Instability of Hopf vector fields on Lorentzian Berger spheres

Constant angle surfaces in Lorentzian Berger spheres

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 so…

Berger 型洛伦兹结构的割迹

Cut loci of Berger type Lorentzian structures

本文研究反德西特空间上沿 Hopf 纤维化形变的 Berger 型洛伦兹结构，通过万有覆盖排除类时环，刻画可容许曲线可达集与最长弧，并确定测地线的割时、割迹及流形的单射半径。

Towards a classification of Lorentzian holonomy groups. Part II: Semisimple, non-simple weak-Berger algebras

The holonomy group of an (n+2)-dimensional simply-connected, indecomposable but non-irreducible Lorentzian manifold (M,h) is contained in the parabolic group $(\mathbb{R} \times SO(n))\ltimes \mathbb{R}^n$. The main ingredient of such a holonomy group is the SO(n)--projection $G:=pr_{SO(n)}(Hol_p(M,h))$ and one may ask whether it has to be a Riemannian holonomy group. In this paper we show that this is always the case, completing our results of the first part math.DG/0305139. We draw consequences for the existence of parallel spinors on Lorentzian manifolds.

Einstein connections with skew-torsion on Berger spheres

The invariant metric affine connections on Berger spheres which are Einstein with skew torsion are determined in both Riemannian and Lorentzian signature. Expressions of such connections are explicitly given. In particular, every Berger sphere with Lorentzian signature admits invariant metric affine connections Einstein with skew-torsion up to $\mathbb{S}^3$. For Riemannian signature, the existence of such connections strongly depends on the dimension of the sphere and on the scale of the deformation used for the Berger metric. In particular, there are Riemaniann Berger spheres, not Einstein, …

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

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

Helix surfaces in the Berger Sphere

We characterize helix surfaces in the Berger sphere, that is surfaces which form a constant angle with the Hopf vector field. In particular, we show that, locally, a helix surface is determined by a suitable 1-parameter family of isometries of the Berger sphere and by a geodesic of a 2-torus in the 3-dimensional sphere.