Hopf fibrations and totally geodesic submanifolds
Carlos Enrique Olmos
Universidad Nacional de Córdoba, Córdoba, ArgentinaAlberto Rodríguez-Vázquez
Université Libre de Bruxelles, Bruxelles, Belgium

Abstract
We classify totally geodesic submanifolds in Hopf–Berger spheres, which constitute a special family of homogeneous spaces diffeomorphic to spheres constructed via Hopf fibrations. As a byproduct of our investigations, we have discovered very intriguing examples of totally geodesic submanifolds: totally geodesic submanifolds isometric to real projective spaces, uncountably many isometric but non-congruent totally geodesic submanifolds, and a totally geodesic submanifold that is not extrinsically homogeneous. Remarkably, all these examples arise in certain Hopf–Berger spheres with positive curvature.
Cite this article
Carlos Enrique Olmos, Alberto Rodríguez-Vázquez, Hopf fibrations and totally geodesic submanifolds. J. Eur. Math. Soc. (2025), published online first
DOI 10.4171/JEMS/1606