Natural connections on the bundle of Riemannian metrics

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$FM,{\cal M}_M$\end{document} be the bundles of linear frames and Riemannian metrics of a manifold M, respectively. The existence of a unique Diff M-invariant connection form on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$J^1{\cal M}_M\times _M FM\rightarrow J^1{\cal M}_M$\end{document}, which is Riemannian with respect to the universal metric on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$J^1{\cal M}_M\times _M TM$\end{document}, is proved. Applications to the construction of universal Pontryagin and Euler forms, are given.