6
$\begingroup$

I am looking for a reference treating the slice theorem for Banach Lie group actions on Banach manifolds, i.e. proving that a smooth, free and proper action of a Banach Lie group $G$ on a Banach manifold $M$ with embedded orbits ensures that the quotient $M/G$ inherits the structure of a Banach manifold, s.t. $M\rightarrow M/G$ becomes a $G$-principal bundle.

I know this is briefly treated in Bourbaki's "Lie groups and Lie algebras". Unfortunately, in the proof they refer to the Bourbaki book "differentiable and analytic manifolds" which is not available to me.

Could anyone please provide me with a reference?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
1
$\begingroup$

Dear Orbicular the theorem on the existence of slices is stated without proof as Theorem 5.2.6 in Critical Point Theory and Submanifold Geometry, LNM 1353, of Palais and Terng (for example see here).
The proof should be adapted without difficulty from that in the finite-dimensional case.
For this case you can look at ``On the existence of slices for actions of non-compact groups'' by Palais (for example see here).

Improve this answer
$\endgroup$
1
  • 1
    $\begingroup$ As far as I can tell the Palais, Terng reference only proves this when the quotient M/G is finite dimensional (this is the upshot of their Fredholm assumption). Is there a better reference? $\endgroup$
    – Jesse Wolfson
    Commented Dec 23, 2020 at 23:42

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .