Division algebras, fibrations of spheres by great spheres and the topological determination of space by the gross behavior of its geodesics
The Blaschke problem asks for the determination of all compact Riemannian manifolds such that the distance from a point to the cut locus is a constant independent of the initial point and direction. It is known that the cut locus is a smooth submanifold, and there is a canonical model for each admissible pair of dimensions of the manifold and cut locus. In the case where the model is the round sphere or real projective space, the given manifold is known to be isometric to the model. In this paper, the authors show that the given manifold is homeomorphic to the model, provided the model is a complex projective space of complex dimension 2, 3 or 4, the quaternion projective plane, or the Cayley plane. The proofs begin with the known fact that the exponential map gives rise to a smooth fiber bundle over the cut locus with total space a round sphere and with fibers great subspheres. A variety of special methods are used to prove the various cases. Division algebras are used in the study of the projective planes. In the complex case, the authors find a homology sphere of codimension 2 fibered in circles over a homology projective space, but can make only limited use of it, since they cannot proceed by induction.
scientific article; zbMATH DE number 3848184
scientific article