The Fine Structure of Transitive Riemannian Isometry Groups. I

Let M be a connected homogeneous Riemannian manifold, G the identity component of the full isometry group of M and H a transitive connected subgroup of G. $G = HL$, where L is the isotropy group at some point of M. M is naturally identified with the homogeneous space H/H ∩ L endowed with a suitable left-invariant Riemannian metric. This paper addresses the problem: Given a realization of M as a Riemannian homogeneous space of a connected Lie group H, describe the structure of the full connected isometry group G in terms of H. This problem has already been studied in case H is compact, semisimple of noncompact type, or solvable. We use the fact that every Lie group is a product of subgroups of these three types in order to study the general case.