Homogeneous Spaces of Nilpotent and Solvable Groups

Abstract

Homogeneous spaces of nilpotent and solvable Lie groups are called nilmanifolds and solvmanifolds respectively. In addition it is assumed that the transitive Lie group is connected. At this time no comprehensive classification of nil- and solvmanifolds is known, as there is no classification of nilpotent and solvable Lie algebras. In this chapter we give a survey of results, which reduce the study of arbitrary nil- and solvmanifolds to the compact case, and also of results about topological construction of compact nil- and solvmanifolds and about transitive actions of Lie groups on them.