# When strictly locally convex hypersurfaces are embedded

@article{Espinar2010WhenSL,
title={When strictly locally convex hypersurfaces are embedded},
author={Jos'e M. Espinar and Harold Rosenberg},
journal={Mathematische Zeitschrift},
year={2010},
volume={271},
pages={1075-1090}
}
• Published 27 February 2010
• Mathematics
• Mathematische Zeitschrift
In this paper we will prove Hadamard–Stoker type theorems in the following ambient spaces: $${\mathcal{M}^n \times \mathbb{R}}$$, where $${\mathcal{M}^n}$$ is a 1/4−pinched manifold, and certain Killing submersions, e.g., Berger spheres and Heisenberg spaces. That is, under the condition that the principal curvatures of an immersed hypersurface are greater than some non-negative constant (depending on the ambient space), we prove that such a hypersurface is embedded and we also study its…
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## References

SHOWING 1-10 OF 16 REFERENCES
Locally convex surfaces immersed in a Killing submersion
• Mathematics
• 2010
We generalize Hadamard-Stoker-Currier Theorems for surfaces immersed in a Killing submersion over a strictly Hadamard surface whose fibers are the trajectories of a unit Killing field. We prove that
Local convexity and nonnegative curvature —Gromov's proof of the sphere theorem
An immersed hypersurface S in a riemannian manifold M will be called e-convex for some e > 0 if all principal curvatures have the same sign and absolute value at least e. Can one characterize all
Rigidity and Convexity of Hypersurfaces in Spheres
• Mathematics
• 1970
We shall consider isometric immersions $$x:{\text M}^{n}\rightarrow\,\,{\text X}^{n+1}$$ of a compact, connected, orientable, n-dimensional $$(n\geq 2),{\text C}^\infty$$ Riemannian manifold \({\text
Locally convex hypersurfaces of negatively curved spaces
A well-known theorem due to Hadamard states that if the second fundamental form of a compact immersed hypersurface M of Euclidean space El (n > 3) is positive definite, then M is imbedded as the
Complete surfaces with positive extrinsic curvature in product spaces
• Mathematics
• 2007
We prove that every complete connected immersed surface with positive extrinsic curvature K in H 2 � R must be properly embedded, homeomorphic to a sphere or a plane and, in the latter case, study
Isometric immersions into 3-dimensional homogeneous manifolds
We give a necessary and sufficient condition for a 2-dimensional Riemannian manifold to be locally isometrically immersed into a 3-dimensional homogeneous Riemannian manifold with a 4-dimensional
On hypersurfaces of hyperbolic space infinitesimally supported by horospheres
This paper is concerned with complete, smooth immersed hypersurfaces of hyperbolic space that are infinitesimally supported by horospheres. This latter condition may be restated as requiring that all
Embedded positive constant r-mean curvature hypersurfaces in Mm × R
• Physics
• 2005
Let M be an m-dimensional Riemannian manifold with sectional curvature bounded from below. We consider hypersurfaces in the (m + 1)-dimensional product manifold M × R with positive constant r-mean