A framed $f$-structure on the tangent bundle of a Finsler manifold
Volume 104 / 2012
Abstract
Let $(M,F)$ be a Finsler manifold, that is, $M$ is a smooth manifold endowed with a Finsler metric $F$. In this paper, we introduce on the slit tangent bundle $\widetilde{TM}$ a Riemannian metric $\widetilde{G}$ which is naturally induced by $F$, and a family of framed $f$-structures which are parameterized by a real parameter $c\neq 0$. We prove that (i) the parameterized framed $f$-structure reduces to an almost contact structure on $IM$; (ii) the almost contact structure on $IM$ is a Sasakian structure iff $(M,F)$ is of constant flag curvature $K=c;$ (iii) if $\mathcal{S}=y^i\delta_i$ is the geodesic spray of $F$ and $R(\cdot,\cdot)$ the curvature operator of the Sasaki–Finsler metric which is induced by $F$, then $R(\cdot,\cdot)\mathcal{S}=0$ iff $(M,F)$ is a locally flat Riemannian manifold.
