hilbert's
question is affirmative, as was proved by Gleason, Montgomery and
Zippin.
For the question (1) we prove.
\medskip
\noindent
{\it Theorem.} Let $G$ be a Lie group which acts on a $C^1$ smooth
manifold $M$ by a $C^1$ smooth proper action. Then there exists a
real analytic structure $\beta$ on $M$, compatible with the given
smooth structure on $M$, such that the action of $G$ on $M_{\beta}$ is
real analytic.
\medskip
Concerning the uniqueness of $\beta$ in Theorem 1 we have (from a
paper by the author and Marja Kankaanrinta).
\medskip
\noindent
{\it Theorem.} Let $M$ and $N$ be real analytic proper $G$-manifolds,
where $G$ is a linear Lie group. Suppose that $M$ and $N$ are
$G$-equivariantly $C^1$ diffeomorphic. Then $M$ and $N$ are
$G$-equivariantly real analytically isomorphic.
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