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.
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References
Kobayashi, S.:Transformation Groups in Differential Geometry, Springer, New York, 1972.
Kobayashi, S. and Nomizu, K.:Foundations of Differential Geometry, Wiley Interscience, New York, 1963.
Zimmer, R. J.:Ergodic Theory and Semisimple Groups, Birkhäuser, Boston, 1984.
Zimmer, R. J.: Ergodic theory and the automorphism group of aG-structure, inGroup Representations, Ergodic Theory, Operator Algebras, and Mathematical Physics, MSRI Publications, Springer, New York, 1987.
Zimmer, R. J.: Lattices in semisimple groups and invariant geometric structures on compact manifolds, inDiscrete Groups in Geometry and Analysis, Progr. Math. 67, Birkhäuser, Boston, 1987.
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Research completed while a member of the University of Chicago Mathematics Department.