A Finsler metric on an *n*-dimensional *C*^{∞} manifold *M* is a function *F* : *TM* → [0, ∞) which has the following properties: (i) *F* is *C*^{∞} on *TM*_{0} = *TM* ∖ {0}, (ii) *F* is positively 1-homogeneous on the fibers of tangent bundle *TM*, (iii) for each *y* ∈ *T*_{x}M, the quadratic form **g**_{y} : *T*_{x}M ⊗ *T*_{x}M → ℝ on *T*_{x}M is positive definite,

$$\begin{array}{}{\displaystyle {\mathbf{g}}_{y}(u,v):=\frac{1}{2}\frac{{\mathrm{\partial}}^{2}}{\mathrm{\partial}s\mathrm{\partial}t}[{F}^{2}(y+su+tv)]{|}_{s,t=0},\phantom{\rule{1em}{0ex}}u,v\in {T}_{x}M.}\end{array}$$

Let *x* ∈ *M* and *F*_{x} := *F*|_{TxM}. To measure the non-Euclidean feature of *F*_{x}, define **C**_{y} : *T*_{x}M ⊗ *T*_{x}M ⊗ *T*_{x}M → ℝ by

$$\begin{array}{}{\displaystyle {\mathbf{C}}_{y}(u,v,w):=\frac{1}{2}\frac{d}{dt}[{\mathbf{g}}_{y+tw}(u,v)]{|}_{t=0},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}u,v,w\in {T}_{x}M.}\end{array}$$

The family **C** := {**C**_{y}}_{y∈TM0} is called the Cartan torsion. It is well known that **C** = 0 if and only if *F* is Riemannian.

Taking a trace of Cartan torsion yields the mean Cartan torsion **I**_{y}. Let (*M*, *F*) be an *n*-dimensional Finsler manifold. For *y* ∈ *T*_{x}M_{0}, define **I**_{y} : *T*_{x}M → ℝ by

$$\begin{array}{}{\displaystyle {\mathbf{I}}_{y}(u)=\sum _{i=1}^{n}{g}^{ij}(y){\mathbf{C}}_{y}(u,{\mathrm{\partial}}_{i},{\mathrm{\partial}}_{i}),}\end{array}$$

where {*∂*_{i}} is a basis for *T*_{x}M at *x* ∈ *M*. The family **I** := {**I**_{y}}_{y∈TM0} is called the mean Cartan torsion.

For *y* ∈ *T*_{x}M_{0}, define the Matsumoto torsion **M**_{y} : *T*_{x}M ⊗ *T*_{x}M ⊗ *T*_{x}M → ℝ by **M**_{y}(*u*, *v*, *w*) := *M*_{ijk}(*y*)*u*^{i}v^{j}w^{k} where

$$\begin{array}{}{\displaystyle {M}_{ijk}:={C}_{ijk}-\frac{1}{n+1}\{{I}_{i}{h}_{jk}+{I}_{j}{h}_{ik}+{I}_{k}{h}_{ij}\},}\end{array}$$

and *h*_{ij} = *g*_{ij} – *F*_{yi}F_{yj} is the angular metric. *F* is said to be C-reducible if **M**_{y} = 0.

#### Lemma 2.1

([7]) A Finsler metric *F* on a manifold *M* of dimension *n* ≥ 3 is a Randers metric if and only if **M**_{y} = 0, ∀*y* ∈ *TM*_{0}.

The horizontal covariant derivative of the Cartan torsion **C** along geodesics gives rise to the Landsberg curvature **L**_{y} : *T*_{x}M ⊗ *T*_{x}M ⊗ *T*_{x}M → ℝ is defined by **L**_{y}(*u*, *v*, *w*) := *L*_{ijk}(*y*)*u*^{i}v^{j}w^{k}, where

$$\begin{array}{}{\displaystyle {L}_{ijk}:={C}_{ijk|s}{y}^{s}.}\end{array}$$(3)

The family **L** := {**L**_{y}}_{y∈TM0} is called the Landsberg curvature. A Finsler metric is called a Landsberg metric if **L** = 0 (see [8, 9]).

The horizontal covariant derivative of the mean Cartan torsion **I** along geodesics gives rise to the mean Landsberg curvature **J**_{y} : *T*_{x}M → ℝ which is defined by **J**_{y}(*u*) := *J*_{i}(*y*)*u*^{i}, where

$$\begin{array}{}{\displaystyle {J}_{i}:={I}_{i|s}{y}^{s}.}\end{array}$$(4)

The family **J** := {**J**_{y}}_{y∈TM0} is called the mean Landsberg curvature. A Finsler metric is called a weakly Landsberg metric if **J** = 0.

A Finsler metric *F* is called *GP*-*reducible* if its Landsberg curvature is given by

$$\begin{array}{}{\displaystyle {L}_{ijk}=\lambda {C}_{ijk}+{a}_{i}{h}_{jk}+{a}_{j}{h}_{ki}+{a}_{k}{h}_{ij},}\end{array}$$(5)

where *a*_{i} and λ are scalar functions on *TM*. By definition, if *a*_{i} = 0 then *F* reduces to a general relatively isotropic Landsberg metric and if λ = 0 then *F* is P-reducible [10]. Thus, the study of this class of Finsler spaces will enhance our understanding of the geometric meaning of C-reducible and P-reducible metrics.

Finsler geometry is a natural generalization of Riemannain one. Study of geometric vector fields has been extended from Riemannian geometry to Finsler geometry. Concurrent vector fields in Finslerian setting have been studied extensively (for example see [11, 12]). Let us consider a vector field *X*^{i}(*x*) in (*M*, *F*). This field is called *concurrent* if it satisfies the following

$$\begin{array}{}{\displaystyle {X}_{|j}^{i}={\delta}_{j}^{i},}\end{array}$$(6)

$$\begin{array}{}{\displaystyle {X}_{,j}^{i}=0.}\end{array}$$(7)

One can also use covariant derivative with respect to Berwald connection to define concurrent vector fields. It is well knwon that the two defintions are the same.

Let us mention a standard example of concurrent vector field. Let *V* be a vector space and (*x*^{i}) be a standard chart of *V*. Suppose that *F* is a Minkowski norm on *V*. Then, the radial vector field
$\begin{array}{}{\displaystyle X={x}^{i}\frac{\mathrm{\partial}}{\mathrm{\partial}{x}^{i}}}\end{array}$ is a concurrent vector field of (*V*, *F*).

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