Skew Hopf fibrations (Q1174901)

Skew Hopf fibrations of the \((2n+1)\)-dimensional sphere are obtained by intersecting the sphere with the planes spanned by \(x\) and \(Jx\), where \(J\) is the almost complex structure on the \((2n+2)\)-dimensional Euclidean space obtained in a standard way from an affine frame \(e\) of the space. (If the frame \(e\) is orthonormal, the corresponding fibration coincides with the standard Hopf fibration.) The main result says that if \(n=2k\), then any analytic fibration \(f\) of the \((2n+1)\)-dimensional sphere coincides with a skew Hopf fibration provided that \(f\) satisfies the following, relatively weak condition: For any great 3-sphere \(S\) spanned by two orthogonal fibres of \(f\), \(f\) is skew Hopf when restricted to \(S\).