Invariant Hyperplane Sections of Vector Fields on the Product of Spheres

Let \(S_{p,q}\) $S_{p,q}$ be the hypersurface in \(\mathbb {R}^{n},\) ${\mathbb{R}}^n,$ where \(n=p+q+1,\) $n=p+q+1,$ defined by the following: \(\begin{aligned}\begin{gathered} S_{p,q}:= \left\{ (x_1,\ldots ,x_{n}) \in \mathbb {R}^{n} ~~ \big | ~~ \left( \sum _{i=1}^{p+1} x_i^2 - a^2 \right) ^2 + \sum _{j=p+2}^n x_j^2 = 1 \right\} \end{gathered}\end{aligned}\) $$\begin{array}{c}\hfill \begin{array}{c}{S}_{p,q}:=\left\{(x_1,\dots ,x_n),\in ,{\mathbb{R}}^n,,,|,,,{\left(\sum _{i=1}^{p+1},x_i^2,-,a^2\right)}^2,+,\sum _{j=p+2}^n,x_j^2,=,1\right\}\end{array}\end{array}$$ where \(a > 1\) $a>1$ . We show that \(S_{p,q}\) $S_{p,q}$ is homeomorphic to the product \(S^p \times S^q\) $S^p\times S^q$ . We classify all degree one and two polynomial vector fields on \(S_{p,q}\) $S_{p,q}$ . We consider the polynomial vector field \(\mathcal {X} = (R_1,...,R_{p+1},R_{p+2},...,R_n)\) $\mathcal{X}=(R_1,...,R_{p+1},R_{p+2},...,R_n)$ in \(\mathbb {R}^{p+q+1}\) ${\mathbb{R}}^{p+q+1}$ which keeps \(S_{p,q}\) $S_{p,q}$ invariant. Then, we study the number of certain invariant algebraic subsets of \(S_{p,q}\) $S_{p,q}$ for the vector field \(\mathcal {X}\) $\mathcal{X}$ if either \(p>1\) $p>1$ or \(q>1\) $q>1$ .