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}
}
  • Rabah Souam, J. Veken
  • Published 2010
  • Mathematics
  • Transactions of the American Mathematical Society
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
Totally umbilical hypersurfaces of product spaces
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 areExpand
Associated Families of Surfaces in Warped Products and Homogeneous Spaces
We classify Riemannian surfaces admitting associated families in three dimensional homogeneous spaces with four-dimensional isometry groups and in a wide family of (semi-Riemannian) warped products,Expand
The classification of totally umbilical surfaces in homogeneous 3-manifolds
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 RiemannianExpand
Compact stable surfaces with constant mean curvature in Killing submersions
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
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. WeExpand
TOTALLY UMBILICAL SLANT SUBMANIFOLDS OF RIEMANNIAN PRODUCT MANIFOLDS
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 ofExpand
Spacelike submanifolds, their umbilical properties and applications to gravitational physics.
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 thatExpand
First stability eigenvalue characterization of CMC Hopf tori into Riemannian Killing submersions
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, theExpand
Ju n 20 19 Functions with isotropic sections
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 measurableExpand
Biharmonic hypersurfaces in a product space $L^m\times \mathbb{R}$
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 eitherExpand
...
1
2
...

References

SHOWING 1-10 OF 16 REFERENCES
Killing vector fields of constant length on Riemannian manifolds
We study the nontrivial Killing vector fields of constant length and the corresponding flows on complete smooth Riemannian manifolds. Various examples are constructed of the Killing vector fields ofExpand
Lagrangian submanifolds of $C^{n}$ with conformal Maslov form and the Whitney sphere
The Lagrangian submanifolds of the complex Euclidean space with conformal Maslov form can be considered as the Lagrangian version of the hypersurfaces of the Euclidean space with constant meanExpand
Totally umbilic surfaces in homogeneous 3-manifolds
We discuss existence and classification of totally umbilic surfaces in the model geometries of Thurston and the Berger spheres. We classify such surfaces in $H^2 \times R$, $S^2 \times R$ and the SolExpand
Higher Order Parallel Surfaces in Bianchi–Cartan–Vranceanu Spaces
Abstract.We give a full classification of higher order parallel surfaces in three-dimensional homogeneous spaces with four-dimensional isometry group, i.e., in the so-called Bianchi–Cartan–VranceanuExpand
Riemannian Geometry
THE recent physical interpretation of intrinsic differential geometry of spaces has stimulated the study of this subject. Riemann proposed the generalisation, to spaces of any order, of Gauss'sExpand
A Panoramic View of Riemannian Geometry
0. Vector fields, tensors 1. Tensor Riemannian duality, the connection and the curvature 2. The parallel transport 3. Absolute (Ricci) calculus, commutation formulas 4. Hodge and the Laplacian,Expand
ON THE CLASSIFICATION AND REGULARITY OF UMBILIC SURFACES IN HOMOGENEOUS 3-MANIFOLDS
We survey existence and classication of totally umbilic surfaces in the model geometries of Thurston and the Berger spheres. We also discuss the regularity of totally umbilic surfaces.
Parallel and semi-parallel hypersurfaces of $$ \mathbb{S}^n $$ × ℝ
AbstractWe give a complete classification of totally umbilical, parallel and semi-parallel hypersurfaces of the Riemannian product space $$ \mathbb{S}^n $$ × ℝ.
ON EXTRINSICALLY SYMMETRIC HYPERSURFACES OF ℍ n ×ℝ
Totally umbilical, semi-parallel and parallel hypersurfaces of ℍ n ×ℝ are completely classified. More examples arise than in the analogous study on the ambient space 𝕊 n ×ℝ.
Differential Equations, Dynamical Systems, and Linear Algebra
This book is about dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. A prominent role is played by theExpand
...
1
2
...