Algebraic classification of the curvature of three-dimensional manifolds with indefinite metric

Abstract

The curvature of a three-dimensional Riemannian manifold with Lorentzian signature is algebraically classified using the fact that the spinor equivalent of the traceless part of the Ricci tensor is a totally symmetric four-index spinor. Following G. S. Hall and M. S. Capocci [J. Math. Phys. 40, 1466 (1999)] it is shown that at each point of the manifold there exists four, possibly complex, null vectors which are analogous to the Debever–Penrose vectors and also satisfy the condition Rablalb=0. It is also shown that a similar conclusion holds for the Cotton–York tensor.