Totally geodesic submanifolds of Hopf-Berger spheres

Início: 17/03/2023 17:00

Término: 17/03/2023 18:00

Palestrante: Carlos Olmos (UNC)

E-mail do Palestrante: olmos@famaf.unc.edu.ar

Resumo: A Hopf-Berger sphere of factor $tau$ is a sphere which is the total space of a Hopf fibration and such that the Riemannian metric is rescaled by a factor $tauneq 1$ in the directions of the fibers. A Hopf-Berger sphere is the usual {it Berger sphere} for the complex Hopf fibration. A Hopf-Berger sphere may be regarded as a geodesic sphere $mathsf{S}_t^m(o)subsetbar M$ of radius $t$ of a rank one symmetric space of non-constant curvature ($bar M$ is compact if and only if $tau <1$). A Hopf-Berger sphere has positive curvature if and only if $tau <4/3$. A standard totally geodesic submanifold of $mathsf{S}_t^m(o)$ is obtained as the intersection of the geodesic sphere with a totally geodesic submanifold of $bar M$. We will speak about the classification of totally geodesic submanifolds of Hopf-Berger spheres. In particular, for quaternionic and octonionic fibrations, non-standard totally geodesic spheres with the same dimension of the fiber appear, for $tau <1/2$. Moreover, there are totally geodesic $mathbb RP^2$, and $mathbb RP^3$ (with some restrictions on $tau$, the dimension and the type of the fibration). On the one hand, as a consequence of the connectedness principle of Wilking, there does not exist a totally geodesic $mathbb RP^4$ in a space of positive curvature which diffeomorphic to the sphere $S^7$. On the other hand, we construct an example of a totally geodesic $mathbb RP^2$ in a Hopf-Berger sphere of dimension $7$ and positive curvature. Natural question: could there exist a totally geodesic $mathbb RP^3$ in a space of positive curvature which diffeomorphic to $S^7$?. This talk is related to a joint work with Alberto Rodríguez-Vázquez.