The warped Sasaki–Matsumoto metric and bundlelike condition

Abstract

Let \documentclass[12pt]{minimal}\begin{document}$\mathbb {F}^n=(M,F)$\end{document}Fn=(M,F) be a Finsler manifold and G be the Sasaki–Matsumoto metric on \documentclass[12pt]{minimal}\begin{document}$TM^\circ$\end{document}TM∘. Bejancu and Farran [“Finsler geometry and natural foliations on the tangent bundle,” Rep. Math. Phys. 58, 131 (2006)] proved that \documentclass[12pt]{minimal}\begin{document}$\mathbb {F}^n=(M,F)$\end{document}Fn=(M,F) is a Riemannian manifold if and only if the Sasaki–Matsumoto metric G on \documentclass[12pt]{minimal}\begin{document}$TM^\circ$\end{document}TM∘ is bundlelike for the vertical foliation. Let \documentclass[12pt]{minimal}\begin{document}$\mathbb {F}^{n_1+n_2}=(M_1\times _fM_2,F)$\end{document}Fn1+n2=(M1×fM2,F) be the warped product Finsler manifold. In this paper the warped Sasaki–Matsumoto metric \documentclass[12pt]{minimal}\begin{document}${}^*\mathbf {G}$\end{document}*G is introduced for the warped product Finsler manifold, and it is shown if the warped function f is not a constant, then \documentclass[12pt]{minimal}\begin{document}${}^*\mathbf {G}$\end{document}*G on \documentclass[12pt]{minimal}\begin{document}$TM^\circ$\end{document}TM∘ is bundlelike for the warped vertical foliation \documentclass[12pt]{minimal}\begin{document}$\mathcal {V}^*(TM^\circ )$\end{document}V*(TM∘) if and only if \documentclass[12pt]{minimal}\begin{document}$\mathbb {F}^{n_1}_1=(M_1,F_1)$\end{document}F1n1=(M1,F1) and \documentclass[12pt]{minimal}\begin{document}$\mathbb {F}^{n_2}_2=(M_2,F_2)$\end{document}F2n2=(M2,F2) are Riemannian manifolds.