On a remark of O'Neill
The remark of O'Neill examined in this paper pertains to a Riemannian submersion \(\pi\) : \(S^ n\to B\), where \(S^ n\) is the unit n- dimensional sphere and B is a Riemannian manifold, with the property that the norm of the integrability tensor A of the horizontal distribution to the fibers of \(\pi\) is at most one. O'Neill stated that the geodesics of B have essentially the same conjugacy properties as \({\mathbb{C}}P^ n\) or \({\mathbb{Q}}P^ n.\) In this paper, this remark is strengthened in two ways. A stronger conclusion is obtained. Namely, the space B is isometric to \({\mathbb{C}}P^ n\), \({\mathbb{Q}}P^ n\), or \(S^ 8(1/2)\). A weaker hypothesis is needed. Namely, it is enough to start with a foliation on \(S^ n\) for which the metric is bundle-like, and for which the norm of A is bounded above by one, in order to conclude that the leaves of the foliation fiber \(S^ n\) over some space B, as above.
scientific article; zbMATH DE number 3967354