Commentarii Mathematici Helvetici

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Volume 82, Issue 1, 2007, pp. 87–131
DOI: 10.4171/CMH/86

Isometric immersions into 3-dimensional homogeneous manifolds

Benoît Daniel[1]

(1) Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil

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.

Keywords: Isometric immersions, constant mean curvature surfaces, homogeneous manifolds, Gauss and Codazzi equations

Daniel Benoît: Isometric immersions into 3-dimensional homogeneous manifolds. Comment. Math. Helv. 82 (2007), 87-131. doi: 10.4171/CMH/86