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

Isotropic Jacobi fields on naturally reductive spaces

“…Remark 5.4. In [7], the simply connected Killing-transversally symmetric spaces are introduced as simply connected Riemannian manifolds equipped with a complete unit Killing vector field ξ such that all reflections with respect to the flow lines of ξ are global isometries. These manifolds are naturally reductive spaces and ξ is G-invariant with respect to a naturally reductive representation (G/G o , g).…”

Geodesic tubes and normal flow space forms

“…In that case we mention the following useful result. Proposition 2.3 [6]. Any complete, contact locally KTS-space (M, g, F ξ ) is transversally modelled on a Hermitian symmetric space M .…”

“…This vector field determines an isometric flow on the manifold and for that reason the corresponding geometry has been called flow geometry or also generalized Sasakian geometry. We refer to [5], [6] for more details and further references. In [7] the authors studied the notion of a normal flow space form.…”

Transversally symmetric immersions

“…Moreover, the submanifold is a locally Killing-transversally symmetric space. The local and global theory of these spaces was developed in [6], [8]. These spaces are in a natural way analogues of symmetric spaces, and the (^-symmetric spaces from Sasakian geometry are interesting examples which are analogues of Hermitian symmetric spaces.…”