Almost-Quaternion Substructures on the Canonical C_n−1 -Bundle over S_2n−1

Nonexistence of Almost-Quaternion Substructures on the Complex Projective Space

“…PROOF OF THEOREM 2. For n even and « ^ 2 the result has already been proved in corollary 5.1 of [2]. For n = 2, the result follows from Lemma 1, because the total Chern class of P 2 (C) is (1 + a) 3 where a is a suitably chosen generator H 2 (P n (C), Z).…”

“…In [2], we have shown that there are no almost-quaternion substructures on even dimensional projective spaces P 2n (C) for n =£ 1. In [1], which appeared after the submission of our paper [2], the following theorem was proved: THEOREM 1. (Glover, Homer and Stong): Ifn is even, the tangent bundle T(P"(C)) does not split into a nontrivial Whitney sum of complex subbundles.…”

“…It is the purpose of this note to prove Theorem 2. As in [2], by an almost-quaternion ^-substructure on P"(C) we mean a 4&-dimensional subbundle £ of the tangent bundle T(P"(C)) which is invariant under the standard almost-complex structure of P"(C), together with two orthogonal (continuous) almost-complex substructure maps F, G defined on the total space of £, satisfying the extra condition F G = -GF, and such that F is the restriction of the standard almost-complex structure map of P n (C) to the total space of £.…”

$G$-Structures on Spheres

“…In the papers [3,5,12,18,19,23] the problem of when the structure group G n in one of the fibrations (1.1)-(1.3) can be reduced to a group G = SU(k) or Sp(k) via a standard inclusion G → G n was solved and the results were expressed in terms of complex and quaternionic James numbers in a way similar to the result quoted above. The following theorem can be regarded as a generalization of these results.…”

“…Considering standard inclusions ρ : G = G k ֒→ G n we get the famous problem on sections of Stiefel manifolds over spheres resolved in [1], [3], [5] and [23]. The other standard inclusions SU(k) ֒→ SO(n), Sp(k) ֒→ SO(n) and Sp(k) ֒→ SU(n) are dealt with in [12], [18] and [19], respectively. In these cases the question was to find a minimal standard subgroup to which G n can be reduced.…”