Isometric actions on the projective planes and embedded generators of homotopy groups
- Thomas Püttmann (Ruhr-Universität Bochum, Fakultät für Mathematik, Germany)
Abstract
We present minimal harmonic embeddings of the sphere 
into the quaternionic projective plane 
and of the sphere 
into the octonionic projective plane 
that represent generators of the homotopy groups 
and 
, 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 
.
The opposite singular orbits in the three cases are the totally geodesic submanifolds
,
,
and 
, respectively.
The related Hopf fibrations 
,
, and
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)
