A locally Killing-transversally symmetric space (locally KTS-space) is a Riemannian manifold admitting a unit Killing vector field such that the local reflections with respect to the flow lines are isometries. Although the class of locally KTS-spaces is much broader than that of Sasakian \(\varphi\)-symmetric spaces, their geometric properties are to a large extent similar, and this leads to a list of analogous characteristic properties which are derived in the paper.
Section 2 collects a series of definitions and results about these spaces which provide the basic material for the further treatment. In Section 3, the authors derive the new characterizations by means of properties of tubes about the flow lines, first using extrinsic geometry and then by considering intrinsic properties. A similar line is followed in Section 4, where the tubular hypersurfaces which are orthogonal to the flow lines are used.