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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

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# On the irreducibility of locally metric connections

Florin Belgun
• Fachbereich Mathematik, Bereich AD, Hamburg University, Bundesstr. 55 (Geomatikum), 20146 Hamburg, Germany
• :
/ Andrei Moroianu
• Laboratoire de Mathématiques, Université de Versailles-St Quentin, UMR 8100 du CNRS, 45 avenue des États-Unis, 78035 Versailles, France
• :
Published Online: 2014-01-05 | DOI: https://doi.org/10.1515/crelle-2013-0128

## Abstract

A locally metric connection on a smooth manifold M is a torsion-free connection D on TM with compact restricted holonomy group ${\mathrm{Hol}}_{0}\left(D\right)$. If the holonomy representation of such a connection is irreducible, then D preserves a conformal structure on M. Under some natural geometric assumption on the life-time of incomplete geodesics, we prove that conversely, a locally metric connection D preserving a conformal structure on a compact manifold M has irreducible holonomy representation, unless ${\mathrm{Hol}}_{0}\left(D\right)=0$ or D is the Levi-Civita connection of a Riemannian metric on M. This result generalizes Gallot's theorem on the irreducibility of Riemannian cones to a much wider class of connections. As an application, we give the geometric description of compact conformal manifolds carrying a tame closed Weyl connection with non-generic holonomy.

Revised: 2013-10-29

Published Online: 2014-01-05

Published in Print: 2016-05-01

Funding Source: Agence Nationale de la Recherche

Award identifier / Grant number: ANR-10-BLAN 0105

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2016, Issue 714, Pages 123–150, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, January 2014