Rabah Souam, Joeri Van der Veken
Published 2010-06-07, updated 2010-07-08Version 2
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.
Comments: minor corrections to v1
Manifolds admitting a metric with co-index of symmetry 4
Parallel and totally umbilical hypersurfaces of the four-dimensional Thurston geometry $\text{Sol}^4_0$
Topological properties of manifolds admitting a $Y^x$-Riemannian metric
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