Infinitesimal homogeneity and bundles

Let Q→M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q\rightarrow M$$\end{document} be a principal G-bundle, and B0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_0$$\end{document} a connection on Q. We introduce an infinitesimal homogeneity condition for sections in an associated vector bundle Q×GV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q\times _GV$$\end{document} with respect to B0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_0$$\end{document}, and, inspired by the well known Ambrose–Singer theorem, we prove the existence of a connection which satisfies a system of parallelism conditions. We explain how this general theorem can be used to prove the known Ambrose–Singer type theorems by an appropriate choice of the initial system of data. We also obtain new applications, which cannot be obtained using the known formalisms, e.g. a classification theorem for locally homogeneous spinors. Finally, we introduce natural local homogeneity and local symmetry conditions for triples [inline-graphic not available: see fulltext] consisting of a Riemannian metric on M, a principal bundle on M, and a connection on P. Our main results concern locally homogeneous and locally symmetric triples, and they can be viewed as bundle versions of the Ambrose–Singer and Cartan theorem.