Helix surfaces in Lorentzian Heisenberg group

TL;DR

This paper classifies and explicitly describes constant angle, minimal, and CMC helix surfaces in the Lorentzian Heisenberg group, revealing their geometric properties and providing new characterizations for spacelike and timelike cases.

Contribution

It provides a complete classification and explicit parametrizations of helix surfaces in the Lorentzian Heisenberg group, including new results on constant angle spacelike and timelike surfaces.

Findings

Constant angle surfaces have constant Gaussian curvature.

Complete classification of minimal and CMC helix surfaces.

Explicit parametrizations of these surfaces.

Abstract

In this work we investigate constant angle surfaces in the Lorentzian Heisenberg group $\htt$. After providing a complete description of the geometry of the ambient space, we perform the full classification of minimal and CMC helix surfaces in $\htt$, giving their explicit parametrizations. In addition, we investigate the constant angle spacelike and timelike surfaces for a Lorentzian metric on the Heisenberg group $H_3$ not treated before in the literature, first showing that such surfaces have constant Gaussian curvature and then obtaining their complete characterization.