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

Abstract

We consider the Anti-de Sitter space $$\mathbb {H}^3_1$$ H 1 3 equipped with Berger-like metrics, that deform the standard metric of $$\mathbb {H}^3_1$$ 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.

Totally geodesic Lagrangian submanifolds of the pseudo‐nearly Kähler SL(2,R)×SL(2,R)$\mathrm{SL}(2,\mathbb {R})\times \mathrm{SL}(2,\mathbb {R})$

“…The second immersion turns SL(2, ℝ) into an anti-de Sitter space with a Berger-like metric stretched in a timelike direction, and the third immersion turns it again into an anti-de Sitter space with a Berger-like metric, but now stretched in a spacelike direction. These metrics have been studied more generally in [3,17].…”