Constant Angle Surfaces in Lorentzian Berger Spheres

Abstract

In this work, we study helix spacelike and timelike surfaces in the Lorentzian Berger sphere S 3 ε , 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 → S 2 (1/2) by −ε 2 . Our main result provides a characterization of the helix surfaces in S 3 ε using the symmetries of the ambient space and a general helix in S 3 ε , with axis the infinitesimal generator of the Hopf fibers. Also, we construct some explicit examples of helix surfaces in S 3 ε .1991 Mathematics Subject Classification. 53B25, 53C50.

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

“…We end this section describing the isometries of H 3 1,τ . Following the idea used in [14] and [17], we observe that the isometry group of H 3 1,τ is the four-dimensional pseudo-unitary group U 1 (2), that can be identified with…”

“…Lorentzian settings allow to more possibilities, as both spacelike and timelike surfaces can be studied. Some examples of the study of the geometry of helix surfaces in Lorentzian spaces are given in [11,12,16,17]. In particular, helix surfaces of the anti-de Sitter space H 3 1 were studied in [12].…”

“…In [13,14,16,17], the second author et al gave an explicit local classification of helix surfaces by means a one-parameter family of isometries of the ambient space and a suitable curve. A similar appproach will be used in this paper in order to find the explicit characterization of the helix surfaces in the anti-de Sitter space.…”

Constant angle null hypersurfaces

“…Several classification results for constant angle semi-Riemannian (nondegenerate) hypersurfaces have been established recently in ambient spaces such as cartesian products [10,13,16,19], warped products [15,22], spaceforms [20,28,30,34] and other geometrically relevant spaces [1,36,37]. One remarkable relation between the distinguished vector field V and the geometry of a constant angle hypersurface M is encapsulated in the notion of canonical principal direction: the tangent component V ∈ Γ(T M ) is a principal direction of M [11,21,23,29].…”