Minimal isoparametric submanifolds of ${\mathbb{S}}^7$ and octonionic eigenmaps

Abstract

We use the octonionic multiplication $\cdot$ of ${\mathbb{S}}^7$ to associate, to each unit normal section $\eta$ of a submanifold $M$ of ${\mathbb{S}}^7,$ an octonionic Gauss map ${\gamma }_{\eta }:M\to {\mathbb{S}}^6,$ ${\gamma }_{\eta }(x)=x^{-1}\cdot \eta (x),$ $x\in M,$ where ${\mathbb{S}}^6$ is the unit sphere of $T_1{\mathbb{S}}^7,$ $1$ is the neutral element of $\cdot$ in ${\mathbb{S}}^7.$ Denoting by $\mathcal{N}(M)$ the vector bundle of normal sections of $M$ we set, for $\eta$ $\in \mathcal{N}(M),$ $S_{\eta }(X)=-{\left({\nabla }_X\eta \right)}^{\top },$ $X\in TM.$ Considering the Hilbert-Schmidt inner product on the vector bundle $\mathcal{S}(M)=\left\{S_{\eta },\eta \in \mathcal{N}(M)\right\}$ and defining the bundle map $\mathcal{B}:\mathcal{N}(M)\to \mathcal{S}(M)$ by $\mathcal{B}(\eta )=S_{\eta },$ we prove that if $M$ is a minimal submanifold of ${\mathbb{S}}^7$ and $\eta \in \mathcal{N}(M)$ is unitary and parallel on the normal connection, then ${\gamma }_{\eta }$ is harmonic if and only if $\eta$ is an eigenvector of ${\mathcal{B}}^{\ast }\mathcal{B}:\mathcal{N}(M)\to \mathcal{N}(M),$ where ${\mathcal{B}}^{\ast }$ is the adjoint of $\mathcal{B}.$ If $M$ is an isoparametric compact minimal submanifold of codimension $k$ of $\mathbb{S}$ then ${\mathcal{B}}^{\ast }\mathcal{B}$ has constant non negative eigenvalues $0\le {\sigma }_1\le \cdots \le {\sigma }_k$ and the associated eigenvectors ${\eta }_1,\cdots ,{\eta }_k$ form an orthonormal basis of $\mathcal{N}(M)$, parallel on the normal connection, such that each ${\gamma }_{{\eta }_j}$ is an eigenmap of $M$ with eigenvalue $7-k+$ ${\sigma }_j.$ Moreover, ${\sigma }_j=\Vert S_{{\eta }_j}{\Vert }^2,$ $1\le j\le k.$