Isometric immersions into 3-dimensional homogeneous manifolds

By Benoît Daniel. 2005

288 citations recorded by the source corpus.

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 spaces S2×R and H2×R. We give some applications to constant mean curvature (CMC) surfaces in these manifolds; in particular we prove the existence of a generalized Lawson correspondence, i.e., a local isometric correspondence between CMC surfaces in homogeneous 3-manifolds

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/16ae9d03e68ee472bbf046a297339280298891d7

How researchers see this paper

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In Applied Mathematics

1. W. H. Meeks
2. L. Hauswirth
3. Eric Toubiana
4. Benoît Daniel
5. H. Rosenberg
6. A. Ros
7. Rabah Souam
8. J. Espinar

In Astronomy and Astrophysics

1. W. H. Meeks
2. L. Hauswirth
3. J. Espinar
4. Rabah Souam
5. Eric Toubiana
6. Iskander A. Taimanov
7. Dmitry Berdinsky
8. H. Rosenberg

In Geometry and Topology

1. W. H. Meeks
2. Benoît Daniel
3. L. Hauswirth
4. H. Rosenberg
5. A. Ros
6. Eric Toubiana
7. Iskander A. Taimanov
8. Dmitry Berdinsky

Citations in the corpus

• Complete surfaces with positive extrinsic curvature in product spaces (2007, methodology): “Therefore, the uniqueness part in [D] gives that the immersion is invariant under a 1-parameter group of transformations (.”
• Totally umbilic surfaces in homogeneous 3-manifolds (2006, methodology): “More precisely we first consider the manifolds having a 4-dimensional isometry group, denoted by M 3 ( κ, τ ) (see section 2).”
• Surfaces in Three-Dimensional Lie Groups (2005, methodology): “nd S3. We note that the approach by Abresch is different and based on a representation of Nil and Sol as line bundles of constant curvature over surfaces of constant curvature (see also a recent paper [3] where this property is used for studying surfaces Nil and Sol). It looks that the failure of suc”
• Stable constant mean curvature surfaces (2008, background): “By work of Daniel [13, 12], the above conjecture is equivalent to the following statements.”
• Half‐space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group (2007, background): “It is known from [8] that a CMC 12 immersion X ∗ = (F ∗, h∗) is locally isometric to a minimal immersion X = (F, h) in Nil3.”