The fine structure of transitive Riemannian isometry groups. I

Let M M be a connected homogeneous Riemannian manifold, G G the identity component of the full isometry group of M M and H H a transitive connected subgroup of G G . G = H L G = HL , where L L is the isotropy group at some point of M M . M M is naturally identified with the homogeneous space H / H ∩ L H/H ∩ L endowed with a suitable left-invariant Riemannian metric. This paper addresses the problem: Given a realization of M M as a Riemannian homogeneous space of a connected Lie group H H , describe the structure of the full connected isometry group G G in terms of H H . This problem has already been studied in case H 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.