Totally umbilical hypersurfaces of manifolds admitting a unit Killing field

Details Totally umbilical hypersurfaces of manifolds admitting a unit Killing field Totally umbilical hypersurfaces of manifolds admitting a unit Killing field

2010-06-07

arxiv.org/abs/1006.1261v2

Mathematics

Differential Geometry

minor corrections to v1

Scientific paper

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.

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