Isometric actions on the projective planes and embedded generators of homotopy groups

We present minimal harmonic embeddings of the sphere 𝕊⁵ into the quaternionic projective plane ℍℙ² and of the sphere 𝕊¹¹ into the octonionic projective plane 𝕆ℙ² that represent generators of the homotopy groups π₅(ℍℙ²) ≈ ℤ₂ and π₁₁(𝕆ℙ²) ≈ ℤ₂₄, respectively. The embeddings parametrize singular orbits of isometric cohomogeneity one actions. In the case of the complex projective plane the analogous singular orbit is the quadric which represents twice a generator of π₂(𝕆ℙ²). The opposite singular orbits in the three cases are the totally geodesic submanifolds ℝℙ² ⊂ 𝕆ℙ², 𝕆ℙ² ⊂ ℍℙ², and ℍℙ² ⊂ 𝕆ℙ², respectively. The related Hopf fibrations 𝕊² → ℝℙ², 𝕊⁵ → 𝕆ℙ², and 𝕊¹¹ → ℍℙ² are realized in the projective planes by intersections of the singular orbits with projective lines. We also show that the above mentioned orbits together with the projective lines provide all orbits that are diffeomorphic to spheres and represent non-trivial elements in the corresponding homotopy groups.

(joint work with A. Rigas, Campinas)