On homogeneous connections with exotic holonomy, Geom.Ded

In [Br], Bryant gave examples of torsion free connections on four-manifolds whose holonomy is exotic, i.e. is not contained on Berger’s classical list of irreducible ho-lonomy representations [Ber]. The holonomy in Bryant’s examples is the irreducible four-dimensional representation of Sl(2,R) (Gl(2,R) resp.) and these connections are called H3-connections (G3-connections resp.). In this paper we give a complete classification of homogeneous G3-connections. The moduli space of these connections is four-dimensional, and the generic homoge-neous G3-connection is shown to be locally equivalent to a left-invariant connection on U(2). Thus, we prove the existence of compact manifolds with G3-connections. This contrasts a result in [Sch] which states that there are no compact manifolds with an H3-connection. §0 Introduction. Since its introduction by Èlie Cartan, the holonomy of a connection has played an important role in differential geometry. Most of the classical results are concerned with the holonomy of Levi Civita connections of Riemannian metrics. In 1955