Invariant Hyperplane Sections of Vector Fields on the Product of Spheres

Abstract

Let $S_{p,q}$ be the hypersurface in ${\mathbb{R}}^{p+q+1}$ defined by the following: $$S_{p,q}:=\left\{(x_1,\dots ,x_{p+1},x_{p+2},\dots ,x_{p+q+1})\in {\mathbb{R}}^{p+q+1}|{\left(\sum _{i=1}^{p+1}{x}_i^2-a^2\right)}^2+\sum _{j=p+2}^{p+q+1}{x}_j^2=1\right\},$$ 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 $q>1$.

Major revision undertaken. 18 pages. Comments are welcome

References in corpus (1)

• The $16$th Hilbert problem on algebraic limit cycles

Cited by in corpus (2)

• Characterization and dynamics of certain classes of polynomial vector fields on the torus
• Quadratic, Homogeneous and Kolmogorov vector fields on $S^1\times S^2$ and $S^2\times S^1$