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

We consider the Anti-de Sitter space \(\mathbb {H}^3_1\) ${\mathbb{H}}_1^3$ equipped with Berger-like metrics, that deform the standard metric of \(\mathbb {H}^3_1\) ${\mathbb{H}}_1^3$ 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.