Helix surfaces for Berger-like metrics on the anti-de Sitter space

TL;DR

This paper classifies helix surfaces in Anti-de Sitter space with Berger-like metrics, showing they have constant Gaussian curvature and providing explicit descriptions involving general helices and hyperbolic fibrations.

Contribution

It introduces a detailed analysis of helix surfaces in Berger-like metrics on Anti-de Sitter space, including their explicit local parametrizations and curvature properties.

Findings

Helix surfaces have constant Gaussian curvature.

Explicit local descriptions involve isometries and general helices.

Helices meet fibers of the hyperbolic Hopf fibration at a constant angle.

Abstract

We consider the Anti-de Sitter space $\mathbb{H}^3_1$ equipped with Berger-like metrics, that deform the standard metric of $\mathbb{H}^3_1$ in the direction of the hyperbolic Hopf vector field. Helix surfaces are the ones forming a constant angle with such vector field. After proving that these surfaces have (any) constant Gaussian curvature, we achieve their explicit local description in terms of a one-parameter family of isometries of the space and some suitable curves. These curves turn out to be general helices, which meet at a constant angle the fibers of the hyperbolic Hopf fibration.