# 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} }

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…

## 9 Citations

Locally convex hypersurfaces immersed in $H^n \times R$

- Mathematics
- 2012

We prove a theorem of Hadamard-Stoker type: a connected locally convex complete hypersurface immersed in $H^n \times R$ (n>1), where $H^n$ is n-dimensional hyperbolic space, is embedded and…

M ay 2 01 2 Locally convex hypersurfaces immersed in H n × R

- 2021

We prove a theorem of Hadamard-Stoker type: a connected locally convex complete hypersurface immersed in H×R (n ≥ 2), where H is n-dimensional hyperbolic space, is embedded and homeomorphic either to…

Locally convex hypersurfaces immersed in \(\mathbb {H}^n\times \mathbb {R}\)

- Mathematics
- 2017

We prove a theorem of Hadamard–Stoker type: a connected locally convex complete hypersurface immersed in \(\mathbb {H}^n\times \mathbb {R}\) (\(n\ge 2\)), where \(\mathbb {H}^n\) is n-dimensional…

Embeddedness, convexity, and rigidity of hypersurfaces in product spaces

- Mathematics
- 2018

We establish the following Hadamard--Stoker type theorem: Let $f:M^n\rightarrow\mathscr{H}^n\times\mathbb R$ be a complete connected hypersurface with positive definite second fundamental form, where…

On a class of immersions of spheres into space forms of nonpositive curvature

- MathematicsGeometriae Dedicata
- 2019

Let $$ M^{n+1} $$ M n + 1 ( $$ n \ge 2 $$ n ≥ 2 ) be a simply-connected space form of sectional curvature $$ -\kappa ^2 $$ - κ 2 for some $$ \kappa \ge 0 $$ κ ≥ 0 , and I an interval not containing…

Weingarten type surfaces in $\mathbb{H}^2\times\mathbb{R}$ and $\mathbb{S}^2\times\mathbb{R}$

- Mathematics
- 2015

In this article, we study complete surfaces $\Sigma$, isometrically immersed in the product space $\mathbb{H}^2\times\mathbb{R}$ or $\mathbb{S}^2\times\mathbb{R}$ having positive extrinsic curvature…

The space of Constant Mean Curvature surfaces in compact Riemannian Manifolds

- Mathematics
- 2011

The main point of this paper is that, under suitable conditions on the mean curvature and the Ricci curvature of the ambient space, we can extend Choi-Schoen's Compactness Theorem to compact embedded…

Weingarten Type Surfaces in ℍ2 × ℝ and 𝕊2 × ℝ

- MathematicsCanadian Journal of Mathematics
- 2017

Abstract In this article, we study complete surfaces $\sum $ , isometrically immersed in the product spaces ${{\mathbb{H}}^{2}}\,\times \,\mathbb{R}$ or ${{\mathbb{S}}^{2\,}}\times \,\mathbb{R}$…

## 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

- Mathematics
- 1986

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

- Mathematics
- 1977

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

- Mathematics
- 2005

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

- Mathematics
- 1989

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…