In this paper, we will study global well-posedness for the cubic defocusing nonlinear Schrödinger equations on the compact Riemannian manifold without boundary, below the energy space, i.e. $s\<1$, under some bilinear Strichartz assumption. We will find some $\tilde{s}\<1$, such that the solution is global for $s\>\tilde{s}$.