( ISSN 2277 - 9809 (online) ISSN 2348 - 9359 (Print) ) New DOI : 10.32804/IRJMSH

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DECOMPOSITION OF BERWALD’S CURVATURE TENSOR FIELD IN A FINSLER SPACE

    3 Author(s):  ANKIT MAURYA, K.B. GUPTA,JITENDRA SINGH

Vol -  10, Issue- 11 ,         Page(s) : 257 - 265  (2019 ) DOI : https://doi.org/10.32804/IRJMSH

Abstract

The covarient derivative of an arbitrary contravariant vector X^i in the sense of Berwald in a Finsler space F_n is given by (1.1) X_((j))^i = ∂ ̇_j X^i- (∂ ̇_j X^i ) G_j^k+ X^k G_kj^i , where G_jk^i (x ,x ̇ ) is the connection parameter introduced by Berwald and is defined by (1.2) G_jk^i = ∂ ̇_jk^2 〖 G〗^i , which is the positively homogeneous of degree zero in its directional arguments . The geodesic deviation has been given in the following form ( Rund [6] ) (1.3) (δ^2 Z^j)/(δu^2 ) + H_k^j (x,x ̇) Z^k = 0, where the vector Z^i is called the “variation vector” and the tensor H_k^j (x,x ̇) is called the “deviation tensor” defined by (1.4) H_k^j (x,x ̇) = K_ihk^j x ̇^i x ̇^h

[1] Kumar , A.  Decomposiation of pseudo projective curvature tensor field  in a Second oreder recurrent Finsler space, Ibid,(8), 59, (1976), 77-82.
[2] Pande, H.D.  A Studie of some Problems in a Finsler Space, Pustaksthan, Gorakhpur (India), (1978).
[3] Pande, H.D.and Khan, T.A. General decomposition of Berwald’s curvature tensor field in a recurrent Finsler Space, Rend. Acad. Naz. dei Lincei. (8), L V  (1973), 680-685.
[4] Pande, H.D. and  Khan, T.A. Decomposition of Berwald’s curvature tensor field in a   second order recurrent Finsler space,Rend- Acad.Naz.dei                     Lincei.(8) L VII    (1974), 565-569.
[5] Pande, H.D.and Khan, T.A. On the decomposition of curvature Tensor Fields in a recurrent Finsler Space, Indian J.Pure Appl. Math. 8, (1977), 418- 424.                     
[6] Rund .H The differential geometry of Finsler spaces, Springr, Verlag, Berlin (1959).

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