In this note, we continue the investigation of a projective K\"ahler manifold
M of semi-negative holomorphic sectional curvature H. We introduce a new
differential geometric numerical rank invariant which measures the number of
linearly independent {\it truly flat} directions of H in the tangent spaces.
We prove that this invariant is bounded above by the nef dimension and bounded
below by the numerical Kodaira dimension of M. We also prove a splitting
theorem for M in terms of the nef dimension and, under some additional
hypotheses, in terms of the new rank invariant