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

TL;DR

This paper classifies polynomial vector fields on the manifolds $S^1\times S^2$ and $S^2\times S^1$, identifying their integrals, Hamiltonian properties, and invariant hyperplanes, with sharp bounds in many cases.

Contribution

It provides a complete characterization of linear, quadratic, cubic Kolmogorov, and homogeneous vector fields on the specified manifolds, including their integrals and invariant structures.

Findings

Characterized all relevant polynomial vector fields on the manifolds.

Constructed first integrals and identified Hamiltonian vector fields.

Established sharp upper bounds for invariant hyperplanes.

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.