Minimal isoparametric submanifolds of [formula omitted] and octonionic eigenmaps.

Abstract We use the octonionic multiplication ⋅ of S 7 to associate, to each unit normal section η of a submanifold M of S 7 , an octonionic Gauss map γ η : M → S 6 , γ η (x) = x − 1 ⋅ η (x) , x ∈ M , where S 6 is the unit sphere of T 1 S 7 , 1 is the neutral element of ⋅ in S 7. Denoting by N (M) the vector bundle of normal sections of M and by E (M) the vector bundle of sections of the vector bundle of endomorphisms of TM eqquiped with the Hilbert–Schmidt metric and defining the bundle homomorphism B : N (M) → E (M) by B (η) = S η , where S η is the second fundamental form of M determined by η , we prove that if M is a minimal submanifold of S 7 and η ∈ N (M) is unitary and parallel on the normal connection, then γ η is harmonic if and only if η is an eigenvector of B ⁎ B : N (M) → N (M) , where B ⁎ is the adjoint of B. If M is an isoparametric compact minimal submanifold of codimension k of S 7 then B ⁎ B has constant non negative eigenvalues 0 ≤ σ 1 ≤ ⋯ ≤ σ k and the associated eigenvectors η 1 , ⋯ , η k form an orthonormal basis of N (M) , parallel on the normal connection, such that each γ η j is an eigenmap of M with eigenvalue 7 − k + σ j. Moreover, σ j = ‖ S η j ‖ 2 , 1 ≤ j ≤ k. [ABSTRACT FROM AUTHOR]