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Totally umbilical hypersurfaces of manifolds admitting a unit Killing field

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.

preprint2010arXivOpen access
Rabah SouamJoeri Van der Veken
math.DG
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Rabah SouamJoeri Van der Veken

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