Nonexistence of weakly almost complex structures on Grassmannians

Abstract:

In this paper we prove that, for $2 \leq k \leq n/2$, the unoriented Grassmann manifold ${G_k}({\mathbb {R}^n})$ admits a weakly almost complex structure if and only if $n = 2k = 4 \; \text {or} \; 6$; for $3 \leq k \leq \frac {n}{2}$, none of the oriented Grassmann manifolds ${\tilde G_k}({\mathbb {R}^n})$—except ${\tilde G_3}({\mathbb {R}^6})$ and a few as yet undecided ones—admits a weakly almost complex structure.