Stable parallelizability of partially oriented flag manifolds

Nonexistence of Almost Complex Structures on Grassmann Manifolds

“…£ is necessarily orientable, since HP" is simply connected. The manifold G6 3 is parallelizable [13]. Therefore, it is weakly almost complex.…”

“…Proof, (i), (ii), and parts of (iii) follow from the observation that in each of the cases the manifold in question can be regarded as the total space of a differentiable bundle with fiber, an oriented Grassmann manifold, or a Grassmann manifold which is not weakly almost complex by Theorem 1.1 or 3.1 (cf. [13] [10] has observed using his ' /-genus' that (G7 3)" is not almost complex. He shows, also, that none of the quaternionic flag manifolds other than HG(1, ... , 1) admits a weakly almost complex structure, using the corresponding negative results of Hsiang and Szczarba [6] for quaternionic Grassmannians.…”

Stable Parallelizability of Partially Oriented Flag Manifolds II

“…G(1Ò Ò 1 j 3Ò 1) has trivial normal bundle (cf. [10], p. 456, or simply note that this is clearly true for the fibre inclusion of any locally trivial smooth fibration of smooth manifolds). Now from Theorem 4(i) and Theorem 7(i), it follows that X is not stably parallelizable.…”

“…Work related to their parallelizability was done by R. Stong [13] and by the present authors [10], [11]. In particular, the problem of determining which of the p.o.…”

“…Furthermore, it is not hard to see that the normal bundle of this embedding is trivial, hence i Ł (ú M ) ¾ ú L where ¾ denotes stable equivalence (for details cf. [10], [11], or Section 6). In particular, if L is not stably parallelizable then M is also not stably parallelizable.…”