Generalized K¨ ahlerian spaces * Svetislav M. Minˇ ci´ c, Mi´ ca S. Stankovi´ c and Ljubica S. Velimirovi´ c Abstract In this paper 1 we investigate generalized K¨ ahlerian spaces and find some relations for curvature tensors in these spaces. Also, we define holo- morphically projective mappings of generalized K¨ ahlerian spaces and ob- tain an invariant geometric object for these mappings. 1 Introduction A generalized Riemannian space GR N in the sense of Eisenhart’s definition [1] is a differentiable N -dimensional manifold, equipped with nonsymmetric ba- sic tensor g ij . Connection coefficients of this space are generalized Cristoffel’s symbols of the second kind. Generally it is Γ i jk 6=Γ i kj . In a generalized Riemannian space one can define four kinds of covariant derivatives [3], [4]. For example, for a tensor a i j in GR N we have a i j | 1 m = a i j,m +Γ i pm a p j - Γ p jm a i p , a i j | 2 m = a i j,m +Γ i mp a p j - Γ p mj a i p , a i j | 3 m = a i j,m +Γ i pm a p j - Γ p mj a i p , a i j | 4 m = a i j,m +Γ i mp a p j - Γ p jm a i p . In the case of the space GR N we have five independent curvature tensors [5] (in [5] R 5 is denoted by ˜ R 2 ): R 1 i jmn =Γ i jm,n - Γ i jn,m +Γ p jm Γ i pn - Γ p jn Γ i pm , R 2 i jmn =Γ i mj,n - Γ i nj,m +Γ p mj Γ i np - Γ p nj Γ i mp , R 3 i jmn =Γ i jm,n - Γ i nj,m +Γ p jm Γ i np - Γ p nj Γ i pm +Γ p nm (Γ i pj - Γ i jp ), R 4 i jmn =Γ i jm,n - Γ i nj,m +Γ p jm Γ i np - Γ p nj Γ i pm +Γ p mn (Γ i pj - Γ i jp ), R 5 i jmn = 1 2 (Γ i jm,n +Γ i mj,n - Γ i jn,m - Γ i nj,m +Γ p jm Γ i pn +Γ p mj Γ i np -Γ p jn Γ i mp - Γ p nj Γ i pm ). * Supported by Grant 04M03C of RFNS trough Math. Inst. SANU. 1 Presented at the IMC “Filomat 2001”, Niˇ s, August 26–30, 2001 2000 Mathematics Subject Classification: 53B05 Keywords: Generalized Riemannian space, K¨ ahlerian space, Generalized K¨ ahlerian space, holo- morphically projective mapping. 167