Quadratic, Homogeneous and Kolmogorov vector fields on $S^1\times S^2$ and $S^2 \times S^1$

Abstract

In this paper, we consider the following two algebraic hypersurfaces $$S^1\times S^2=\{(x_1,x_2,x_3,x_4)\in \mathbb{R}^4:(x_1^2+x_2^2-a^2)^2 + x_3^2 + x_4^2 -1=0;~ a>1\}$$ and $$S^2\times S^1=\{(x_1,x_2,x_3,x_4)\in \mathbb{R}^4:(x_1^2+x_2^2+x_3^2-b^2)^2+x_4^2-1=0;~ b>1\}$$ embedded in $\mathbb{R}^4$. We study polynomial vector fields in $\mathbb{R}^4$ separately, having $S^1\times S^2$ and $S^2\times S^1$ invariant by their flows. We characterize all linear, quadratic, cubic Kolmogorov and homogeneous vector fields on $S^1\times S^2$ and $S^2\times S^1$. We construct some first integrals of these vector fields and find which of the vector fields are Hamiltonian. We give upper bounds for the number of the invariant meridian and parallel hyperplanes of these vector fields. In addition, we have shown that the upper bounds are sharp in many cases.