Title:
Totally geodesic submanifolds and Hopf fibrations
Abstract: The classification of transitive Lie group actions on
spheres was obtained by Borel, Montgomery, and Samelson in the
forties. As a consequence of this, it turns out that apart from the
round metric there are other Riemannian metrics in spheres which are
invariant under the action of a transitive Lie group. These other
homogeneous metrics in spheres can be constructed by modifying the
metric of the total space of the complex, quaternionic or octonionic
Hopf fibration in the direction of the fibers.
In this talk, I will report on a joint work with Carlos Olmos
(Universidad Nacional de Córdoba), where we classified totally
geodesic submanifolds in Hopf-Berger spheres. These are those
Riemannian homogeneous spheres obtained by rescaling the round metric
of the total space of Hopf fibrations by a positive factor in the
direction of the fibers.