Isometric actions on the projective planes and embedded generators of homotopy groups
- Thomas PΓΌttmann (Ruhr-UniversitΓ€t Bochum, FakultΓ€t fΓΌr Mathematik, Germany)
Abstract
We present minimal harmonic embeddings of the sphere π5 into the quaternionic projective plane ββ2 and of the sphere π11 into the octonionic projective plane πβ2 that represent generators of the homotopy groups Ο5(ββ2) β β€2 and Ο11(πβ2) β β€24, respectively. The embeddings parametrize singular orbits of isometric cohomogeneity one actions. In the case of the complex projective plane the analogous singular orbit is the quadric which represents twice a generator of Ο2(πβ2). The opposite singular orbits in the three cases are the totally geodesic submanifolds ββ2 β πβ2, πβ2 β ββ2, and ββ2 β πβ2, respectively. The related Hopf fibrations π2 β ββ2, π5 β πβ2, and π11 β ββ2 are realized in the projective planes by intersections of the singular orbits with projective lines. We also show that the above mentioned orbits together with the projective lines provide all orbits that are diffeomorphic to spheres and represent non-trivial elements in the corresponding homotopy groups.
(joint work with A. Rigas, Campinas)