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Hopf fibrations and totally geodesic submanifolds (2302.11711v3)

Published 23 Feb 2023 in math.DG

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.

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