On the holonomy groups of Weyl manifolds

TL;DR

This paper classifies the local holonomy groups of Weyl connections, focusing on the reducible, non-closed case, and establishes that non-closed Einstein-Weyl product structures only occur in four dimensions.

Contribution

It completes the classification of holonomy groups of Weyl connections by analyzing the reducible, non-closed case and identifies dimension 4 as unique for non-closed Einstein-Weyl products.

Findings

Holonomy groups of reducible, non-closed Weyl connections are classified.

Non-closed Einstein-Weyl product structures exist only in dimension 4.

The structure of non-closed conformal products is characterized in this context.

Abstract

We classify the possible local holonomy groups of Weyl connections. The Berger-Simons theorem and the Merkulov-Schwachh\"ofer classification of holonomy groups of irreducible torsion-free connections leaves us with the remaining case, where the Weyl connection $D$ is reducible and non-closed. In this case, it was shown by F. Belgun and A. Moroianu that the Weyl structure is an adapted Weyl structure of a non-closed conformal product. Furthermore we prove that non-closed Einstein-Weyl product structures only exist in dimension $4$.