Totally umbilical hypersurfaces of manifolds admitting a unit Killing field

Souam, Rabah; Van der Veken, Joeri

doi:10.1090/S0002-9947-2012-05472-9

Totally umbilical hypersurfaces of manifolds admitting a unit Killing field
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by Rabah Souam and Joeri Van der Veken PDF

Trans. Amer. Math. Soc. 364 (2012), 3609-3626 Request permission

Abstract:

We prove that a Riemannian product of type $\mathbb {M}^n \times \mathbb {R}$ admits totally umbilical hypersurfaces, which are neither horizontal nor vertical, if and only if $\mathbb {M}^n$ 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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