Hopf fibrations and totally geodesic submanifolds
Differential Geometry
2023-12-11 v3
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. In particular, we stand out the following three: 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 only arise in certain Hopf-Berger spheres with positive curvature.
Keywords
Cite
@article{arxiv.2302.11711,
title = {Hopf fibrations and totally geodesic submanifolds},
author = {Carlos E. Olmos and Alberto Rodríguez-Vázquez},
journal= {arXiv preprint arXiv:2302.11711},
year = {2023}
}
Comments
Minor changes. To appear in JEMS (Journal of the European Mathematical Society)