On Generalized BP-Recurrent Finsler Space of The Third Order
Keywords:
Cartan’s second curvature tensor, Generalized BP-trirecurrent space, Berwald’s covariant derivative of third orderAbstract
In this paper, we introduce a Finsler space which Cartan's second curvature tensor \(P_{jkh}^i\) satisfies the generalized trirecurrent property in sense of Berwald's, this space characterized by the following condition
\(B_n B_m B_l P_{jkh}^i = a_{lmn} P_{jkh}^i + b_{lmn} (δ_h^i g_jk - δ_k^i g_{jh})- 2 c_{lm} B_r (δ_h^i C_{jkn} - δ_k^i C_{jhn}) y^r\)
\( - 2 d_{ln} B_r (δ_h^i C_{jkm} - δ_k^i C_{jhm}) y^r - 2 μ_l B_n B_r (δ_h^i C_{jkm} - δ_k^i C_{jhm}) y^r, P_{jkh}^i ≠ 0\),
Where Bn Bm Bl is Berwald's covariant differential operator of third order with respect to xl , xm and xn , successively. Br is Berwald's covariant differential operator of first order with respect to xr, \(a_{lmn}\) and \(b_{lmn}\) are non-zero covariant tensor fields of third order, \(c_{lm}\) and \(d_{ln}\) are non-zero covariant tensor fields of second order, μl is non-zero covariant vector fields and \(C_{jkn} \) is (h)-torsion tensor, such space is called as a generalized Ph- trirecurrent Finsler space and we denote by GBP-TIR-RFn . we obtained the necessary and sufficient condition for some tensors to be generalized trirecurrent space.
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