On the existence of a fundamental domain for Riemannian transformation groups

Abstract

This is just a generalization of the classical concept of fundamental domain, usually formulated for G discrete. In this work we show that fundamental domains with nice properties exist for a large class of nondiscrete transformation groups: We assume that M is a complete, connected Riemannian manifold and that G is a closed subgroup of the group of isometries of M and show that the question can be reduced to asking whether a certain principal fiber bundle admits a sufficiently large cross-section. The technique we use to reduce the problem has other applications to the theory of transformation groups; therefore we devote the later sections to working out some miscellaneous differential-geometric results that will be useful in further work. 2. Preliminaries. All manifolds, maps, action of groups, curves, etc., will be differentiability class Co unless mentioned otherwise. We