TX_1~ABS:AT/ADD:TX_2~ABS:AT 35 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2023, 7 (2): 35-41 ReseaRch aRticle On Generalized Recurrent Finsler Spaces of Higher Order with Berwald’s Curvature Tensor A. M. A. Al-Qashbari1, Salem Saleh2,3 1Department of Mathematics, Faculty of Education-Aden, Aden University, Aden, Yemen, 2Department of Computer Science, Cihan University-Erbil, Kurdistan Region, Iraq, 3Department of Mathematics, Hodeidah University, Hodeidah, Yemen ABSTRACT In this article, we presented the recurrent of higher order in Finsler space Fn for projective tensor K jkh i which satisfies a generalized five recurrence with respect to Berwald’s connection parameters Gkh i , and we have some theorems and some identities also we get more results in a recurrent and generalized 5-recurrent Fn by using the sense of Berwald’s curvature tensor. Keywords: Generalized K -five-recurrent, Berwald’s derivative of fifth order, curvature tensor K jkh i , curvature tensor Rjkh i , Weyl projective curvature tensor Wjkh i INTRODUCTION The generalized recurrent Finsler space used the sense of Berwald curvature tensor discussed by Abdallah,[1] AL-Qashbari and Qasem,[2] and some others. A n-dimensional Riemannian space of recurrent was introduced and studied by Rund[3]. Some properties for Weyl’s curvature tensor studied by Al-Qashbari[4],[15],[17],[19],[20],[21],[22],[23],[24], Ahsan and Ali[5], Abu-Donia et al.,[6] Emamian et al.[7],[13],[14],[16],[18] and Qasem et al.[8],[25],[26] The generalized birecurrent, trirecurrent Finsler space, and higher order recurrent are studied. Furthermore, Awed[9] introduced the curvature tensors for the space-time of general relativity. The decomposability of certain generalized K -recurrent Finsler space have been studied by Baleedi[10], Al-Qufail,[11] and Pandey et al.[12] and others. Berwald’s covariant derivative k j iT of an arbitrary tensor filed   Tj i with respect to xk is given by k j i k j i r j i k r j r rk i r i jk rT T T G T G T G� � � �� � � �   (1.1) Berwald’s covariant derivative of the metric function and the vector yi vanish identically, i.e. (a) kF = 0�and (b) k jy = 0. (1.2) But Berwald’s covariant derivative of the metric tensor gij does not vanish and is given by  k ij ijklh h h h ijkg C y y C� � � �� � � �2 2 (1.3) The vectors yi and yi satisfy the following relations (a) g y yij j i  = , (b) y y Fi i� = 2 , (c) � j k j ky y� and (d) �h k ik ihg g  � (1.4) The two sets of quantities gij and its associate tensor gij are related by[11] g g if i k ifij jk i k � �� ���,������ ������ ������� ���,������ �� � � � � 1 0 ����� �������i k� � � � (1.5) The tensor Cijk defined by C g Fijk i jk i j k� � � � � � 1 2 1 4 2    � � � (1.6) is known as (h) hv-torsion tensor. The torsion tensor      Cik h and its associate torsion tensor Cijk are related by (a) C y C yjk i j kj i j � �= = 0 , (b) C y C y C yijk i ijk j ijk k � � � �= = = 0 and (c) �h k ijk hijC C  � (1.7) The tensor     Krkj i as defined above is called Cartan’s fourth Corresponding Author: Salem Saleh, Department of Computer Science, Cihan University-Erbil, Kurdistan Region, Iraq. E-mail: s_wosabi@yahoo.com Received: July 17, 2023 Accepted: September 02, 2023 Published: September 20, 2023 DOI: 10.24086/cuesj.v7n2y2023.pp35-41 Copyright © 2023 Adel M. A. Al-Qashbari, Salem Saleh. This is an open- access article distributed under the Creative Commons Attribution License (CC BY-NC-ND 4.0). Cihan University-Erbil Scientific Journal (CUESJ) Al-Qashbari and Saleh: On generalized recurrent finsler spaces of higher order 36 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2023, 7 (2): 35-41 curvature tensor, this tensor is positively homogeneous of degree zero in the directional argument (a) ( ) ( )= ∂ Γ + ∂ Γ + Γ Γ − ∂ Γ + ∂ Γ + Γ Γ  * * * * * * * *   i i i l i m i i l i m rkj j kr l rj k mj kr k jr l rk j mk jrK G G and (b) K Kjkh i jhk i� �    (1.8) The tensor K jkh i   satisfies the relation too. (a) K y Hjkh i j kh i     = and (b) H K y Kjkh i jkh i m j mkh i� � �( � )� (1.9) The curvature tensor K jkh i  satisfies the following identities known as Bianchi identities[27] K K Kjkh i hjk i khj i� � �� �0 (1.10) Ricci tensor Kjk, the curvature vector Kj and the curvature scalar K are given by: (a) K Kjki i jk=  , (b) K y Kjk k j    = and (c) K y Hjk j k    = (1.11) The quantities Hjkh i   and Hkh i   form the components of tensors and they called h-curvature tensor of Berwald and torsion tensor respectively are defined as (a) H G G G G G h k H h kjkh i j kh i kh r rj i rhj i k r jkh i� � � � � �� � � �� ��� / * � / ��: (b) H G G C h kkh i h k i k r rh i� � � �� /� (1.12) They are also related by: (a) H y Hjkh i j kh i  = and (b) H Hjkh i j kh i� � (1.13) The tensor Hh i   , called the deviation tensor, given by: H G G y G G G Gh i h i r h i r hs i s s i h s� � � � � �2 2� � � � �� �� (1.14) The curvature tensor Rjkh i , Ricci tensor Rjk, tensor Hkh, curvature vector Hk and scalar curvature H are connected by the following: (a) H y Hjk i j k i      = , (b) H Hjk jkr r=   , (c)     H Hj ji i= , (d) R y Hjkh i j kh i   = , (e) R y Hjk j k  = , (f)., (j)    R Ri i = and (h) R Rjki i jk= (1.15) also connected by: (a)      H Hkh k h� � , (b) H y Hkh k h   = and (d) H y H n Hk k k k � �� � �� �1 (1.16) The Weyl curvature tensor denoted as Wjkh i is defined by W H n H H x n H Hjkh i jkh i j i kh hk i j kh j hk� � � �� � � � � � �� � � � ( ) ( ) �� � � 1 1    �� � k i jh hj r j hr h i jk n nH H x H n nH ( ) ��� � � � ( ) � �� �2 2 1 1 � � � � � � �� � � � �   HH x Hkj r j kr� �� �� � � � �� � � �  � � (1.17) The tensors Wjkh i , Wjk i and Wjk satisfies the following identities. (a) W y Wjkh i j kh i  = , (b) W y Wjk i j k i  = and (c) W Wjki i jk= (1.18) Notations:   K jkh i : Cartan’s 4th Curvature Tensor; Rjkh i : Cartan’s 3th Curvature Tensor; Hjkh i : Berwald Curvature Tensor; Kkh: K-Ricci Tensor, Kk: Curvature Vector and K: Scalar Curvature. ON GENERALIZED –BK-FIVE-RECURRENT SPACE In this, proposal is defined as     r s n m l is derivative of fifth order, for projective curvature tensor K jkh i which is defined as:     r s n m l jkh i lmnsr jkh i lmnsr h i jk k i jhK K g g       � � �� �� � � � � �� � � ��� �2 2� � � � � �lmns q q h i jkr k i jhr lmnr q q h i jks k iy C C y C� � � � �  CC jhs� �� � �� � � ��2 2� � � � � �� � � � � �lmn r q q h i jks k i jhs lmsr q q h i jkn k iy C C y C   CC jhn� � � �� � � ��� �2 2� � � � �lms r q q h i jkn k i jhn lmr s q q h i jkny C C y C� �� � � �    ��k i jhnC� � � �� � � ���2 2� � � � � �lm r s q q h i jkn k i jhn lnsr p q h i jkmy C C y C� �� � � �    kk i jhmC� � (2.1) � �� � � �� �2 2� � � ��� � � � � �lns r p q h i jkm k i jhm lnr s q q h i jkm ky C C y C    ii jhmC� �� � �� �� ��� �2 2� � � � � �ln r s q q h i jkm k i jhm lsr n q q h i jkm k iy C C y C C    � jjhm� �� � �� � � ��� �2 2� � � � � �ls r n q q h i jkm k i jhm lr s n q q h i jkm k iy C C y C      � CC jhm� � � �� ��� �2 � � �l r s n q q h i jkm k i jhmy C C    � � � � �. Where ≠ 0i jkhK and     r s n m l is derivative of five order with respect to xl, xm, xn, x, and xr, respectively, the quantities λlmnsr and μlmnsr are non-null covariant vectors field. Result 2.1. Every generalized K -recurrent is generalized K -five recurrent. Definition 2.1. A Finsler space for the tensor K jkh i is known to satisfy the condition (2.1), and will be called generalized-five recurrent space. We shall call such Finsler space a generalized K -five-recurrent space and is denoted by G K -FIRFn. Transvecting (2. 1) by yj, using (1.2b), (1.9a), (1.4a) and (1.7b), we get     r s n m l kh i lmnsr kh i lmnsr h i k k i hH a H b y y� � � � � �(� �)�� � �� � (2.2) Transvecting (2.2) by yk, using (1.2b), (1.15a), (1.4b) and (1.4c), we get     r s n m l h i lmnsr h i lmnsr h i h iH a H b F y y� � � � �(� � �)�� � �� 2 (2.3) Thus, we conclude Theorem 2.1. In G K -FIRFn, Berwald derivative of the five orders for torsion tensor Hkh i and the deviation tensor Hh i   are given by the conditions (2.2) and (2.3), respectively. Putting i=h in (2.2) and (2.3), using (1.4b), (1.4c), (1.5), (1.15b), and (1.15c), we get     r s n m l k lmnsr k lmnsr kH a H n b y� � � �( )�� � �1 and (2.4)     r s n m l lmnsr lmnsrH a H n b F� � � �� ( )�� � �1 2 (2.5) Al-Qashbari and Saleh: On generalized recurrent finsler spaces of higher order 37 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2023, 7 (2): 35-41 The conditions (2.4) and (2.5), show that the curvature vector Hk and the curvature scalar H cannot vanish because one of these would imply almnsr=0 and blmnsr=0, that is a contradiction. Thus, we conclude, Theorem 2.2. In G K -FIRFn the curvature vector WK and the curvature scalar W are given by the conditions (2.4) and (2.5), respectively, are non-vanishing. Contracting the indices i and h in (2.1), using (1.11a), (1.4c) and (1.7c), we get       r s n m l jk lmnsr jk lmnsr jk lmns K K n g n� � � � �� � �� � � �� �� � � 1 2 1� � qq q jkry C� � �� � � �� � � �� � � �2 1 2 1 2 1 n y C n y C n lmnr q q jks lmn r q q jks lm � � � � � �� �    ssr q q jkny C � �� � � �� � � �� � �2 1 2 1 2 1 n y C n y C n lms r q q jkn lmr s q q jkn lm r � � � � � �       s q q jkny C (2.6) � �� � � �� � � �� � �2 1 2 1 2 1 n y C n y C n lnsr p q jkm lns r p q jkm lnr � �� �� � �    ss q q jkmy C � �� � � �� � � �� � 2 1 2 1 2 1 n y C n y C n ln r s q q jkm lsr n q q jkm ls r n � � �        qq q jkmy C � � �� �� � � �� �� �2 1 2 1n y C n y Clr s n q q jkm l r s n q q jkm� �       Thus, we conclude Theorem 2.3. In G K -FIRFn, Berwald derivative of the five orders for Ricci tensor Kjk is given by the condition (2.6). Transvecting (2.6) by yj, using (1.2b), (1.11b), (1.4a), and (1.7b), we get     r s n m l j lmnsr j lmnsr jK K n y� � � � �� � �� �� �1 (2.7) Thus, we get Theorem 2.4. In G K -FIRFn, Berwald derivative of the five orders for the curvature vector Kj is given by the condition (2.7). DIVERGENCE OF K-TENSOR AND OTHER CURVATURE TENSORS In this section, we will obtain the necessary and sufficient conditions for tensors to be interpreted to generalized recurrent in   G K -FIRFn. It is known that curvature tensor Rjkh i and curvature tensor K jkh i  are connected by the formula[27] R K C Hjkh i jkh i jp i hk p� � ��. (3.1) Taking derivative of 5th order of (3.1), with respect to xl, xm, xn, xs, and xr, successively, we get:               r s n m l jkh i r s n m l jkh i r s n m l jp i hk pR K C H           � � � � (3.2) Using the condition (2.1) in (3.2), we get:     r s n m l jkh i lmnsr jkh i lmnsr h i jk k i jhR K g g      � � �� �� � � � � �� � � ��� �2 2� � � � � �lmns q q h i jkr k i jhr lmnr q q h i jks k iy C C y C� � � � �  CC jhs� �� � �� � � ��2 2� � � � � �� � � � � �lmn r q q h i jks k i jhs lmsr q q h i jkn k iy C C y C   CC jhn� � � �� � � ��� �2 2� � � � �lms r q q h i jkn k i jhn lmr s q q h i jkny C C y C� �� � � �    ��k i jhnC� � � �� � � ���2 2� � � � � �lm r s q q h i jkn k i jhn lnsr p q h i jkmy C C y C� �� � � �    kk i jhmC� � (3.3) � �� � � �� �2 2� � � ��� � � � � �lns r p q h i jkm k i jhm lnr s q q h i jkm ky C C y C    ii jhmC� �� � �� � � ��� �2 2� � � � � �ln r s q q h i jkm k i jhm lsr n q q h i jkm k i jy C C y C C     hhm� �� � �� � � ��� �2 2� � � � � �ls r n q q h i jkm k i jhm lr s n q q h i jkm k iy C C y C      � CC jhm� � � �� � ��� �2 � � �l r s n q q h i jkm k i jhm r s n m l jp i hky C C C H        � � �� � � �� � p� � By using (3.1), the above equation can be written as:     r s n m l jkh i lmnsr jkh i lmnsr jp i hk p lmns h i jkR a R a C H b g      � � � � ��� ��k i jhg   � �� � � ��� �2 2� � � � � �lmns q q h i jkr k i jhr lmnr q q h i jks k iy C C y C� � � � �  CC jhs� �� � �� � � ��2 2� � � � � �� � � � � �lmn r q q h i jks k i jhs lmsr q q h i jkn k iy C C y C   CC jhn� � � �� � � ��� �2 2� � � � �lms r q q h i jkn k i jhn lmr s q q h i jkny C C y C� �� � � �    ��k i jhnC� � � �� � � ���2 2� � � � � �lm r s q q h i jkn k i jhn lnsr p q h i jkmy C C y C� �� � � �    kk i jhmC� � � �� � � �� �2 2� �� � � � � �lns r p q h i jkm k i jhm lnr s q q h i jkm k i jy C C y C C    hhm� �� (3.4) � �� � � ��� �2 2� � � � � �ln r s q q h i jkm k i jhm lsr n q q h i jkm k i jy C C y C C     hhm� �� � �� � � ��� �2 2� � � � � �ls r n q q h i jkm k i jhm lr s n q q h i jkm k iy C C y C      � CC jhm� � � �� � ��� �2 � � �l r s n q q h i jkm k i jhm r s n m l jp i hky C C C H        � � �� � � �� p� � . This shows that,     r s n m l jkh i lmnsr jkh i lmns h i jk k i jhR a R b g g   � � �� �� � � �� � � ��� �2 2� � � � � �lmns q q h i jkr k i jhr lmnr q q h i jks k iy C C y C� � � � �  CC jhs� �� � �� � � ��2 2� � � � � �� � � � � �lmn r q q h i jks k i jhs lmsr q q h i jkn k iy C C y C   CC jhn� � � �� � � ��� �2 2� � � � �lms r q q h i jkn k i jhn lmr s q q h i jkny C C y C� �� � � �    ��k i jhnC� � � �� � � ���2 2� � � � � �lm r s q q h i jkn k i jhn lnsr p q h i jkm k iy C C y C� �� �    CC jhm� � (3.5) Al-Qashbari and Saleh: On generalized recurrent finsler spaces of higher order 38 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2023, 7 (2): 35-41 � �� � � �� �2 2� � � ��� � � � � �lns r p q h i jkm k i jhm lnr s q q h i jkm ky C C y C    ii jhmC� �� � �� � � ��� �2 2� � � � � �ln r s q q h i jkm k i jhm lsr n q q h i jkm k i jy C C y C C     hhm� �� � �� � � ��� �2 2� � � � � �ls r n q q h i jkm k i jhm lr s n q q h i jkm k iy C C y C      � CC jhm� � � �� ��� �2 � � �l r s n q q h i jkm k i jhmy C C    � � � � �. If and only if     r s n m l jp i hk p lmnsr jp i hk pC H a C H         � � � � � (3.6) Thus, we conclude Theorem 3.1 In G K -FIRFn, Cartan’s fourth curvature tensor  Rjkh i is generalized five recurrent Finsler space if and only if the condition C Hjp i hk p    � � is five recurrent Finsler space. Transvecting (3.4) by yj, using (1.2b), (1.15d), (1.4a), (1.7a) and (1.7b), we get     r s n m l kh i lmnsr kh i lmns h i k k i hH a H b y y          � � �� �� � (3.7) The following is derived. Theorem 3.2. In G K -FIRFn, Berwald derivative of the five orders for the torsion tensor Hkh i is generalized five recurrent Finsler space. Contracting the indices i and h in (3.4), using (1.15h), (1.5), (1.4b), (1.4c), and (1.7c), we get:     r s n m l jk lmnsr jk lmnsr jp i ik p lmns jR a R a C H n b g� � � � �� � � � � �� �1 kk � �� � � �� � � �� � � � �2 1 2 1 2 1 n y C n y C n lmns q q jkr lmnr q q jks lmn � � � � � � �    r q q jksy C � � �� � � �� � � �� � � � � 2 1 2 1 2 1 n y C n y C n lmsr q q jkn lms r q q jkn lmr � � � � � �     s q q jkny C � �� � � �� � � �� � �2 1 2 1 2 1 n y C n y C n lm r s q q jkn lnsr p q jkm lns r � � �      � pp q jkmy C (3.8) � �� � � �� � � �� � �2 1 2 1 2 1 n y C n y C n lnr s q q jkm ln r s q q jkm lsr n � � �       qq q jkmy C � �� � � �� ��2 1 2 1n y C n y Cls r n q q jkm lr s n q q jkm� �     � � � �� � � � �� �2 1n y C C Hl r s n q q jkm r s n m l jp i ik p�         � � � � . This shows that       r s n m l jk lmnsr jk lmns jk lmns q R a R n b g n y � � � � � � � �� � � �� � 1 2 1 �� qq jkrC � (3.9) � �� � � �� � � �� � �2 1 2 1 2 1 n y C n y C n lmnr q q jks lmn r q q jks lmsr � � � � � �   qq q jkny C � �� � � �� � � �� � �2 1 2 1 2 1 n y C n y C n lms r q q jkn lmr s q q jkn lm r � � � � � �       s q q jkny C � �� � � �� � � �� � 2 1 2 1 2 1 n y C n y C n y lnsr p q jkm lns r p q jkm lnr s q � � � �     qq jkmC � �� � � �� � � �� � 2 1 2 1 2 1 n y C n y C n ln r s q q jkm lsr n q q jkm ls r n � � �        qq q jkmy C � �� � � �� �2 1 2 1n y C n y Clr s n q q jkm l r s n q q jkm� � �� �      � �� �. If and only if     r s n m l jp i ik p lmnsr jp i ik pC H a C H         � � � � � (3.10) This following is derived Theorem 3.3. In G K -FIRFn, R-Ricci tensor Rjk is generalized five recurrent Finsler space if and only if the condition C Hjp i ik p    � � is five recurrent Finsler space. For a Riemannian space V4, when n=4, the equation (3.8), shows that      r s n m l jk lmnsr jk lmns jk lmns q q jkrR a R b g y C� �� � � �� � � � 3 6 6 � � � ��lmnr q q jksy C� � � � � � � �6 6 6 6 � � � � � ��� � �lmn r q q jks lmsr q q jkn lms r q q jkny C y C y C     ���lmr s q q jkny C� �  � � � � 6 6 6 6 � � � � �lm r s q q jkn lnsr p q jkm lns r p q jkmy C y C y C� � �� � �      ��lnr s q q jkmy C  � � ��� � �6 6 6� � �ln r s q q jkm lsr n q q jkm ls r n q q jkmy C y C y C       � � � � �6 6� �� �lr s n q q jkm l r s n q q jkmy C y C      � � (3.11) If and only if     r s n m l jp i ik p lmnsr jp i ik pC H a C H         � � � � � (3.12) The following is derived. Theorem 3.4. In G K -FIRFn, For a Riemannian space V4, when n=4, the R-Ricci tensor Rjk is given by condition (3.11) if and only if the condition C Hjp i ik p    � � is five recurrent Finsler space. Transvecting (3.9) by yk, using (1.2b), (1.15f), (1.4a) and (1.7b), we get     r s n m l j lmnsr j lmns jR a R n b y� �� �� � �� �1 (3.13) Thus, we conclude Theorem 3.5. In G K -FIRFn, Berwald derivative of the five order for curvature vector Rj is generalized five recurrent Finsler space. COMPOUND DERIVATIONS OF TENSOR Ki jkh In this section, we present the relation between the curvature tensor K jkh i  and Wely’s projective tensor Wjkh i . It is Al-Qashbari and Saleh: On generalized recurrent finsler spaces of higher order 39 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2023, 7 (2): 35-41 known that Cartan’s third curvature tensor Rjkh i and Wely’s projective tensor    Wjkh i are connected by the formula[1] W R R g Rjkh i jkh i k i jh jk h i� � �� �1 3 � � � � (4.1) Using (3.1) in (4.1), we get, W K C H R g Rjkh i jkh i jp i hk p k i jh jk h i� � � �� �� � � �� 1 3 � (4.2) Taking derivative of 5th order of (4.2), with respect to xl, xm, xn, xs, and xr, successively, we get               r s n m l jkh i r s n m l jkh i r s n m l jp i hk pW K C H       � � � � � �� �1 3 �B B B B Br s n m l k i jh jk h iR g R� (4.3) Using (2.1) and substituting the condition (4.2) in (4.3), we get     r s n m l jkh i lmnsr jkh i lmnsr jp i hk p lmnsr k iW W C H� �� � � � �� � � � 1 3 RR g Rjh jk h i�� �� � � �� � � �� �� � � � �� � � � � �lmnsr h i jk k i jh lmns q q h i jkr k i jhrg g y C C2  � ��� �� � � �2 2� � � � � �lmnr q q h i jks k i jhs lmn r q q h i jks k iy C C y C C� � �   jjhs� � � �� � � ��2 2� � � ��� � � � � �lmsr q q h i jkn k i jhn lms r q q h i jkn k iy C C y C C   jjhn� � � �� � � ��� �2 2� � � � �lmr s q q h i jkn k i jhn lm r s q q h i jkny C C y C� �� � �     ��k i jhnC� � � �� � � ��2 2� � � � � �� �� � � � � �lnsr p q h i jkm k i jhm lns r p q h i jkmy C C y C   kk i jhmC� � � �� � � ��� �2 2� � � � � �lnr s q q h i jkm k i jhm ln r s q q h i jkm ky C C y C    � �� ii jhmC� �� � � �� �� � � ��� �2 2� � � � � �lsr n q q h i jkm k i jhm ls r n q q h i jkm ky C C y C     ii jhmC� �� � �� � � ��� � �2 2� � � � �lr s n q q h i jkm k i jhm l r s n q q h i jkmy C C y C      � � ��k i jhmC� � � � � � �� �� � �         r s n m l jp i hk p r s n m l k i jh jk h iC H R g R 1 3 � (4.4) This shows that     r s n m l jkh i lmnsr jkh i lmnsr h i jk k i jhW W g g     � � �� �� � � � � �� � � ���2 2� � � � � �lmns q q h i jkr k i jhr lmnr q q h i jks k i jhsy C C y C C� �  �� � �� �� � � �2 2� � � � � �lmn r q q h i jks k i jhs lmsr q q h i jkn k iy C C y C C  � � � � jjhn� � � �� � � �2 2� � � � � �lms r q q h i jkn k i jhn lmr s q q h i jkn k iy C C y C� �� � �   � CC jhn� � (4.5) � �� � � �2 2�� � � � � �lm r s q q h i jkn k i jhn lnsr p q h i jkm ky C C y C� �� � � �    ii jhmC� � � �� � � �2 2� � � �� �� � � � � �lns r p q h i jkm k i jhm lnr s q q h i jkm ky C C y C   � ii jhmC� � � �� � � �2 2� �� � � � � �ln r s q q h i jkm k i jhm lsr n q q h i jkm k iy C C y C C    � � jjhm� �� � � �� �� � � �2 2� �� � � � � �ls r n q q h i jkm k i jhm lr s n q q h i jkm ky C C y C      ii jhmC� � � �� �2� �� � �l r s n q q h i jkm k i jhmy C C    � If and only if     r s n m l jp i hk p lmnsr jp i hk pC H C H   � � � � �� (4.6) and     r s n m l k i jh jk h i lmnsr k i jh jk h iR g R R g R       � � ��� � � �� � (4.7) The following is derived Theorem 4.1. In G K -FIRFn, Wely’s projective tensor Wjkh i is generalized five recurrent Finsler space if and only if the tensors C Hjp i hk p� � and �k i jh jk h iR g R�� � are five recurrent Finsler space. Transvecting (4.4) by yj, using (1.2b), (1.18a), (1.4a),(1.7a), (1.15e) and (1.7b), we get     r s n m l kh i lmnsr kh i lmnsr k i h k h iW W H y R� � � ��� � �� �� � � 1 3 � �� � � �� �� � � � � � �� � � �lmnsr h i k k i h r s n m l k i h k h iy y H y R 1 3      (4.8) This shows that     r s n m l kh i lmnsr kh i lmnsr h i k k i hW W y y     � � �� �� � � � (4.9) If and only if     r s n m l k i h k h i lmnsr k i h k h iH y R H y R              � � ��� � � �� � (4.10) This following is derived. Theorem 4.2. In G K -FIRFn, the torsion curvature tensor Wkh i is generalized five recurrent Finsler space if and only if the tensor        �k i h k h iH y R�� � is five recurrent Finsler space. Transvecting (4.5) by yk, using (1.2b), (1.18b), (1.4c), and (1.4b), we get     r s n m l h i lmnsr h i lmnsr i h h iW W y H F R� ��� � �� �� � 1 3 2 Al-Qashbari and Saleh: On generalized recurrent finsler spaces of higher order 40 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2023, 7 (2): 35-41 � �� � � �� �� � � �� �lmnsr h i i h r s n m l i h h iF y y y H F R2 21 3      (4.11) This shows that     r s n m l h i lmnsr h i lmnsr h i i hW W F y y� �� � �� �� � � 2 (4.12) If and only if     r s n m l i h h i lmnsr i h h iy H F R y H F R� �� ��� � � �� �2 2� (4.13) This following is derived Theorem 4.3. In G K -FIRFn, the curvature tensor Wh i is generalized five recurrent Finsler space if and only if the tensor � �y H F Ri h h i�� �2 is five recurrent Finsler space. Contracting the indices i and h in (4.4), using (1.18c), (1.5), (1.4b), (1.4c), and (1.15j), we get     r s n m l jk lmnsr jk lmnsr jp i ik p lmnsr jk W a W a C H R g � � � �� � � � � � 1 3 � jjkR� � (4.14) � �� � � �� � � �� �n b g n y C n y Clmns jk lmns q q jkr lmnr q q j1 2 1 2 1� � � �� �� �  kks � �� � � �� � � �� � 2 1 2 1 2 1 n y C n y C n lmn r q q jks lmsr q q jkn lms r � � � � � �    � q q jkny C � � �� � � �� � � �� � 2 1 2 1 2 1 n y C n y C n lmr s q q jkn lm r s q q jkn lnsr �� � � � �     ��p q jkmy C � �� � � �� � � �� � 2 1 2 1 2 1 n y C n y C n lns r p q jkm lnr s q q jkm ln r s � � � �      qq q jkmy C �� �� � � �� � � �� � 2 1 2 1 2 1 n y C n y C n lsr n q q jkm ls r n q q jkm lr s n � � �        q q jkmy C � �� � � � � � 2 1 1 3 n y C C Hl r s n q q jkm r s n m l jp i ik p r s n m ��              � � l jk jkR g R�� �. This shows that       r s n m l jk lmnsr jk lmns jk lmns q W a W n b g n y � � � � � � � �� � � �� � 1 2 1 �� qq jkrC (4.15) � �� � � �� � � �� � 2 1 2 1 2 1 n y C n y C n lmnr q q jks lmn r q q jks lmsr �� � � � � �    qq q jkny C � �� � � �� � � �� � 2 1 2 1 2 1 n y C n y C n lms r q q jkn lmr s q q jkn lm r s q � � �        yy Cq jkn� � �� � � �� � � �� � 2 1 2 1 2 1 n y C n y C n lnsr p q jkm lns r p q jkm lnr s q � � � � �     yy Cq jkm � �� � � �� � � �� � 2 1 2 1 2 1 n y C n y C n ln r s q q jkm lsr n q q jkm ls r n � � �        qq q jkmy C � �� � � �� �2 1 2 1n y C n y Clr s n q q jkm l r s n q q jkm� �      � If and only if     r s n m l jk jk lmnsr jk jkR g R R g R�� � � �� �� (4.16) and     r s n m l jp i ik p lmnsr jp i ik pC H a C H       � � � � � (4.17) This following is derived Theorem 4.4. In G K -FIRFn, the Ricci Tensor Wjk is generalized five recurrent Finsler space if and only if the tensors C Hjp i ik p� � and (Rjk–gjkR) are five recurrent Finsler space. CONCLUSION AND RECOMMENDATIONS The generalized K -five recurrent space is satisfied in the condition (2.1). In G K FIRFn − , the  -derivative of the five order for Ricci Tensor Kjk and curvature vector Kj are given by (2.6) and (2.7), respectively. In G K - FIRFn, the necessary and sufficient condition of curvature tensor  Rjkh i is generalized five recurrent Finsler space, if and only if the condition C Hjp i hk p    � � is five recurrent Finsler space and Ricci tensor Rjk is generalized five recurrent Finsler space, if and only if the condition C Hjp i ik p    � � is five recurrent Finsler space. In  G K - FIRFn, the Wely’s projective tensor Wjkh i is generalized five recurrent Finsler space if and only if the tensors C Hjp i hk p� � and �k i jh jk h iR g R�� � are five recurrent Finsler space. In  G K - FIRFn, the curvature tensor Wkh i is generalized five recurrent Finsler space if and only if the tensor        �k i h k h iH y R�� � is five recurrent Finsler space. Finally, Ricci tensor Wjk is generalized five recurrent Finsler space if and only if the tensors C Hjp i ik p� � and (Rjk–gjkR) are five recurrent Finsler space. The authors call the need for research and study in generalized K - higher recurrent Finsler spaces and interlard it with the properties of special spaces for Finsler space. REFERENCES 1. A. A. A. Abdallah, On Generalize BR-Recurrent Finsler Space, M. Sc. Dissertation. University of Aden, Aden, Yemen, 2017. 2. M. A. AL-Qashbari, F. Y. A. Qasem. Study on generalized Br-trirecurrent finsler space. Journal of Yemen engineer, Faculty of Engineering, University of Aden, vol. 15, pp. 79-89, 2017. 3. H. Rund. The Differential Geometry of Finsler Spaces, Springer- Verlag, Berlin, 1959, 2nd ed., Nauka, Moscow, 1981. 4. M. A. AL-Qashbari. 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