In 1955, Berger 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.
Description
On the Incompleteness of Berger's List of Holonomy Representations
%0 Generic
%1 chi1995incompleteness
%A Chi, Quo-Shin
%A Merkulov, Sergey A.
%A Schwachhöfer, Lorenz J.
%D 1995
%K bergers holonomy incompleteness representations
%T On the Incompleteness of Berger's List of Holonomy Representations
%U http://arxiv.org/abs/dg-ga/9508014
%X In 1955, Berger 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.
@misc{chi1995incompleteness,
abstract = {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.},
added-at = {2013-12-23T06:28:58.000+0100},
author = {Chi, Quo-Shin and Merkulov, Sergey A. and Schwachhöfer, Lorenz J.},
biburl = {https://www.bibsonomy.org/bibtex/23dcaa6b8debad1d105611bc4c56c0e71/aeu_research},
description = {On the Incompleteness of Berger's List of Holonomy Representations},
interhash = {da0f2dd9afea597106e6dac135970ee2},
intrahash = {3dcaa6b8debad1d105611bc4c56c0e71},
keywords = {bergers holonomy incompleteness representations},
note = {cite arxiv:dg-ga/9508014Comment: 20 pages, AMS-LaTeX, no figures},
timestamp = {2013-12-23T06:28:58.000+0100},
title = {On the Incompleteness of Berger's List of Holonomy Representations},
url = {http://arxiv.org/abs/dg-ga/9508014},
year = 1995
}