The Blaschke conjecture and great circle fibrations of spheres

Abstract. We construct an explicit diffeomorphism taking any fibration of a sphere by great circles into the Hopf fibration. We use elementary differential geometry, and no surgery or K-theory, to carry out the construction—indeed the diffeomorphism is a local (differential) invariant, algebraic in derivatives. This result is new only for 5 dimensional spheres, but our new method of proof is elementary. 1. Introduction. “Notice that the classification of fibrations of spheres by great circles is an interesting but almost untouched subject... ” Arthur L. Besse [1] pg. 135. In studying the Blaschke conjecture (see Besse [1]) and in the theory o