# Isometric immersions into 3-dimensional homogeneous manifolds

@article{Daniel2005IsometricII, title={Isometric immersions into 3-dimensional homogeneous manifolds}, author={Beno{\^i}t Daniel}, journal={Commentarii Mathematici Helvetici}, year={2005}, volume={82}, pages={87-131} }

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 isometry group. The condition is expressed in terms of the metric, the second fundamental form, and data arising from an ambient Killing field. This class of 3-manifolds includes in particular the Berger spheres, the Heisenberg group Nil3, the universal cover of the Lie group PSL2(R) and the product… Expand

#### Figures from this paper

#### 228 Citations

A characterization of constant mean curvature surfaces in homogeneous 3-manifolds

- Mathematics
- 2005

Abstract It has been recently shown by Abresch and Rosenberg that a certain Hopf differential is holomorphic on every constant mean curvature surface in a Riemannian homogeneous 3-manifold with… Expand

Constant mean curvature hypersurfaces on symmetric spaces, minimal graphs on semidirect products and properly embedded surfaces in hyperbolic 3-manifolds

- Mathematics
- 2015

We prove results concerning the geometry of hypersurfaces on different ambient spaces. First, we define a generalized Gauss map for a hypersurface Mn−1 ⊆ N, where N is a symmetric space of dimension… Expand

Constant mean curvature surfaces in 3-dimensional Thurston geometries

- Mathematics, Physics
- 2010

This is a survey on the global theory of constant mean curvature surfaces in Riemannian homogeneous 3-manifolds. These ambient 3-manifolds include the eight canonical Thurston 3-dimensional… Expand

Jacobi osculating rank and isotropic geodesics on naturally reductive 3-manifolds

- Mathematics
- 2007

Abstract We study the Jacobi osculating rank of geodesics on naturally reductive homogeneous manifolds and we apply this theory to the 3-dimensional case. Here, each non-symmetric, simply connected… Expand

The Gauss map of surfaces in ~PSL_2(R)

- Physics, Mathematics
- 2013

We define a Gauss map for surfaces in the universal cover of the Lie group PSL_2(R) endowed with a left-invariant Riemannian metric having a 4-dimensional isometry group. This Gauss map is not… Expand

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… Expand

Spinorial characterizations of surfaces into three-dimensional homogeneous manifolds

- Mathematics
- 2010

Abstract We give spinorial characterizations of isometrically immersed surfaces into three-dimensional homogeneous manifolds with four-dimensional isometry group in terms of the existence of a… Expand

Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds

- Mathematics
- 2014

Abstract We study curvature functionals for immersed 2-spheres in non-compact, three-dimensional Riemannian manifold ( M , h ) without boundary. First, under the assumption that ( M , h ) is the… Expand

Compact stable constant mean curvature surfaces in homogeneous 3-manifolds

- Mathematics
- 2012

We classify the stable constant mean curvature spheres in the homogeneous Riemannian 3-manifolds: the Berger spheres, the special linear group and the Heisenberg group. We show that all of them are… Expand

Equivariant Willmore surfaces in conformal homogeneous three spaces

- Mathematics
- 2014

Abstract The complete classification of homogeneous three spaces is well known for some time. Of special interest are those with rigidity four which appear as Riemannian submersions with geodesic… Expand

#### References

SHOWING 1-10 OF 19 REFERENCES

Invariant surfaces of the Heisenberg groups

- Mathematics
- 1999

SummaryWe fix a left-invariant metric g in the eisenberg group,H3, and give a complete classification of the constant mean curvature surfaces (including minimal) which are invariant with respect to… Expand

Isometric immersions into S^n x R and H^n x R and applications to minimal surfaces

- Mathematics
- 2004

We give a necessary and sufficient condition for an n-dimensional Riemannian manifold to be isometrically immersed in S^n x R or H^n x R in terms of its first and second fundamental forms and of the… Expand

FOR CONSTANT MEAN CURVATURE SURFACES IN S 2 R AND H 2 R

- Mathematics
- 2003

A basic tool in the theory of constant mean curvature (cmc) surfaces 2 in space forms is the holomorphic quadratic dierential dis- covered by H. Hopf. In this paper we generalize this dierential to… Expand

Complete surfaces of constant mean curvature-1 in the hyperbolic 3-space

- Mathematics
- 1993

In the study of minimal surfaces in the euclidean 3-space, the Weierstrass representation plays an important role. Bryant [Br] showed that an analogue of the Weierstrass-representation formula holds… Expand

Riemannian Geometry

- Nature
- 1927

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's… Expand

Minimal disks bounded by three straight lines in Euclidean space and trinoids in hyperbolic space

- Mathematics
- 2003

Following Riemann's idea, we prove the existence of a minimal disk in Euclidean space bounded by three lines in generic position and with three helicoidal ends of angles less than $\pi$. In the case… Expand

A characteristic property of spheres

- Mathematics
- 1962

SummaryWe prove: Let S be a closed n-dimensional surface in an(n+1)-space of constant curvature (n ≥ 2); k1 ≥ ... ≥ kn denote its principle curvatures. Let φ(ξ1, ..., ξn) be such that
$${}_{\partial… Expand

The geometries of 3-manifolds

- Mathematics
- 1983

The theory of 3-manifolds has been revolutionised in the last few years by work of Thurston [66-70]. He has shown that geometry has an important role to play in the theory in addition to the use of… Expand

GLOBAL PROPERTIES OF CONSTANT MEAN CURVATURE SURFACES IN H 2 ◊R

- Mathematics
- 2006

We discuss some aspects of the global behavior of surfaces in H 2 ◊ R with constant mean curvature H (known as H-surfaces). We prove a maximum principle at infinity for complete properly embedded… Expand

TRIUNDULOIDS: EMBEDDED CONSTANT MEAN CURVATURE SURFACES WITH THREE ENDS AND GENUS ZERO

- Mathematics
- 2000

We construct the entire three-parameter family of embedded constant mean curvature surfaces with three ends and genus zero. They are classified by triples of points on the sphere whose distances are… Expand