Quadratic, Homogeneous and Kolmogorov vector fields on $S^1\times S^2$ and $S^2 \times S^1$
Supriyo Jana, Soumen Sarkar
TL;DR
The work classifies polynomial vector fields in $\mathbb{R}^4$ that preserve the 3D manifolds $S^1\times S^2$ and $S^2\times S^1$, deriving explicit forms for linear, quadratic, and cubic Kolmogorov and homogeneous fields on each surface. Using extactic polynomials and Darboux theory, it shows nonexistence of Lotka–Volterra and Hamiltonian cases in key degrees, while proving that all such fields possess rational first integrals, and Type-$n$ fields yield two independent integrals given by $x_1^2+x_2^2$ and $x_3^2+x_4^2$ on $S^1\times S^2$ (with analogous structures on $S^2\times S^1$). The paper also establishes sharp bounds on the number of invariant meridian and parallel hyperplanes via extactic polynomials and provides explicit extremal constructions. An application demonstrates that there is no polynomial diffeomorphism with linear components between the two hypersurfaces, highlighting intrinsic differences in their polynomial dynamics.
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.