The Holonomy of Spherically Symmetric Projective Finsler Metrics of Constant Curvature

In this paper, we investigate the holonomy group of n-dimensional projective Finsler metrics of constant curvature. We establish that in the spherically symmetric case, the holonomy group is maximal, and for a simply connected manifold it is isomorphic to D i f f o ( S n - 1 ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}i\!f \hspace{-3pt} f_o({\mathbb {S}}<^>{n-1})$$\end{document} , the connected component of the identity of the group of smooth diffeomorphism on the n - 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n-1}$$\end{document} -dimensional sphere. In particular, the holonomy group of the n-dimensional standard Funk metric and the Bryant-Shen metrics are maximal and isomorphic to D i f f o ( S n - 1 ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}i\!f \hspace{-3pt} f_o({\mathbb {S}}<^>{n-1})$$\end{document} . These results are the firsts describing explicitly the holonomy group of n-dimensional Finsler manifolds in the non-Berwaldian (that is when the canonical connection is non-linear) case.