Torsion-free \(G_{3}\)-connections that are not analytic
In 1955 \textit{M. Berger} [Bull. Soc. Math. Fr. 83, 279--330 (1955; Zbl 0068.36002)] classified the possible irreducible Riemannian holonomies. Berger also, partially classified the possible non-Riemannian holonomies of torsion free connections. However, his classification omits a finite number of possibilities, which are referred to as exotic holonomies. \textit{R. Bryant} [Proc. Symp. Pure Math. 53, 33--88 (1991; Zbl 0758.53017)] gave examples of torsion free connections on four-dimensional manifolds whose holonomy is not contained on Berger's classical list of irreducible holonomy representations. The holonomy in Bryant's examples is the irreducible four-dimensional representation of \(SL(2,\mathbb{R})\) (\(GL(2,\mathbb{R})\), respectively) and these connections are called \(H_{3}\)-connections (\(G_{3}\)-connections, respectively). A complete classification of homogeneous \(G_{3}\)-connections was given by \textit{L. J. Schwachhöfer} [Geom. Dedicata 62, 193--208 (1996; Zbl 0860.53015)]. The moduli space of the \(H_{3}\) family is analytic and finite-dimensional whereas the moduli space of the generic (nondegenerate) analytic \(G_{3}\)-connections is parametrized by four functions in three variables. Up to now, all the known torsion-free \(G_{3}\)-connections have been analytic. In this paper, the author give examples of torsion-free \(G_{3}\)-connections that are not analytic. Actually, he provides an explicit PDE-solving approach to construct a family of smooth torsion-free \(G_{3}\)-connections that are not equivalent to any analytic ones. More exactly, it is proven: There are smooth torsion-free \(G_{3}\)-connections that are not equivalent to any analytic ones, although each of these connections is locally homogeneous (in fact, is left-invariant on a local solvable group) away from a hypersurface of the base manifold.
scientific article; zbMATH DE number 2116887