Compact Homogeneous Spaces

Abstract

In this chapter we shall study homogeneous spaces G/H, under the assumption of their compactness. This assumption is trivially satisfied if the group G is compact, but it imposes very strong restrictions on the subgroup H in the case when G is not compact. The problem of describing the Lie subgroups H ⊂ G such that G/H is compact (uniform Lie subgroups) is considered in §1. In connection with this problem we consider the real analogue of a remarkable fibre bundle introduced by Tits in the case of complex compact homogeneous spaces (Tits 1962, see also Part IV of Encycl. Math. Sc. 10). In §2 we give a survey of results on classification of transitive actions on compact manifolds with a finite fundamental group. The remaining part of this chapter is devoted to the study of compact homogeneous spaces of general type, based on certain natural fiberings of these spaces, which make it possible to clarify their topological structure. The majority of results of this chapter hold, sometimes with obvious changes, for a wider class of homogeneous spaces than the compact ones, namely: for plesi-compact homogeneous spaces. A homogeneous space M = G/H of a Lie group G is called plesicompact (Gorbatsevich 1988), if there exists a uniform subgroup P of G, containing H and such that the space P/H has a finite P-invariant measure.