Gromov hyperbolicity of negatively curved Finsler manifolds

Let (M, F) be a closed C∞ Finsler manifold. The lift of the Finsler metric F to the universal covering space defines an asymmetric distance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widetilde d}$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widetilde M}$$\end{document}. It is well-known that the classical comparison theorem of Aleksandrov does not exist in the Finsler setting. Therefore, it is necessary to introduce new Finsler tools for the study of the asymmetric metric space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\widetilde M, \widetilde d)}$$\end{document}. In this paper, by using the geometric flip map and the unstable-stable angle introduced in [2], we prove that if (M, F) is a closed Finsler manifold of negative flag curvature, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\widetilde M, \widetilde d)}$$\end{document} is an asymmetric δ-hyperbolic space in the sense of Gromov.