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• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 14th 2021

## 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.

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