Closedness of the tangent spaces to the orbits of proper actions

TL;DR

This paper proves that for proper Banach-Lie group actions on Banach manifolds, the tangent spaces of orbits are closed, enabling a natural manifold structure on the quotient in the Hilbert case.

Contribution

It establishes the closedness of tangent spaces to orbits under proper Banach-Lie group actions, facilitating manifold structures on quotients.

Findings

Tangent maps have closed range for proper Banach-Lie group actions.

Orbits' tangent spaces are closed in Banach manifolds.

Quotients by free proper actions on Hilbert manifolds are manifolds.

Abstract

In this note we show that for any proper action of a Banach--Lie group $G$ on a Banach manifold $M$, the corresponding tangent maps $\g \to T_x(M)$ have closed range for each $x \in M$, i.e., the tangent spaces of the orbits are closed. As a consequence, for each free proper action on a Hilbert manifold, the quotient $M/G$ carries a natural manifold structure.