Totally umbilical hypersurfaces of manifolds admitting a unit Killing field

TL;DR

This paper characterizes totally umbilical hypersurfaces in Riemannian products involving a Euclidean line and provides conditions for totally geodesic surfaces in three-manifolds with a unit Killing field.

Contribution

It offers a complete description of totally umbilical hypersurfaces in warped product manifolds and establishes criteria for totally geodesic surfaces in three-manifolds with a unit Killing field.

Findings

Riemannian product M x R admits totally umbilical hypersurfaces iff M is locally warped.

Complete description of totally umbilical hypersurfaces in such warped products.

Necessary and sufficient conditions for totally geodesic surfaces in three-manifolds with a unit Killing field.

Abstract

We prove that a Riemannian product of type M x R (where R denotes the Euclidean line) admits totally umbilical hypersurfaces if and only if M has locally the structure of a warped product and we give a complete description of the totally umbilical hypersurfaces in this case. Moreover, we give a necessary and sufficient condition under which a Riemannian three-manifold carrying a unit Killing field admits totally geodesic surfaces and we study local and global properties of three-manifolds satisfying this condition.