Slice theorem and orbit type stratification in infinite dimensions

TL;DR

This paper extends the classical slice theorem to infinite-dimensional settings using advanced tools like the Nash--Moser theorem, and shows that the orbit type decomposition forms a stratification.

Contribution

It generalizes the slice theorem to infinite-dimensional manifolds with group actions, introducing graded Riemannian metrics and establishing stratification of orbit types.

Findings

Established a slice theorem for locally convex Lie group actions on manifolds.

Proved that the orbit type decomposition is a stratification.

Applied inverse function and Nash--Moser theorems to specific infinite-dimensional cases.

Abstract

We establish a general slice theorem for the action of a locally convex Lie group on a locally convex manifold, which generalizes the classical slice theorem of Palais to infinite dimensions. We discuss two important settings under which the assumptions of this theorem are fulfilled. First, using Gl\"ockner's inverse function theorem, we show that the linear action of a compact Lie group on a Fr\'echet space admits a slice. Second, using the Nash--Moser theorem, we establish a slice theorem for the tame action of a tame Fr\'echet Lie group on a tame Fr\'echet manifold. For this purpose, we develop the concept of a graded Riemannian metric, which allows the construction of a path-length metric compatible with the manifold topology and of a local addition. Finally, generalizing a classical result in finite dimensions, we prove that the existence of a slice implies that the decomposition of the manifold into orbit types of the group action is a stratification.