Hopf fibrations and totally geodesic submanifolds

Abstract: A Hopf-Berger sphere of factor $\tau$ is the total space of a Hopf fibration such that the Riemannian metric is rescaled by a factor $\tau\neq 1$ in the directions of the fibers. If the Hopf fibration is the complex one, a Hopf-Berger sphere of $\tau <1$ is the usual Berger sphere. Any Hopf-Berger sphere may be regarded as a geodesic sphere $\mathsf{S}_t^m(o)\subset\bar M$ 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 $\bar M$ that contains the center $o$. In this talk we will refer to our recent 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 $\mathbb RP^3$ (under 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. Could there exist a totally geodesic $\mathbb RP^3$ in a space of positive curvature which diffeomorphic to $S^7$?.

This talk is based on a joint work with Alberto Rodríguez-Vázquez.

differential geometry

Audience: researchers in the topic

( video )

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Virtual seminar on geometry with symmetries

Series comments: Description: Research seminar in Lie group actions in Differential geometry.

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