# 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}
}
• Benoît Daniel
• Published 2005
• Mathematics
• Commentarii Mathematici Helvetici
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
228 Citations

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#### 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 toExpand
Isometric immersions into S^n x R and H^n x R and applications to minimal surfaces
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 theExpand
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 toExpand
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 holdsExpand
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
Minimal disks bounded by three straight lines in Euclidean space and trinoids in hyperbolic space
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 caseExpand
A characteristic property of spheres
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 {}_{\partialExpand
The geometries of 3-manifolds
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 ofExpand
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 embeddedExpand
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 areExpand