Spinorial representation of surfaces in Lorentzian homogeneous spaces of dimension 3

Abstract

We find a spinorial representation of a Riemannian or Lorentzian surface in a Lorentzian homogeneous space of dimension $3.$ We in particular obtain a representation theorem for surfaces in the $\mathbb{L}(\kappa ,\tau )$ spaces. We then recover the Calabi correspondence between minimal surfaces in ${\mathbb{R}}^3$ and maximal surfaces in ${\mathbb{R}}_1^3$, and obtain a new Lawson type correspondence between CMC surfaces in ${\mathbb{R}}_1^3$ and in the 3-dimensional pseudo-hyperbolic space ${\mathbb{H}}_1^3.$

References in corpus (4)

• Spinorial Representation of Submanifolds in Riemannian Space Forms
• Spinor representation of Lorentzian surfaces in R^{2,2}
• A generalized Weierstrass representation of Lorentzian surfaces in ${\mathbb{R}}^{2,2}$ and applications
• Complex and Lagrangian surfaces of the complex projective plane via Kählerian Killing Spin${}^c$ spinors