denote the Grassmann manifold of all k-dimensional (left) F-vector subspace of Fn
for F = ℝ, the reals, C, the complex numbers, or H the quaternions. The problem of determining which of the Grassmannians bound was addressed by the author in . Partial results were obtained in  for the case F = ℝ, including a sufficient condition, due to A. Dold, on n and k for ℝ Gn,k
to bound. Here, we show that Dold's condition is also necessary, and obtain a new proof of sufficiency using the methods of this paper, which cover the complex and quaternionic cases as well.