# 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 Nil_{3}, the
universal cover of the Lie group PSL_{2}(R) and the
product spaces S^{2}×R and H^{2}×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