An attempt is made to introduce dipole structure into gravitational theory. We do this by trying to produce torsion tensors with the same form as the electric-field tensor for an electric dipole. First, we construct Ricci tensors for space-times with vanishing Christoffel connections. Then we discuss the conditions to put on the tensors. Each condition gives differential equations; however, none of the solutions provides an entirely satisfactory dipole structure. Finally, we briefly discuss how to "force" axially symmetric space-times having dipole structure.