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Totally umbilical hypersurfaces of manifolds admitting a unit Killing field

Authors: Rabah Souam and Joeri Van der Veken
Journal: Trans. Amer. Math. Soc. 364 (2012), 3609-3626
MSC (2010): Primary 53B25, 53C40, 53C42
Published electronically: February 20, 2012
MathSciNet review: 2901226
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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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Additional Information

Rabah Souam
Affiliation: Institut de Mathématiques de Jussieu, CNRS UMR 7586, Université Paris Diderot, Paris 7, “Géométrie et Dynamique”, Site Chevaleret, Case 7012, 75205, Paris Cedex 13, France

Joeri Van der Veken
Affiliation: Departement Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, Box 2400, BE-3001 Leuven, Belgium

Keywords: Totally umbilical, totally geodesic, product manifold, Killing field, warped product
Received by editor(s): June 29, 2010
Published electronically: February 20, 2012
Additional Notes: The second author is a post-doctoral researcher supported by the Research Foundation, Flanders (F.W.O.)
This work was done while the second author visited the Université Paris Diderot, Paris 7 supported by a grant of the Research Foundation, Flanders (F.W.O.)
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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