# Berger's theorem on holonomy

Can someone clarify to me what the correct hypothesis of Berger's theorem are (if at all what I write is correct)?

Theorem: assume $M$ is a Riemannian manifold, with irreducible reduced holonomy group; moreover that $M$ is not homogeneous, and it is compact. Then its holonomy must be one of the well-known list (http://en.wikipedia.org/wiki/Holonomy#The_Berger_classification).

In particular, I'd like to understand

i. In which sense $M$ must be something trivial (no Lie groups, etc.)?

ii. Is simply-connectedness already implied by the above hypothesis or can it be omitted?

Could you point me to some good references, other than the original paper?

If $\mathcal{M}$ is complete, locally irreducible and non-locally symmetric then its restricted holonomy group must be in the Berger list. The restricted holonomy group is the subgroup of the full holonomy group generated by null-homotopic loops, and is the identity component.
• Thanks. So, in my original post, compactness can be dropped, while $M$ must not be homogeneous, is that correct? – jj_p Jun 23 '13 at 18:20
• No problem. Compactness can be dropped for the weaker condition of completeness. The space $\mathcal{M}$ can be homogeneous, as long as it is not locally symmetric (symmetric implies homogeneous, locally symmetric implies locally homogeneous). – William Shears Jun 23 '13 at 18:42