# Totally umbilical hypersurfaces of manifolds admitting a unit Killing field

@article{Souam2010TotallyUH, title={Totally umbilical hypersurfaces of manifolds admitting a unit Killing field}, author={Rabah Souam and Joeri Van der Veken}, journal={Transactions of the American Mathematical Society}, year={2010}, volume={364}, pages={3609-3626} }

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… Expand

#### 14 Citations

Totally umbilical hypersurfaces of product spaces

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Given a Riemannian manifold M, and an open interval I ⊂ R, we characterize nontrivial totally umbilical hypersurfaces of the product M×I — as well as of warped products I ×ω M — as those which are… Expand

Associated Families of Surfaces in Warped Products and Homogeneous Spaces

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- Bulletin of the Belgian Mathematical Society - Simon Stevin
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The classification of totally umbilical surfaces in homogeneous 3-manifolds

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We obtain an exhaustive classification of totally umbilical surfaces in unimodular and non-unimodular simply-connected 3-dimensional Lie groups endowed with arbitrary left-invariant Riemannian… Expand

Compact stable surfaces with constant mean curvature in Killing submersions

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A Killing submersion is a Riemannian submersion from a 3-manifold to a surface, both connected and orientable, whose fibers are the integral curves of a Killing vector field, not necessarily unitary.… Expand

On the classification of Killing submersions and their isometries

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A Killing submersion is a Riemannian submersion from an orientable 3-manifold to an orientable surface whose fibers are the inte- gral curves of a unit Killing vector field in the 3-manifold. We… Expand

TOTALLY UMBILICAL SLANT SUBMANIFOLDS OF RIEMANNIAN PRODUCT MANIFOLDS

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In this paper we show that every totally umbilical slant submanifold of di- mension ≥ 4 is an extrinsic sphere. We also provide the non trivial examples of totally umbilical slant submanifolds of… Expand

Spacelike submanifolds, their umbilical properties and applications to gravitational physics.

- Physics
- 2017

We give a characterization theorem for umbilical spacelike submanifolds of arbitrary dimension and co-dimension immersed in a semi-Riemannian manifold. Letting the codimension arbitrary implies that… Expand

First stability eigenvalue characterization of CMC Hopf tori into Riemannian Killing submersions

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- 2013

We find out upper bounds for the first eigenvalue of the stability operator for compact constant mean curvature orientable surfaces immersed in a Riemannian Killing submersion. As a consequence, the… Expand

Ju n 20 19 Functions with isotropic sections

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We prove a local version of a recently established theorem by Myroshnychenko, Ryabogin and the second named author. More specifically, we show that if n ≥ 3, f : S → R is an even bounded measurable… Expand

Biharmonic hypersurfaces in a product space $L^m\times \mathbb{R}$

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In this paper, we study biharmonic hypersurfaces in a product of an Einstein space and a real line. We prove that a biharmonic hypersurface with constant mean curvature in such a product is either… Expand

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