Constant angle surfaces in Lorentzian Berger spheres

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