On homogeneous connections with exotic holonomy (Q1922799)

\textit{M. Berger} [Bull. Soc. Math. Fr. 83, 279-330 (1955; Zbl 0068.36002)] classified the irreducible Riemannian holonomies and partially the non-Riemannian holonomies of torsion free connections. \textit{R. L. Bryant} [Proc. Symp. Pure Math. 53, 33-88 (1991; Zbl 0758.53017)] found some torsion free connections on 4-manifolds whose holonomy is the irreducible 4-dimensional representation of \(\text{SL}(2,\mathbb{R})\) (called \(H_3\)-connections) and of \(\text{GL}(2,\mathbb{R})\) (called \(G\)-connections). The \(H_3\)-connections and \(G_3\)-connections are not contained in Berger's list (their holonomy is exotic). The author gives a complete classification of homogeneous \(G_3\)-connections. Such a connection has a symmetry group of dimension at most 4, and every generic homogeneous \(G_3\)-connection is locally equivalent to a left-invariant connection on a 4-dimensional Lie group. The classification of these connections is realised by the orbit space of some polynomials which can be completely classified.