Substantial Riemannian Submersions of S(15) with 7- dimensional fibres

TL;DR

This paper proves that certain Riemannian submersions of the 15-sphere with 7-dimensional fibers are equivalent to the Hopf fibration, leading to a new form of the Diameter Rigidity theorem for the Cayley plane.

Contribution

It establishes the congruence of substantial Riemannian submersions of S(15) with the Hopf fibration and derives a related rigidity theorem for the Cayley plane.

Findings

Substantial Riemannian submersions of S(15) with 7-dimensional fibers are congruent to the Hopf fibration.

A weaker form of the Diameter Rigidity theorem for the Cayley plane is proven.

The result is stronger than Wilhelm's recent Radius Rigidity Theorem.

Abstract

In this paper we show that a substantial Riemannian submersion of S(15) with 7- dimensional fibres is congruent to the standard Hopf fibration. As a consequence we prove a slightly weak form of of the Diameter Rigidity theorem for the Cayley plane which is considerably stronger than the very recent Radius Rigidity Theorem of Wilhelm.