zbMATH — the first resource for mathematics

Isometric immersions into 3-dimensional homogeneous manifolds. (English) Zbl 1123.53029
The author provides 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 author expresses the above stated condition in terms of the metric, the second fundamental form, and information that comes out from an ambient Killing field. Applications to constant mean curvature surfaces in these manifolds are given. The proofs of the theorems are given in a selfcontained manner. This provides an essential contribution to the subject.

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A35 Non-Euclidean differential geometry
53B25 Local submanifolds
Full Text: DOI