On homogeneous connections with exotic holonomy

✍ Scribed by Lorenz J. Schwachhöfer

• Publisher: Springer
• Year: 1996
• Tongue: English
• Weight: 735 KB
• Volume: 62
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00147811

✦ Synopsis

In Proc. Symp. Pure Math. 53 (1991), 33-88, 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 holonomy representations. The holonomy in Bryant's examples is the irreducible four-dimensional representation of S1(2, ]~) (GI(2, JR) resp.) and these connections are called//3connections (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 homogeneous 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 by Schwachh6fer (Trans. Amer Math. Soc. 345 (1994), 293-321) which states that there are no compact manifolds with an H3-connection.