Abstract: For the MIMO timevarying channel with memory, the feedback delay is inevitable in limited feedback precoding systems. To overcome this problem, the column spaces spanned by principal right singular matrix of the singular value decomposition of the channels are modeled as points on a Grassmannian manifold Gn,p of p dimensional subspaces in the n dimensional Euclidean space, so that the timevarying channel can be tracked and predicted by the geodesic of the Grassmannian manifold. More specifically, by exploiting the differential geometric properties of the Grassmannian manifold, the multidimensional dynamic focused codebook with high resolution on the tangent space of Grassmannian manifold is proposed in this paper. The computer simulation results show that the system performance of the proposed Grassmannian predictive precoding using the dynamic focused codebook with high resolution is obviously superior to memoryless limited feedback precoding which has a shortcoming of feedback delay, as well as the limited feedback predictive precoding using the fixed codebook.