eu ON SPECIAL CONCIRCULAR R-LIE-RECURRENCE IN SPECIAL FINSLER SPACES

In this paper we discuss a special concircular R-Lie-recurrence in special Finsler spaces such as R-recurrent, R-symmetric, R-birecurrent and R-bisymmetric. Apart from other theorems, it is being proved that an R-recurrent Finsler space can not admit a special concircular R-Lie-recurrence while a non-flat R-symmetric Finsler space Fn(n > 2) admitting a special concircular R-Lie-recurrence is necessarily of constant Riemannian curvature. AMS Subject Classification: 53B40.


Introduction
In 1982, P. N. pandey [5] introduced the concept of Lie-recurrence in a Finsler Space.In 1992, K. L. Duggal [3] studied the Lie-recurrence in a Riemannian space with its application to fluid space time but he used the term curvature inheriting symmetry in place of Lie-recurrence.He also used the theory to the study of fluid space time.Since then both the terms (Lie-recurrence and curvature inheriting symmetry) are in use (see [1] and [11]- [14]).P. N. Pandey and vaishali pandey [12] discussed a K-curvature inheritance, K-projective Lierecurrence and special concircular K-Lie-recurrence in a Finsler space.
The authors (P.N. pandey and Vaishali Pandey) [13] studied a K-Lierecurrence in a Finsler space.C. K. Mishra and Gautam Lodhi [1] discussed curvature inheriting symmetry and Ricci-inheriting symmetry in a Finsler space and obtained some results.In this paper we have discussed a special concircular R-Lie-recurrence in special Finsler spaces such as R-recurrent, R-symmetric, Rbirecurrent and R-bisymmetric.

Preliminaries
Let F n be an n-dimensional Finsler space equipped with a metric function F satisfying the requisite conditions [2].The relation between the metric tensor g ij of the Finsler space F n and the metric function F are given by (a) where ∂i ≡ ∂ ∂y i .The Cartan h-covariant derivative of an arbitrary vector field T i with respect to connection coefficients Γ * i jk is given by where ∂ k ≡ ∂ ∂x k .The Cartan v-covariant derivative of an arbitrary vector field T i is given by where C i rk = g ij C jrk .The tensor C jrk is called Cartan tensor and defined as C jrk = 1 2 ∂j g rk .The Ricci commutation formula for h-covariant derivative is given by where This tensor is skew-symmetric in last two lower indices and positively homogeneous of degree zero in y i .Cartan curvature tensor K i jkh , Cartan h-curvature tensor R i jkh are related by where 5) by y j and using the fact C i jm y j = 0, we get The tensor H i kh is connected with Berwald deviation tensor Berwald deviation tensor satisfies the following: where y i = g ij y j and H is scalar curvature.The commutation formula for the operators of partial differentiation with respect to y k and h-covariant differentiation is given by Let us consider an infinitesimal transformation generated by a contravariant vector field v i (x j ) which depends on position coordinates only.ǫ appearing in (10) is an infinitesimal constant.The Lie-derivative of an arbitrary tensor T i j with respect to the infinitesimal transformation (10) is given by [4] The commutation formula for the operators £ and ∂h is given by where Ω is any geometrical object.An infinitesimal transformation (10) is Lierecurrence or H-Lie-recurrence if the Lie-derivative of Berwald curvature tensor H i jkh of the Finsler space satisfies where φ is a non-zero scalar field [5].In view of this concept, the infinitesimal transformation ( 10) is called R-Lie-recurrence if the Lie-derivative of Cartan h-curvature tensor satisfies [13] £R i jkh = φR i jkh , φ = 0. ( A vector field v i in the Finsler space F n is said to be special concircular if where ρ = ρ(x) [13].

Special Concircular R-Lie-Recurrence
Theorem 1.An R-recurrent Finsler space F n (n > 2) can not admit a special concircular R-Lie-recurrence.
Proof.Let us consider a Finsler space admitting the infinitesimal transformation (10) generated by a special concircular vector field v i (x j ).Differentiating (15a) covariantly with respect to x h ,we get where ρ h = ρ |h .Taking skew-symmetric part of ( 16) and utilizing commutation formula (4) and using (15b), we have Contraction of indices i and h in (17) gives where R rk is Ricci tensor defined as R rk = R h rkh .From equations ( 17) and (18), we may write which implies Let the Finsler space F n be R-recurrent characterized by where λ m are components of a non-zero covariant vector field [7].Contracting the indices i and h in (21), we have From ( 19), ( 20), ( 21) and ( 22), we get Transvecting ( 23) by y m and using ( 6) and R mk y m = H k , we have Transvecting (24) by y i and using y i H i kh = 0 [8], we get H k y h = H h y k , which implies for H h y h = (n − 1)H and y h y h = F 2 .In view of ( 25), (24) may be rewritten as where R = H F 2 .In view of Berwald theorem [2], equation (26) implies that R is a constant and the space F n (n > 2) is of constant Riemannian curvature.Differentiating (26) covariantly, we find for y k|m = 0. Transvecting (21) by y j and using equation ( 6), we get H i kh|m = λ m H i kh , which in view of (27), implies λ m = 0, a contradiction.Therefore, an R-recurrent Finsler space F n (n > 2) can not admit a special concircular infinitesimal transformation.Definition 2. A Finsler space F n (n > 2) be R-symmetric characterized by [6] R i jkh|m = 0. (28) Theorem 3. A special concircular R-Lie-recurrence in a non-flat Rsymmetric Finsler space F n (n > 2) is an H-Lie-recurrence and the R-symmetric Finsler space is necessarily of constant Riemannian curvature.
Proof.Then, equation (20) implies equation (23).Adapting the above proceedure, we may show that equation (23) implies that the space F n (n > 2) is of constant Riemannian curvature if it is non-flat.Suppose that the special concircular transformation ( 10) is a Lie-recurrence in the R-symmetric Finsler space F n (n > 2).Then we have equation (14).In view of equation ( 11), equation ( 14) may be written as Using equations ( 16), (28) and the fact that the curvature tensor R i hjk is positively homogeneous of degree zero in y j , we get φ = 2ρ if the space is non-flat.Since ρ is independent of y i and φ = 2ρ, φ is also independent of y i .Transvecting equation ( 14) by y j and using (6), we get Differentiating (29) partially with respect to y j and using the fact that φ is independent of y j , we get £H i jkh = φH i jkh , which shows that the special concircular R-Lie-recurrence is an H-Lie-recurrence.
where a lm are components of a non-zero covariant tensor of type (0, 2) and R i jkh = 0 (see [8] and [10]).Theorem 5. A birecurrent Finsler space F n (n > 2) admitting a special concircular R-Lie-recurrence necessarily satisfies the conditions φ = 3ρ and v r a mr = ρ m .

4 .
Special Concircular R-Lie-Recurrence in a Birecurrent Finsler Space Definition 4. A birecurrent Finsler space F n characterized by