1995/08/30 by Quo-Shin Chi, Sergey A. Merkulov, Lorenz Schwachhöfer +1 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Affine connection #Affine transformation #Algebra over a field #Connection (principal bundle) #Geometric and Algebraic Topology #Geometry #Holonomy #Mathematics #Pure mathematics #Series (stratigraphy) #Torsion (gastropod) #dg-ga #math.DG
paper · pdf · doi:10.1007/s002220050104
published as Invent. Math. 126 (1996), 391-411 · 20 pages, AMS-LaTeX, no figures
arxiv created 1995/08/30 · openalex publication_date 1996/10/02 · arxiv updated 2015/06/25 · openalex created_date 2020/11/23 · openalex updated_date 2026/07/29
In 1955, Berger \citeBer 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.