Let $M$ be a manifold with all structures smooth which admits a metric $g$. Let $\Gamma$ be a linear connection on $M$ such that the associated covariant derivative $\nabla$ satisfies $\nabla g=g\otimes w$ for some 1-form $w$ on $M$. Then one refers to the above setup as a Weyl structure on $M$ and says that the pair $(g,w)$ fits $\Gamma$. If $\sigma :M\to ℝ$ and if $(g,w)$ fits $\Gamma$, then $(e^{\sigma g},w+d\sigma )$ fits $\Gamma$. Thus if one thinks of this as a map $g\mapsto w$, then $e^{\sigma g}\mapsto w+d\sigma$.In this paper, the author attempts to apply Weyl’s idea above to Finsler spaces. A Finsler fundamental function $L:TM\to ℝ$ satisfies...

Some problems concerning to Liouville distribution and framed $f$-structures are studied on the normal bundle of the lifted Finsler foliation to its normal bundle. It is shown that the Liouville distribution of transversally Finsler foliations is an integrable one and some natural framed $f(3,\epsilon )$-structures of corank 2 exist on the normal bundle of the lifted Finsler foliation.

In this paper we study isometry-invariant Finsler metrics on inner product spaces over $ℝ$ or $ℂ$, i.e. the Finsler metrics which do not change under the action of all isometries of the inner product space. We give a new proof of the analytic description of all such metrics. In this article the most general concept of the Finsler metric is considered without any additional assumptions that are usually built into its definition. However, we present refined versions of the described results...

In this paper, the standard almost complex structure on the tangent bunle of a Riemannian manifold will be generalized. We will generalize the standard one to the new ones such that the induced $(0,2)$-tensor on the tangent bundle using these structures and Liouville $1$-form will be a Riemannian metric. Moreover, under the integrability condition, the curvature operator of the base manifold will be classified.

The author previously studied with and [Publ. Math. 42, 139-144 (1993; Zbl 0796.53022)] the diffeomorphisms between two Finsler spaces $F^n=(M^n,L)$ and ${\overline{F}}^n=(M^n,\overline{L})$ which map the geodesics of $F^n$ to geodesics of ${\overline{F}}^n$ (geodesic mappings).Now, he investigates the geodesic mappings between a Finsler space $F^n$ and a Riemannian space ${\overline{ℝ}}^n$. The main result of this paper is as follows: if $F^n$ is of constant curvature $K$ and the mapping $F^n\to {\overline{ℝ}}^n$ is a strongly geodesic mapping then $K=0$ or $K\ne 0$ and $\overline{L}=e^{\varphi \left(x\right)}L$.

Some special linear connection introduced in the Finsler space by Ichijyō has the property that the curvature tensors under conformal changes remain invariant. Two Ichijyō manifolds $(M,E,\nabla )$ and $(M,\overline{E},\overline{\nabla })$ are said to be conformally equivalent if $\overline{E}=(exp{\sigma }^v)E$, $\sigma \in C^{\infty }\left(M\right)$.It is proved, that in this case, the following assertions are equivalent: 1. $\sigma$ is constant, 2. $h_{\nabla }=h_{\overline{\nabla }}$, 3. $S_{\nabla }=S_{\overline{\nabla }}$, 4. $t_{\nabla }=t_{\overline{\nabla }}$.It is also proved (when the above conditions are satisfied) that1. If $(M,E,\nabla )$ is a generalized Berwald manifold, then $(M,\overline{E},\overline{\nabla })$ is also a generalized Berwald...