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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 14th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: ## Classification of holonomy groups of affine connections Any [[closed Lie subgroup]] of $GL(V)$ occurs as the holonomy group of some [[affine connection]] (with [[torsion]], in general). See Hano–Ozeki \cite{HanoOzeki}. Holonomy groups of locally symmetric connections can be classified using [[Élie Cartan]]'s [[classification of symmetric spaces]]. For [[Levi-Civita connections]], holonomy groups were classified by [[Marcel Berger]] \cite{Berger}. The case of torsion-free [[affine connections]] that are not locally symmetric and are not [[Levi-Civita connections]] was treated by Merkulov and Schwachhöfer \cite{MerkulovSchwachhofer}. A complete list of exotic holonomy groups (for the metric and nonmetric cases) can be found in \cite{MerkulovSchwachhofer2}. ## References * {#HanoOzeki} J. Hano, H. Ozeki, _On the holonomy groups of linear connections_, Nagoya Math. J. 10, 97-100 (1956). [doi](https://doi.org/10.1017/S0027763000000106). * {#Berger} [[Marcel Berger]], Sur les groupes d'holonomie homogènes de variétés à connexion affine et des variétés riemanniennes. Bulletin de la Soci&#233;t&#233; math&#233;matique de France 79:null (1955), 279-330. [doi](http://dx.doi.org/10.24033/bsmf.1464). * {#MerkulovSchwachhofer} [[Sergei Merkulov]], Lorenz Schwachhöfer. Classification of Irreducible Holonomies of Torsion-Free Affine Connections. Annals of Mathematics 150:1 (1999), 77–149. [doi](http://dx.doi.org/10.2307/121098). * {#MerkulovSchwachhofer2} [[Sergei Merkulov]], Lorenz Schwachhöfer. Addendum to Classification of Irreducible Holonomies of Torsion-Free Affine Connections. Annals of Mathematics 150:3 (1999), 1177–1179. [doi](http://dx.doi.org/10.2307/121067). <a href="https://ncatlab.org/nlab/revision/diff/holonomy+group/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/holonomy+group/2">v2</a>, <a href="https://ncatlab.org/nlab/show/holonomy+group">current</a>

   Added:

   Classification of holonomy groups of affine connections

   Any closed Lie subgroup of $GL(V)$ occurs as the holonomy group of some affine connection (with torsion, in general). See Hano–Ozeki \cite{HanoOzeki}.

   Holonomy groups of locally symmetric connections can be classified using Élie Cartan’s classification of symmetric spaces.

   For Levi-Civita connections, holonomy groups were classified by Marcel Berger \cite{Berger}.

   The case of torsion-free affine connections that are not locally symmetric and are not Levi-Civita connections was treated by Merkulov and Schwachhöfer \cite{MerkulovSchwachhofer}. A complete list of exotic holonomy groups (for the metric and nonmetric cases) can be found in \cite{MerkulovSchwachhofer2}.

   References

   • {#HanoOzeki} J. Hano, H. Ozeki, On the holonomy groups of linear connections, Nagoya Math. J. 10, 97-100 (1956). doi.

   • {#Berger} Marcel Berger, Sur les groupes d’holonomie homogènes de variétés à connexion affine et des variétés riemanniennes. Bulletin de la Société mathématique de France 79:null (1955), 279-330. doi.

   • {#MerkulovSchwachhofer} Sergei Merkulov, Lorenz Schwachhöfer. Classification of Irreducible Holonomies of Torsion-Free Affine Connections. Annals of Mathematics 150:1 (1999), 77–149. doi.

   • {#MerkulovSchwachhofer2} Sergei Merkulov, Lorenz Schwachhöfer. Addendum to Classification of Irreducible Holonomies of Torsion-Free Affine Connections. Annals of Mathematics 150:3 (1999), 1177–1179. doi.

   diff, v2, current