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

TL;DR

This paper investigates the properties of minimal isoparametric submanifolds in the 7-sphere using octonionic multiplication, introducing octonionic Gauss maps and characterizing harmonicity and eigenmaps through eigenvalues of a specific bundle map.

Contribution

It establishes a novel connection between octonionic Gauss maps and eigenvalues of a bundle map for minimal submanifolds in , extending classical geometric analysis with octonionic structures.

Findings

Octonionic Gauss maps are harmonic iff associated normal sections are eigenvectors of a bundle map.

Eigenvalues of the bundle map are constant on isoparametric minimal submanifolds.

Each eigenmap corresponds to an eigenvalue related to the norm of the shape operator.

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 $\gamma_{\eta}:M\rightarrow\mathbb{S}^{6},$ $\gamma_{\eta}(x)=x^{-1}\cdot\eta(x),$ $x\in M,$ 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}, \ \text{}\eta\in\mathcal{N}(M)\right\} $ and defining the bundle map $\mathcal{B} :\mathcal{N}(M)\rightarrow\mathcal{S}(M)$ 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 $\mathcal{B}^{\ast}\mathcal{B}:\mathcal{N}(M)\rightarrow\mathcal{N}(M),$ where $\mathcal{B}^{\ast}$ is the adjoint of $\mathcal{B}.$ If $M$ is an isoparametric compact minimal submanifold of codimension $k$ of $\mathbb{S}% ^{7}$ then $\mathcal{B}^{\ast}\mathcal{B}$ has constant non negative eigenvalues $0\leq\sigma_{1}\leq\cdots\leq\sigma_{k}$ 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, $\sigma_{j}=\Vert S_{\eta_{j}}\Vert^{2},$ $1\leq j\leq k.$