Isometric immersions into 3-dimensional homogeneous manifolds

TL;DR

This paper establishes a comprehensive criterion for when a 2D Riemannian manifold can be locally immersed into certain 3D homogeneous spaces, and explores applications to constant mean curvature surfaces, including a generalized Lawson correspondence.

Contribution

It provides a necessary and sufficient condition for isometric immersion into 3D homogeneous manifolds with 4D isometry group, extending understanding of CMC surfaces in these spaces.

Findings

Derived a condition involving metric, second fundamental form, and ambient Killing field data.

Proved the existence of a generalized Lawson correspondence for CMC surfaces.

Included applications to Berger spheres, Nil(3), PSL(2,R), S^2 x R, and H^2 x R.

Abstract

We give a necessary and sufficient condition for a 2-dimensional Riemannian manifold to be locally isometrically immersed into a 3-dimensional homogeneous 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 space Nil(3), the universal cover of the Lie group PSL(2,R) and the product spaces S^2 x R and H^2 x 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.