# Constant angle surfaces in Lorentzian Berger spheres

## TL;DR

This paper characterizes helix spacelike and timelike surfaces in Lorentzian Berger spheres, using symmetries and explicit examples, advancing understanding of geometric structures in Lorentzian manifolds.

## Contribution

It provides a new characterization of helix surfaces in Lorentzian Berger spheres and constructs explicit examples, enriching the geometric theory of these Lorentzian manifolds.

## Key findings

- Characterization of helix surfaces using symmetries of the Lorentzian Berger sphere
- Explicit examples of helix surfaces in Lorentzian Berger spheres
- Use of the infinitesimal generator of Hopf fibers as an axis

## Abstract

In this work, we study helix spacelike and timelike surfaces in the Lorentzian Berger sphere $\s_{\varepsilon}^3$, that is the three-dimensional sphere endowed with a $1$-parameter family of Lorentzian metrics, obtained by deforming the round metric on $\s^3$ along the fibers of the Hopf fibration $\s^3\to \s^2({1}/{2})$ by $-\varepsilon^2$. Our main result provides a characterization of the helix surfaces in $\s_{\varepsilon}^3$ using the symmetries of the ambient space and a general helix in $\s_{\varepsilon}^3$, with axis the infinitesimal generator of the Hopf fibers. Also, we construct some explicit examples of helix surfaces in $\s_{\varepsilon}^3$.