Associated Families of Surfaces in Warped Products and Homogeneous Spaces

TL;DR

This paper classifies Riemannian surfaces with associated families in certain homogeneous and warped product spaces, showing such families exist only in product spaces with minimal surfaces, and not in Heisenberg groups.

Contribution

It provides a classification of surfaces with associated families under a rotating structure vector field in specific geometric settings, identifying conditions for their existence.

Findings

Associated families exist only in product spaces with minimal surfaces.

No associated families with rotating structure vector field in the Heisenberg group.

Classification applies to spaces with four-dimensional isometry groups.

Abstract

We classify Riemannian surfaces admitting associated families in three dimensional homogeneous spaces with four-dimensional isometry groups and in a wide family of (semi-Riemannian) warped products, with an extra natural condition (namely, rotating structure vector field). We prove that, provided the surface is not totally umbilical, such families exist in both cases if, and only if, the ambient manifold is a product and the surface is minimal. In particular, there exists no associated families of surfaces with rotating structure vector field in the Heisenberg group.