On the geometry of orthonormal frame bundles II

We study the geometry of orthonormal frame bundles OM over Riemannian manifolds (M, g). The former are equipped with some modifications \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde g_c$$\end{document} of the Sasaki-Mok metric \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde g$$\end{document} depending on one real parameter c  ≠  0. The metrics \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde g_c$$\end{document} are “strongly invariant” in some special sense. In particular, we consider the case when (M, g) is a space of constant sectional curvature K. Then, for dim M  >  2, we find always, among the metrics \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde g_c$$\end{document} , two strongly invariant Einstein metrics on OM which are Riemannian for K  >  0 and pseudo-Riemannian for K  <  0. At least one of them is not locally symmetric. We also find, for dim M  ≥  2, two invariant metrics with vanishing scalar curvature.