On the spinorial representation of spacelike surfaces into 4-dimensional Minkowski space

Abstract

We prove that an isometric immersion of a simply connected Riemannian surface M in four-dimensional Minkowski space, with given normal bundle E and given mean curvature vector H \in Γ(E), is equivalent to a normalized spinor field φ\in Γ(ΣE \otimes ΣM) solution of a Dirac equation Dφ=H\cdotφon the surface. Using the immersion of the Minkowski space into the complex quaternions, we also obtain a representation of the immersion in terms of the spinor field. We then use these results to describe the flat spacelike surfaces with flat normal bundle and regular Gauss map in four-dimensional Minkowski space, and also the flat surfaces in three-dimensional hyperbolic space, giving spinorial proofs of results by J.A. Galvez, A. Martinez and F. Milan.

References in corpus (5)

• Surfaces in the four-space and the Davey--Stewartson equations
• Spinorial Characterization of Surfaces into 3-dimensional homogeneous Manifolds
• Spinorial Characterizations of Surfaces into 3-dimensional pseudo-Riemannian Space Forms
• Geometric invariants and principal configurations on spacelike surfaces immersed in R^3,1
• Spinorial Representation of Surfaces into 4-dimensional Space Forms

Cited by in corpus (12)

• Spinorial Representation of Submanifolds in Riemannian Space Forms
• Spinor representation of Lorentzian surfaces in R^{2,2}
• Spinorial representation of submanifolds in metric Lie groups
• 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
• Spinorial representation of submanifolds in $S{L}_n(\mathbb{C})/SU(n)$
• Immersion in Sn by complex spinors
• Timelike surfaces into 4-dimensional Minkowski space via spinors
• Constant angle surfaces in 4-dimensional Minkowski space
• Immersion in Rn by Complex irreducible Spinors
• Spinor representation in isotropic 3-space via Laguerre geometry
• Spinorial representation of surfaces in Lorentzian homogeneous spaces of dimension 3