In the present paper, we have studied some properties of a semi-symmetric non-metric connection in HSU-unified structure manifold and HSU-Kahler manifold. Some new results on such manifolds been obtained.

The aim of this paper is to characterize a Riemannian 3-manifold M3 equipped with semi-symmetric metric ξ-connection ∇˜ ρ-Einstein and gradient solitons. existence soliton in an admitting ensured by constructing non-trivial example, hence some our results are verified. By using standard tensorial technique, we prove that the scalar curvature (M3,∇˜) satisfies Poisson equation ΔR=4(2−σ−6ρ)ρ.

We study lightlike hypersurfaces of a semi-Riemannian space form M̃(c) admitting a semi-symmetric non-metric connection. First, we construct a type of lightlike hypersurfaces according to the form of the structure vector field of M̃(c), which is called a ascreen lightlike hypersurface. Next, we prove a characterization theorem for such an ascreen lightlike hypersurface endow with a totally geodes...

we classify the paracontact riemannian manifolds that their rieman-nian curvature satisfies in the certain condition and we show that thisclassification is hold for the special cases semi-symmetric and locally sym-metric spaces. finally we study paracontact riemannian manifolds satis-fying r(x, ξ).s = 0, where s is the ricci tensor.

We study Einstein lightlike hypersurfaces M of a Lorentzian space form M̃(c) admitting a semi-symmetric non-metric connection subject to the conditions; (1) M is screen conformal and (2) the structure vector field ζ of M̃ belongs to the screen distribution S(TM). The main result is a characterization theorem for such a lightlike hypersurface.