The author studies the holonomy group of a simply connected indecomposable and reducible Lorentzian spin manifold under the condition that they admit parallel spinors. He shows that there are only two possible situations: either the manifold is a so-called Brinkmann wave or it has Abelian holonomy and is a pp-manifold – a generalization of a plane-wave. The author gives also sufficient conditions for a Brinkmann wave to have as holonomy the semidirect product of holonomy group of a Riemannian manifold and , and gives examples starting with Kähler and hyper-Kähler manifolds.
@article{701694,
title = {Lorentzian manifolds with special holonomy and parallel spinors},
booktitle = {Proceedings of the 21st Winter School "Geometry and Physics"},
series = {GDML\_Books},
publisher = {Circolo Matematico di Palermo},
address = {Palermo},
year = {2002},
pages = {[131]-159},
mrnumber = {MR1972431},
zbl = {1042.53033},
url = {http://dml.mathdoc.fr/item/701694}
}
Leistner, Thomas. Lorentzian manifolds with special holonomy and parallel spinors, dans Proceedings of the 21st Winter School "Geometry and Physics", GDML_Books, (2002), pp. [131]-159. http://gdmltest.u-ga.fr/item/701694/