Isometric immersions into 3-dimensional homogeneous manifolds

Abstract

Abstract. 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. Mathematics Subject Classification (2000). Primary: 53C42. Secondary: 53A35, 53B25.

The Gauss Map of Minimal Surfaces in the Heisenberg Group

“…Because of the local isometric correspondence between CMC 1 2 surfaces in H 2 × R and minimal surfaces in the 3-dimensional Heisenberg group Nil 3 (see [Dan07]), it is natural to study the same problem for minimal surfaces in Nil 3 (endowed with a left-invariant Riemannian metric). The Lie group Nil 3 is a 3-dimensional homogeneous manifold with a 4-dimensional isometry group; hence it is one of the most simple 3-manifolds apart from space-forms.…”

On the Gauss curvature of compact surfaces in homogeneous 3-manifolds

“…In this section we follow the notation given in [4]. Let N be a simply connected homogeneous Riemannian 3-manifold whose isometry group has dimension 4.…”

“…Then there exists a Riemannian submersion Π : N → M 2 (κ), where M 2 (κ) is a 2-dimensional simply connected space form of constant curvature κ, with totally geodesic fibers and there exists a unit Killing field ξ on N which is vertical with respect to Π. We will assume that N is oriented, and then we can define a vectorial product ∧, such that if {e 1 , e 2 } are linearly independent vectors at a point p, then {e 1 , e 2 , e 1 ∧ e 2 } is the orientation at p. If∇ denotes the Riemannian connection on N , the properties of ξ imply (see [4]) that for any vector field V ,…”

Cylinders as left invariant CMC surfaces in Sol3 and E(κ,τ)-spaces diffeomorphic to Vržina1 2018Differential Geometry and its Applications2000View full textAdd to dashboardCite

“…Only in H 2 × R existence of unduloids with a horizontal axis is known. They have been constructed by [MT14] using the Daniel correspondence from [Dan07], which relates MCH -surfaces in H 2 ×R with mean curvature H > 1/2 to minimal surfaces in a Berger sphere. So, for PSL 2 (R) and Nil 3 existence of horizontal unduloids is an open problem.…”