<正> §1.引言 假设非平坦的 n 维黎曼空间的曲率张量 R_(hijk)在每一点恒满足下列关系(?)其中(?)是某一向量场,同时也满足

A K_n is a Riemannian n-space whose curvature tensor satisfiesand(?)or(?)where κ_1 is a vector field.The purpose of this paper is to give some characteriza-tions of certain K-spaees.We prove the following theorems:1.In all K-spaces other than a flat extension of V_2,the Rieei tensor R_(lj)is given bywhere p is a scalar and λ_i is a null parallel vector field.In such a space, R_(lj)isgiven by R_(lj)=ρ(?), if and only if κ_1 is a null vector.2. κ_i is the plane generated by R_(ij)if and only if the rank of R_(ij) is 1 and κ_iis a null vector.3.The necessary and sufficient condition that a V_n be a conformally flatK-spaee is that the Riemannian curvature is given by(?)where s is a scalar and κ_i is a null parallel vector field.4.A conformally flat space is a K-space if and only if it admits a nullparallel vector field.5.A K-space is a subprojective space if and only if it is conformally flat.6.A K-space which admits n-3 independent parallel vector field is a har-monic space if and only if it is an Einstein space which admits at least twonull parallel vector fields.7.A K_(2p_1)-space which is null extension of V_p(q>1)is harmonic if andonly if it is an Einstein space.8.Every K_(2p)-space which is null extension of V_p is harmonic.