On locally homogeneous \(G\)-structures
A theorem of \textit{I. M. Singer} [Commun. Pure Appl. Math. 13, 685-697 (1960; Zbl 0171.42003)] states that a Riemannian manifold \((M,g)\) is locally homogeneous if and only if it is infinitesimally homogeneous, that is, for each pair of points \(p,q\in M\) there exists an isometry \(F\) between \(T_pM\) and \(T_qM\) which preserves the covariant derivatives of the Riemann curvature tensor up to order \(k_M+ 1\), for an integer \(k_M\leq n(n- 1)/2- 1\), \(n= \dim M\). In the present paper, the author extends the notion of infinitesimal homogeneity to invariant connections on \(G\)-structures and proves a generalization of Singer's theorem. The general results are then applied to discuss the cases of conformal and Weyl structures.
scientific article; zbMATH DE number 1240982