Bundle automorphisms for fiberG-structures of finite type

Abstract

In this paper we study geometric settings where a Lie group preserving a measurable field of measurable Riemannian metrics on the fibers of a smooth fiber bundle must actually preserve a measurable field of smooth Riemannian metrics. For ergodic actions on bundles with compact fiber this will imply that the standard fiber is a homogeneous space for a compact Lie group. In particular we show this conclusion holds for a semisimple Lie group of higher real rank (or a lattice subgroup) preserving a finite measure and either a field of connections or pseudo-Riemannian metrics when the fiber is compact and of ‘low’ dimension.