Classification of Killing-transversally symmetric spaces

Abstract

We treat Killing-transversally symmetric spaces (briefly, KTS-spaces), that is, Riemannian manifolds equipped with a complete unit Killing vector field such that the reflections with respect to the flow lines of that field can be extended to global isometries. Such manifolds are homogeneous spaces equipped with a naturally reductive homogeneous structure and they provide a rich set of examples of reflection spaces. We prove that each simply connected reducible KTS-space $M$ is a Riemannian product of a symmetric space $M^{\prime}$ and a special kind of KTS-space $M^{\prime\prime}$, called a contact KTS-space. Such a particular manifold $M^{\prime\prime}$ is an irreducible, odd-dimensional principal $G^{1}$-bundle over a Hermitian symmetric space. The main purpose of the paper is to give a classification of this special class of manifolds $M^{\prime\prime}$.