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 $H/H \cap L$ 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.