Let M be a pseudo-Riemannian maniflod and H be the holonomy group of M at the point m. It is possible that M\_m has an H-invariant subspace if M\_m is indecomposable. Does it admit a decomposition M\_m=M\_1+M\_2 which M\_i is H-invariant? 



Decomposition which is invariant under the action of holonomy group
Tom Krantz proved that if the holonomy group $H$ of an indecomposable pseudo-Riemannian manifold preserves a non-trivial decomposition $T\_xM=V\_1\oplus V\_2$,
then there exists also an $H$-invariant decomposition $T\_xM=U\_1\oplus U\_2$ into the direct sum of two totally isotropic subspaces, in particular, the manifold must have netral signature. The papers can be found on 
<http://www.mathematik.uni-dortmund.de/~tkrantz/Web/Publikationen.html>


