Slice theorem and orbit type stratification in infinite dimensions

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 Glöckner's inverse function theorem, we show that the linear action of a compact Lie group on a Fréchet space admits a slice. Second, using the Nash--Moser theorem, we establish a slice theorem for the tame action of a tame Fréchet Lie group on a tame Fréchet 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.

References in corpus (6)

• Overview of the Geometries of Shape Spaces and Diffeomorphism Groups
• Orbispaces as differentiable stratified spaces
• Fundamentals of submersions and immersions between infinite-dimensional manifolds
• Projective unitary representations of infinite dimensional Lie groups
• Closedness of the tangent spaces to the orbits of proper actions
• No slices on the space of generalized connections

Cited by in corpus (6)

• Identification is Pointless: Quantum Coordinates, Localisation of Events, and the Quantum Hole Argument
• The Heterotic-Ricci flow and its three-dimensional solitons
• Normal Form of Equivariant Maps and Singular Symplectic Reduction in Infinite Dimensions with Applications to Gauge Field Theory
• Short survey on the existence of slices for the space of Riemannian metrics
• Singular symplectic cotangent bundle reduction of gauge field theory
• Normal form of equivariant maps in infinite dimensions