论文标题
球体乘积的矢量场的不变超平面部分
Invariant Hyperplane Sections of Vector Fields on the Product of Spheres
论文作者
论文摘要
令$ s_ {p，q} $为$ \ mathbb {r}^{p+q+1} $中的高度表面：$ s_ s_ {p，q}：= \ left \ left \ lbrace（x_1，x_1，\ ldots，x__ {p+1}，x_ {p+1}，x______________ {p+2} \ in \ mathbb {r}^{p+q+1} \ big | \ left（\ sum_ {i = 1}^{p+1} x_i^2 -a^2 \ right）^2+\ sum_ {j = p+2}^{p+q+q+q+q+1} x_j^2 = 1 = 1 \ right \ right \ rbrace \ rbrace，$ $ a> 1 $。我们表明$ s_ {p，q} $对产品$ s^p \ times s^q $是同型。我们在$ s_ {p，q} $上对所有一级和两个多项式向量字段进行了分类。我们考虑多项式矢量字段$ \ mathcal {x} =（r_1，...，r_ {p+1}，r_ {p+2}，...，r_ {p+q+q+q+q+q+q+q+q+1}）$ in $ \ mathbb {r}^r}^{p+q+q+q+q+q+q+q+q+q+q+q+1} $保持$ s $ s_ p，p，p，q {p，q {p，q {p，q {然后，我们研究vector field $ \ mathcal {x} $的$ s_ {p，q} $的某些不变代数子集的数量，如果$ p> 1 $或$ q> 1 $。
Let $S_{p,q}$ be the hypersurface in $\mathbb{R}^{p+q+1}$ defined by the following: $$ S_{p,q} := \left\lbrace (x_1,\ldots,x_{p+1},x_{p+2},\ldots,x_{p+q+1}) \in \mathbb{R}^{p+q+1} \big| \left( \sum_{i=1}^{p+1} x_i^2 - a^2 \right)^2 + \sum_{j=p+2}^{p+q+1} x_j^2 = 1 \right\rbrace,$$ where $a > 1$. We show that $S_{p,q}$ is homeomorphic to the product $S^p \times S^q$. We classify all degree one and two polynomial vector fields on $S_{p,q}$. We consider the polynomial vector field $\mathcal{X} = (R_1,...,R_{p+1},R_{p+2},...,R_{p+q+1})$ in $\mathbb{R}^{p+q+1}$ which keeps $S_{p,q}$ invariant. Then we study the number of certain invariant algebraic subsets of $S_{p,q}$ for the vector field $\mathcal{X}$ if either $p>1$ or $q>1$.
