Helix surfaces in the Berger Sphere

TL;DR

This paper characterizes helix surfaces in the Berger sphere, showing they are locally determined by isometries and geodesics, advancing understanding of geometric structures in this specific manifold.

Contribution

It provides a new characterization of helix surfaces in the Berger sphere using isometries and geodesics, which was not previously known.

Findings

Helix surfaces are determined by a 1-parameter family of isometries.

They are characterized by geodesics of a 2-torus in the sphere.

The paper offers a local description of these surfaces.

Abstract

We characterize helix surfaces in the Berger sphere, that is surfaces which form a constant angle with the Hopf vector field. In particular, we show that, locally, a helix surface is determined by a suitable 1-parameter family of isometries of the Berger sphere and by a geodesic of a 2-torus in the 3-dimensional sphere.