A study of recurrent Finsler spaces of higher

In the present communication, we have derived Bianchi and Veblen identities along with a few more related results in a recurrent and generalized n th -recurrent Finsler space with Cartan’s curvature tensor field. A Finsler space whose Cartan's third curvature tensor satisfies the condition | | | | ( ) , where and | | | | are h-covariant differentiation (Cartan's second kind covariant differential operator) with respect to to nth order, and is recurence tensors fields.

Cartan in his second postulate, represented the variation of an arbitrary vector field under the infinitesimal change of its line element ( ) to ( ) by means of covariant (absolute) differential given by (1.1) ( ) , where  c) .
The function defined by (1.4c) is the connection parameter of Cartan, this is symmetric in the lower indices and and positively homogeneous of degree zero in the directional argument and satisfies : (1.5) .The equations (1.4a) and (1.4b) give two processes of covariant differentiation called v-covariant differentiation (Cartan's first kind covariant differentiation) and h-covariant differentiation (Cartan's second kind covariant differentiation), respectively.So | and | are respectively v-covariant derivative an -covariant derivative of the vector field .We note that this notation for covariant differentiation was used by Cartan and followed by Rund and Matsumoto calls these derivatives as " -covariant derivative " and " -covariant derivative ", respectively and his symbols for covariant differentiations are similar to that of Cartan with the only difference that

On Generalized -Recurrent Finsler Space of N th order
Let us consider a Finsler space whose Cartan's third curvature tensor satisfies the following condition (2.1) where and | | | | are h-covariant differentiation (Cartan's second kind covariant differential operator) with respect to to nth order, and are recurrence tensors fields.Definition 2.1.A Finsler space whose Cartan's third curvature tensor satisfies the condition (2.1), where and are non-null covariant tensors fields, is called a generalized -nth order space and the tensor will be called generalized h-nth tensor.We shall denote this space briefly by -.Since the metric tensor is a covariant constant, the transvecting of the condition (2.1) by , using (1.11a), (1.16a) and (1.7b), we get (2.2) ) .Conversely, the transvection of the condition (2.2) by , yields the condition (2.1).Thus, the condition (2.2) is equivalent to the condition (2.1).Therefore a generalized -nth order space may characterized by the condition (2.2).Therefore, we conclude Theorem 2.1.The generalized -nth order space may characterized by condition (2.2).
Let us consider an -, which is characterized by the condition (2.1).Contracting the indices and in (2.1), using (1.17a), (1.6)The conditions (2.3), (2.4), (2.5), and (2.6), show that, Ricci tensor , the curvature vector , the deviation tensor and the curvature scalar (all for Cartan's third curvature tensor ) of a generalized -nth order space cannot vanish, because the vanishing of them imply the vanishing of the covariant tensors fields , i.e. , a contradiction.Thus, we conclude Theorem 2.3.In -, Ricci tensor , the curvature vector , the deviation tensor and the curvature scalar ( all for Cartan's third curvature tensor ) are non-vanishing.Transvecting the condition (2.3) by , using ( ) .The condition (2.7), shows that, the curvature vector of a generalized -nth order space can not vanish, because the vanishing of it would imply the vanishing of the covariant tensors fields , i.e. , a contradiction.Transvecting the condition (2.7) by , using (1.12b), (1.24b) and (1.8), we get (2.8) . Thus, we conclude Theorem 2.4.In -, the curvature vector and the curvature scalar are nonvanishing.Now, we have seen that in a generalized -nth order space, Ricci tensor ( of Cartan's third curvature tensor ) satisfies the condition (2.3).Conversely, if Ricci tensor of a Finsler space satisfy the condition (2.3), then it need not be the space is a generalized -nth order space.However, the converse is true if the dimension of a Finsler space is 3 or the space is -like.The proof of this fact is follows: We know that the associate curvature tensor ( of Cartan's third curvature tensor ) for three dimensioned Finsler space is of the form (2.9) , where .This gives the h-covariant derivative of Ricci tensor in generalized -nth order space.The h-covariant derivative of the condition (2.9) with respect to to nth order and using (1.11a), we get . In view of the condition (2.15), the above equation implies ) .In view of the condition (2.9), the above equation implies ) .
This shows that, the associate curvature tensor ( of Cartan's third curvature tensor ) is a generalized h-recurrent.In view of (1.16a), the above condition implies (2.1).That is, the h-covariant derivative of the condition (2.9) with respect to to nth order and in view of (1.8a), gives (2.1).This shows that, a three dimensional Ricci generalized -nth order space is necessarily generalized -recurrent space.
Matsumote [28] introduced a Finsler space ( ) for which the tensor satisfying (2.9) and called it -like Finsler space .If we consider a -like Ricci generalized -recurrent space and applied the same process as above we may show that (2.16) ) .This, leads to Theorem 2.7.In -, is Ricci generalized -recurrent space, but the converse need not be true.However, if the space is -like, then the converse is also true.Now, taking the h-covariant derivative for (1.13), with respect to to nth order, we get (2.17) Using the conditions (2.1), (1.20) and (1.19a) in (2.17 ) .Thus, we conclude of degree two in the directional argument.Eliminating from (1.1) and in terms of the absolute differential of , Cartan deduced ( By using(1.13),(1.20)and (1.19a), the above equation can be written as ( of Cartan coincides with | of Matsumoto due to this change we have an extra in the first term of the right hand side of the equation (1.5).K. Yano denoted | and | by ̇ and , respectively.