$\mathbb{S}^{7}$的极小等参子流形与八元本征映射
Minimal isoparametric submanifolds of $\mathbb{S}^{7}$ and octonionic eigenmaps
摘要：
我们使用$\mathbb{S}^{7}$的八元数乘法$\cdot$将八元数高斯映射$γ_η:M\rightarrow\mathbb{S}^{6},$ $γ_η(x)=x^{-1}\cdotη(x),$ $x\in M,$与$\mathbb{S}^{7},$的子流形$M$的每个单位法截面$η$相关联其中$\mathbb{S}^{6}$是$T_{1}\mathbb{S}^{7},$的单位球面$1$是$\mathbb{S}^{7}.$中$\cdot$的中性元用$\mathcal{N}(M)$表示我们设置的$M$的法截面的向量丛，考虑向量丛$\mathcal{S}(M)=\left\{S_η, \ \text{}η\in\mathcal{N}(M)\right\} $上的Hilbert-Schmidt内积，并由$\mathcal{B}(η)=S_η,$定义丛映射$\mathcal{B} :\mathcal{N}(M)\rightarrow\mathcal{S}(M)$，证明了：若$M$是$\mathbb{S}^{7}$的极小子流形，且$η\in\mathcal{N}(M)$在法联络上是酉平行的，则$γ_η$是调和的当且仅当$η$是$\mathcal{B}^{\ast}\mathcal{B}:\mathcal{N}(M)\rightarrow\mathcal{N}(M),$的特征向量其中$\mathcal{B}^{\ast}$是$\mathcal{B}.$的伴随如果$M$是$\mathbb{S}% ^{7}$的余维$k$的等参紧致极小子流形则$\mathcal{B}^{\ast}\mathcal{B}$具有常数非负特征值$0\leqσ_{1}\leq\cdots\leqσ_{k}$和相关联的特征向量$η_{1},\cdots,η_{k}$形成$\mathcal{N}(M)$的标准正交基，平行于法联络，使得每个$γ_{η_{j}}$是具有本征值PDMATHABNQPDMATHABNR的$M$的本征映射。此外，$σ_{j}=\Vert S_{η_{j}}\Vert^{2},$ $1\leq j\leq k.$
英文摘要：
We use the octonionic multiplication $\cdot$ of $\mathbb{S}^{7}$ to associate, to each unit normal section $η$ of a submanifold $M$ of $\mathbb{S}^{7},$ an octonionic Gauss map $γ_η:M\rightarrow\mathbb{S}^{6},$ $γ_η(x)=x^{-1}\cdotη(x),$ $x\in M,$ 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 $η$ $\in\mathcal{N}(M),$ $S_η(X)=-\left(\nabla_{X}η\right) ^{\top},$ $X\in TM.$ Considering the Hilbert-Schmidt inner product on the vector bundle $\mathcal{S}(M)=\left\{S_η, \ \text{}η\in\mathcal{N}(M)\right\} $ and defining the bundle map $\mathcal{B} :\mathcal{N}(M)\rightarrow\mathcal{S}(M)$ by $\mathcal{B}(η)=S_η,$ we prove that if $M$ is a minimal submanifold of $\mathbb{S}^{7}$ and $η\in\mathcal{N}(M)$ is unitary and parallel on the normal connection, then $γ_η$ is harmonic if and only if $η$ 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 $0\leqσ_{1}\leq\cdots\leqσ_{k}$ and the associated eigenvectors $η_{1},\cdots,η_{k}$ form an orthonormal basis of $\mathcal{N}(M)$, parallel on the normal connection, such that each $γ_{η_{j}}$ is an eigenmap of $M$ with eigenvalue $7-k+$ $σ_{j}.$ Moreover, $σ_{j}=\Vert S_{η_{j}}\Vert^{2},$ $1\leq j\leq k.$
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
