Connections with exotic holonomy

Abstract:

Berger [Ber] partially classified the possible irreducible holonomy representations of torsion free connections on the tangent bundle of a manifold. However, it was shown by Bryant [Bry] that Berger’s list is incomplete. Connections whose holonomy is not contained on Berger’s list are called exotic. We investigate a certain 4-dimensional exotic holonomy representation of $Sl(2,\mathbb {R})$. We show that connections with this holonomy are never complete and do not exist on compact manifolds. We give explicit descriptions of these connections on an open dense set and compute their groups of symmetry.