On Generalized Ricci-Recurrent Trans-Sasakian Indefinite Finsler Manifolds

In this paper generalized Ricci-recurrent trans-Sasakian indefinite Finsler manifolds are studied.These structures are established on the 〖(M^0)〗^h and 〖(M^0)〗^v vector subbundles, where M is an (2n+1) dimensional C^∞ manifold, M^0=〖(M^0)〗^h⊕ 〖(M^0)〗^v is a non-empty open submanifold of TM. F^* is the fundamental Finsler function and〖 F〗^(2n+1)=(M,M^0,F^*) is an indefinite Finsler manifold. We use the Sasaki Finsler metric G=G^H+G^V=g_ij^(F^* ) dx^i⊗dx^j+g_ij^(F^* ) δy^i ⊗δy^i . Furthermore, we give some formulas for α-Sasakian and β-Kenmotsu Finsler manifolds with pseudo-Finsler metric. It is also provided that If ((M^0 )^h,ϕ^H,ξ^H,η^H,G^H ) and (〖(M^0 )^v,ϕ〗^V,ξ^V,η^V,G^V ) are one of the (ε)- α-Sasakian,(ε)-Sasakian, (ε)- β-Kenmotsu and (ε)-Kenmotsu manifolds, which are generalized Ricci- recurrent with cyclic Ricci tensor and non-zero A^H (ξ^H ),〖 A〗^V (ξ^V ) everywhere, then they are Einstein and Ricci symmetric manifolds, where α,β are constant functions defined on (M^0 )^h and (M^0 )^v.