Differentiable fibrations of the (2n-1)-sphere by great (n-1)-spheres and their coordinatization over quasifields
The author presents a variation of the proof of the following theorem of \textit{H. Gluck}, \textit{F. Warner} and \textit{C. T. Yang}: Theorem. A locally trivial differentiable fibre bundle with total space \(S^{2n-1}\) whose fibres are great (n-1)-spheres is topologically equivalent to one of the classical Hopf-fibrations [cf. Duke Math. J. 50, 1041-1076 (1983; Zbl 0534.53039)]. With each fibering \(\pi\) of \(S^{2n-1}\) by great (n-1)- spheres is associated a quasifield with additive group (\({\mathbb{R}}^ n,+)\) and linear left-multiplication. Relations to the theory of translation planes are briefly indicated. A characterization of locally trivial differentiable fibre bundles by differentiability properties of the associated quasifield is given. Finally Buchanan's theorem can be applied to the characteristic map of the associated differentiable quasifield. Cerf's result on diffeomorphisms of S 3 is used to show that even differentiable equivalence holds in the theorem in the case \(n=4\).
scientific article; zbMATH DE number 4058361