# Einstein metrics on spheres

We know that a closed oriented manifold $$M$$ carries a Lorentzian metric iff the Euler characteristic vanishes. My question concerns the existence of those Lorentzian metrics on odd-dimensional spheres $$S^{2k+1}$$ which are Einstein. What do we know about the holonomy of those metrics, are those always irreducible or indecomposable holonomies? Does there always exists a parallel spinor on those spaces?

• What does "special holonomy metric" mean? Jun 21 at 19:23
• @RyanBudney Special holonomy in the (complete) Lorentzian settings means non-irreducible (here $SO_0(1,n)$), indecomposable, which admits a parallel null line. 2 days ago
• Apologies, I am not familiar with this terrminology. Does non-irreducible mean reducible, i.e. is that a double-negative? 2 days ago
• The Holonomy representation is irreducible if there is no non-trivial invariant subspace of the tangent space. The representation is called indecomposable if there is no non-degenerated non-trivial invariant subspace. In particular, there a non-irreducible but indecomposable representations. yesterday
• But one has to be aware that in the non-Riemannian setting, there is in general no unique decomposition of the tangent space into invariant irreducible subspaces. yesterday