The Automorphism Group of a Banach Principal Bundle with {1}-structure

TL;DR

This paper demonstrates that the automorphism group of a Banach principal bundle with a {1}-structure can be structured as a Banach-Lie group acting smoothly, especially when infinitesimal automorphisms are complete, with applications to affine Banach manifolds.

Contribution

It establishes conditions under which the automorphism group of a Banach principal bundle with a {1}-structure forms a Banach-Lie group, extending to affine Banach manifolds.

Findings

Automorphism group can be given a Banach-Lie group structure.

Smooth action of automorphism group on the principal bundle.

Application to geodesically complete affine Banach manifolds.

Abstract

A {1}-structure on a Banach manifold M (with model space E) is an E-valued 1-form on M that induces on each tangent space an isomorphism onto E. Given a Banach principal bundle P with connected base space and a {1}-structure on P, we show that its automorphism group can be turned into a Banach-Lie group acting smoothly on P provided the Lie algebra of infinitesimal automorphisms consists of complete vector fields. As a consequence we show that the automorphism group of a connected geodesically complete affine Banach manifold M can be turned into a Banach-Lie group acting smoothly on M.