# References for actions of infinite-dimensional Banach-Lie groups on infinite-dimensional Banach manifolds

I am starting to study infinite-dimensional manifolds, specifically, Banach manifolds. I found some interesting introductory texts in which the mathematical background is developed with some detail. However, I am not able to find some organic treatment of the differential geometry of the orbits of a smooth (analytic) action of an infinite-dimensional Banach-Lie group $\mathcal{G}$ on Banach manifolds $\mathcal{M}$. I am particularly interested in the case of non-proper actions, and I would like to know if and under what assumptions the orbits are Banach manifolds.

Are there articles/books developing the subject?

Thank You.

EDIT

Here and in "Bourbaki: Lie groups and Lie algebras, chapters 2 and 3", I found that, whenever the isotropy subgroup $\mathcal{G}_{m}$ at $m\in\mathcal{M}$ is a split Lie subgroup of $\mathcal{G}$ (that is, a subgroup which is a Lie group in the subspace topology), then $\mathcal{G}/\mathcal{G}_{m}$ is an analytic Banach manifold, and $\pi\colon\mathcal{G}\rightarrow\mathcal{G}_{m}$ is a submersion. Clearly, there is a bijection $\gamma$ from $\mathcal{G}/\mathcal{G}_{m}$ to the orbit $\mathcal{G}\cdot m$ through $m\in\mathcal{M}$. What do I have to do to ensure that the bijection $\gamma$ turns $\mathcal{G}\cdot m$ into a Banach manifold?

• I would also like a nice source for this. Many standard texts for Banach manifolds do not talk much about Banach Lie groups and their actions. – user98602 Apr 12 '16 at 21:14

## 1 Answer

Here are some partial answers: It is known that Banach Lie group actions on finite-dimensional manifolds are quite restricted. What I mean by this is: Due to a theorem by Omori, see

Hideki Omori, On Banach-Lie groups acting on finite dimensional manifolds, Tohoku Math. J. (2) 30 (2), 223-250, 1978

if a Banach-Lie group acts smoothly, effectively and transitively on a finite-dimensional manifold, then it automatically is finite-dimensional. Hence for many manifolds which come up, one can only consider actions which do not satisfy these requirements or one is forced outside of the class of Banach Lie groups (e.g. if one considers diffeomorphism groups of finite-dimensional manifolds).

Quotient theorems, or smooth structures on orbits for infinite-dimensional group actions are in general much harder to establish than in the finite-dimensional case. Something along the lines you ask for can be done using the results on submersions in the infinite-dimensional setting e.g. in

Helge Glöckner: Fundamentals of submersions and immersions between infinite-dimensional manifolds. arXiv:1502.05795

Also in your setting a version of Godements theorem (see references in the last paper) is available and could be helpful.