2 transformation under the infinitesimal coordinate trans- formation where | ϵ | ≪ 1. To first order in | ϵ | , the equation (2) reads By making use of the covariant derivatives with respect to Riemannian connection, we can write the above equation as where “ | ” denotes the covariant derivative. Any vec- tor field V$_{µ}$ satisfies equation (5) is called Killing vector. Thus, the problem of finding all isometries of a given met- ric g$_{µν}$ ( x ) is reduced to find the dimension of the linear space formed by Killing vectors. In Riemann geometry, by making use of the covariant derivative, one could obtain the Ricci identities or inter- change formula where R σ ρ νµ is the Riemannian curvature tensor. And the first Bianchi identity for the Riemannian curvature tensor gives Deducing from the equation (6) and (7), we obtain Thus, all the derivatives of V$_{µ}$ will be determined by the linear combinations of V$_{µ}$ and V$_{µ}$$_{|}$$_{ν}$ . Once the V$_{µ}$ and V$_{µ}$$_{|}$$_{ν}$ at an arbitrary point of Riemannian space is given, then V$_{µ}$ and V$_{µ}$$_{|}$$_{ν}$ at any other point is determined by integration of the system of ordinary differential equa- tions. Therefore, the dimension of linear space formed by Killing vector can be at most n ( n +1) 2 in n dimensional Riemannian space. If a metric admits that the maximum number n ( n +1) 2 of Killing vectors, its Riemann space must homogeneous and isotropic (or the space is isotropic for every point). Such space is called maximally symmetry space. In Riemann geometry, the Schur’s lemma tells us that a Riemannian space with at least 3 dimension is maximally symmetry space if and only if its sectional cur- vature is constant. Also, one can check that a 2 dimen- sional Riemannian space is maximally symmetry space if and only if its sectional curvature is constant. Thus, the maximal symmetry of a given metric is an intrinsic property, and not depending on the choice of coordinate system. One special maximally symmetry space is the Minkowskian space. The Killing equation (5) of a given Minkowskian metric η$_{µν}$ ( x ) reduces to The solution of the above equation is where Q$_{µν}$ = η$_{ρµ}$Q ρ ν is an arbitrary constant skew- symmetric matrix and C µ is an arbitrary constant vector. Thus, substituting the solution (10) into the coordinate transformation (3) we obtain One should find that the term δ µ $_{ν}$+ ϵQ µ ν in above equation is just the Lorentz transformation matrix and the term ϵC µ is related to the spacetime translation. Expanding the matrix δ µ ν + ϵQ µ ν and the vector ϵC µ near identity, one could obtain the famous Poincare algebra. Other two types of maximally symmetry space are spherical and hyperbolic case. Without loss of gener- ality, we set its constant sectional curvature to be ± 1 for spherical and hyperbolic case respectively. The length element of spherical and hyperbolic case is given in a unified form where the · denotes the inner product with respect to Minkowskian metric and k = ± 1 for spherical and hy- perbolic case respectively. The metric is given as where x$_{µ}$ ≡ η$_{µν}$x $^{ν}$. The Christoffel symbols of the above length element is given as Thus, the Killing equation (5) now reads The solution of the above equation is where the index of Q and C are raise and lower by Minkowskian metric η µν and its matrix reverse η$_{µν}$ . Instead of defining an inner product structure over the tangent bundle in Riemann geometry, Finsler geometry is base on the so called Finsler structure F with the prop- erty F ( x, λy ) = λF ( x, y ) for all λ > 0, where x represents position and y ≡ dx dτ represents velocity. The Finsler met- ric is given as[19] III. KILLING VECTORS IN FINSLER SPACE ¯ x µ = x µ + ϵV $^{µ}$, (3) V $_{κ}$∂g$_{µν}$ ∂x κ + g$_{κµ}$ ∂V κ ∂x ν + g$_{κν}$ ∂V κ ∂x µ = 0 . (4) V$_{µ}$$_{|}$$_{ν}$ + V$_{ν}$$_{|}$$_{µ}$ = 0 , (5) V$_{ρ}$$_{|}$$_{µ}$$_{|}$$_{ν}$ − V$_{ρ}$$_{|}$$_{ν}$$_{|}$$_{µ}$ = − V$_{σ}$R σ $_{ρ νµ}$, (6) R σ ρ νµ + R σ ν µρ + R σ µ ρν = 0 . (7) V$_{ρ}$$_{|}$$_{µ}$$_{|}$$_{ν}$ = V$_{σ}$R σ ν $_{µρ}$. (8) ∂V$_{µ}$ ∂x ν + ∂V$_{ν}$ ∂x µ = 0 . (9) V µ = Q µ $_{ν}$x ν + C $^{µ}$, (10) ¯ x µ = ( δ µ ν + ϵQ µ $_{ν}$) x ν + ϵC $^{µ}$. (11) ds 2 = √ (1 + k ( x · x ))( dx · dx ) − k ( x · dx ) 2 1 + k ( x · x ) , (12) g$_{µν}$ = ( η$_{µν}$ 1 + k ( x · x ) − k x$_{µ}$x$_{ν}$ (1 + k ( x · x )) 2 ) , (13) γ ρ µν = − k x$_{µ}$δ ρ ν + x$_{ν}$δ ρ µ 1 + k ( x · x ) . (14) ∂V$_{µ}$ ∂x ν + ∂V$_{ν}$ ∂x µ + 2 k 1 + k ( x · x ) ( x$_{µ}$V$_{ν}$ + X$_{ν}$V$_{µ}$ ) = 0 . (15) V µ ≡ g $^{µν}$V$_{ν}$ = Q µ $_{ν}$x ν + C µ + k ( x · C ) x $^{µ}$, (16) g$_{µν}$ ≡ ∂ ∂y µ ∂ ∂y ν ( 1 2 F 2 ) . (17)