Totally umbilical hypersurfaces of manifolds admitting a unit Killing field (Q2839946)

The authors study the following two questions:NEWLINENEWLINE (1) When does a Riemannian product manifold of the type \(M^n\times\mathbb R\) admit totally umbilical hypersurfaces and what are they?NEWLINENEWLINE (2) When does a Riemannian three-manifold with a unit Killing vector field admit totally umbilical surfaces and what are they?NEWLINENEWLINE The first part of the paper gives a complete answer to the first problem. More precisely, the authors show that \(M^n\times\mathbb R\) admits totally umbilical hypersurfaces which are neither horizontal nor vertical if and only if \(M^n\) itself has locally the structure of a warped product. Furthermore, they present a parametrization for totally umbilical hypersurfaces of such a manifold. Well-known examples of three-dimensional Riemannian manifolds admitting a unit Killing vector field are Riemannian products of the type \(M^n\times\mathbb R\), the unit sphere \(S^3\), Berger spheres and the Thurston spaces \(\widetilde{SL}(2,\mathbb R)\) and \(\mathrm{Nil}_3\), see [\textit{V. N. Berestovskij} and \textit{Yu. G. Nikonorov}, Sib. Mat. Zh. 49, No. 3, 497--514 (2008); translation in Sib. Math. J. 49, No. 3, 395--407 (2008; Zbl 1164.53346)]. The second part of the paper provides necessary and sufficient conditions for such three-manifolds to admit totally geodesic surfaces. The authors describe the totally geodesic surfaces and study local and global properties of Riemannian three-manifolds with a unit Killing vector field that admit totally geodesic surfaces which are neither orthogonal nor tangent to this field.