7

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?

jj_p
  • 2,500
  • 1
  • 15
  • 31

1 Answers1

8

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.

The above is a 'topology free' formulation, because it makes sense using only information in any neighbourhood of a point (good enough for calculus). If you want a global statement then the fundamental group of the manifold needs to be considered. The full holonomy group can have several components (the fundamental group surjects onto the component group) and the holonomy list will be different for each manifold, depending on its fundamental group. To avoid this, a common assumption is that your space is simply-connected. This means that the full holonomy group is the same as the restricted one. It also means that you only have to demand that your manifold be irreducible and non-symmetric, without the word 'locally'. It's a matter of taste/application whether you use the 'local' or 'simply-connected global' wording of the Berger-Simons Theorem.

The manifold must not be locally symmetric because in that instance, holonomy groups can occur that aren't on the list. There is a lot more to be said on this topic and it certainly won't fit into a single answer here, so I recommend three well-established and excellent books: Kobayashi-Nomizu 'Foundations of Differential Geometry Vol I', Besse 'Einstein Manifolds', and Salamon 'Riemannian Geometry and Holonomy Groups'.

William Shears
  • 400
  • 1
    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
  • 2
    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
  • 1
    Can you elaborate a little bit more on "The manifold must not be locally symmetric because in that instance, holonomy groups can occur that aren't on the list"? Or at least can you provide me with a good reference to read about that? Thank you in advance – Carlos Cabezas Jul 14 '23 at 11:12