On the Incompleteness of Berger's List of Holonomy Representations

Details On the Incompleteness of Berger's List of Holonomy Representations On the Incompleteness of Berger's List of Holonomy Representations

1995-08-30

arxiv.org/abs/dg-ga/9508014v1

Invent. Math. 126 (1996), 391-411

Mathematics

Differential Geometry

20 pages, AMS-LaTeX, no figures

Scientific paper

In 1955, Berger \cite{Ber} gave a list of irreducible reductive representations which can occur as the holonomy of a torsion-free affine connection. This list was stated to be complete up to possibly a finite number of missing entries. In this paper, we show that there is, in fact, an infinite family of representations which are missing from this list, thereby showing the incompleteness of Berger's classification. Moreover, we develop a method to construct torsion-free connections with prescribed holonomy, and use it to give a complete description of the torsion-free affine connections with these new holonomies. We also deduce some striking facts about their global behaviour.

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