Immersions of Lorentzian surfaces in R2,1R2,1

We study whether a given Lorentzian surface (M,g)(M,g) can be immersed as the hypersurface of codimension one into the pseudo-Euclidean space R2,1R2,1. Using the methods of para-complex geometry and real spinor representations we succeed in proving the equivalence between the data of a spacelike conformal immersion of (M,g)(M,g) into R2,1R2,1 and two spinors satisfying a Dirac-type equation on the surface. We generalize in this way with new technics a result of Friedrich [Th. Friedrich, On the spinor representation of surfaces in euclidean 3-Space, J. Geom. Phys. 28 (1–2) (1998) 143–157] to the pseudo-Riemannian context. Moreover we give a geometrically invariant representation of such surfaces using two Dirac spinors.