On Affine Connections whose Holonomy is a Tensor Representation

Abstract

In 1955, Berger [4] gave a list of irreducible reductive representations which can occur as the holonomy of a torsion-free affine connection. While this list was stated to be complete in the case of metric connections, the situation in the general case remained unclear. The (non-metric) representations which are missing from this list are called exotic. In recent years, it has been determined that exotic holonomies do exist. Thus, Berger’s classification is yet to be completed in the non-metric case. In this paper, we investigate certain holonomy representations of reductive Lie groups whose semi-simple part is not simple.