NONEXISTENCE OF WEAKLY ALMOST COMPLEX STRUCTURES ON GRASSMANNIANS

In this paper we prove that, for 2 less-than-or-equal-to k less-than-o r-equal-to n/2, the unoriented Grassmann manifold Gk (R(n)) admits a w eakly almost complex structure if and only if n = 2k = 4 or 6; for 3 l ess-than-or-equal-to k less-than-or-equal-to n/2, none of the oriented Grassmann manifolds G(k)(R(n))-except G3(R6) and a few as yet undecid ed ones-admits a weakly almost complex structure.