# Hopf fibrations and totally geodesic submanifolds

## TL;DR

This paper classifies totally geodesic submanifolds in Hopf-Berger spheres, revealing new examples including real projective spaces and non-homogeneous submanifolds, primarily in positively curved cases.

## Contribution

It provides a classification of totally geodesic submanifolds in Hopf-Berger spheres and uncovers novel examples with unique geometric properties.

## Key findings

- Identification of totally geodesic submanifolds isometric to real projective spaces
- Existence of uncountably many non-congruent totally geodesic submanifolds
- Discovery of a totally geodesic submanifold that is not extrinsically homogeneous

## 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.