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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 14th 2021


    Classification of holonomy groups of affine connections

    Any closed Lie subgroup of GL(V)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}.


    • {#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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