Invariant Hyperplane Sections of Vector Fields on the Product of Spheres
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866917558292054016 |
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| author | Benny, Joji Sarkar, Soumen |
| author_facet | Benny, Joji Sarkar, Soumen |
| contents | 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 $q>1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2205_08825 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Invariant Hyperplane Sections of Vector Fields on the Product of Spheres Benny, Joji Sarkar, Soumen Dynamical Systems Classical Analysis and ODEs 34A34, 34C14, 34C40, 34C45, 58J90 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 $q>1$. |
| title | Invariant Hyperplane Sections of Vector Fields on the Product of Spheres |
| topic | Dynamical Systems Classical Analysis and ODEs 34A34, 34C14, 34C40, 34C45, 58J90 |
| url | https://arxiv.org/abs/2205.08825 |