Fields of totally isotropic subspaces and almost complex structures (Q760023)

Our aim is to prove global existence of a differentiable field of totally isotropic planes over the sphere \(S_ 4\), to obtain from this result a counterexample nullifying a conjecture about some manifolds or fibre bundles. Namely if the complexified tangent bundle \(T_ C(M)\) of a 2r dimensional real \(C^{\infty}\) manifold M is a Whitney sum \(T_ C(M)=\eta \oplus \eta '\) where \(\eta\), \(\eta\) ' are r-complex subbundles, such that for any x belonging to M, \(\eta 'x={\bar \eta}x\), then M owns an almost complex structure. This paper intends to establish that this statement is erroneous: it is well known that \(S_ 4\) doesn't admit any almost complex structure.