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## A framed $f$-structure on the tangent bundle of a Finsler manifold

### Tom 104 / 2012

Annales Polonici Mathematici 104 (2012), 23-41 MSC: Primary 53B40; Secondary 53C60. DOI: 10.4064/ap104-1-3

#### Streszczenie

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.

#### Autorzy

• Esmaeil PeyghanDepartment of Mathematics
Faculty of Science
Arak University
Arak 38156-8-8349, Iran
e-mail
• Chunping ZhongSchool of Mathematical Sciences
Xiamen University
Xiamen 361005, China
e-mail

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