Asymptotic linking of volume-preserving actions of ${\mathbb R}^k$

TL;DR

This paper generalizes Arnold's asymptotic linking theory from flows to actions of higher-dimensional Euclidean groups, extending linking concepts and the Biot-Savart formula to higher dimensions and more complex actions.

Contribution

It introduces a framework for asymptotic linking of volume-preserving actions of ${f R}^k$ and ${f R}^ ext{ell}$, extending classical linking theory to higher dimensions and actions.

Findings

Extended Arnold's linking theory to ${f R}^k$ actions.

Generalized the Biot-Savart formula to higher dimensions.

Linked volume-preserving actions with submanifolds in ${f R}^n$.

Abstract

We extend V. Arnold's theory of asymptotic linking for two volume preserving flows on a domain in ${\mathbb R}^3$ and $S^3$ to volume preserving actions of ${\mathbb R}^k$ and ${\mathbb R}^\ell$ on certain domains in ${\mathbb R}^n$ and also to linking of a volume preserving action of ${\mathbb R}^k$ with a closed oriented singular $\ell$-dimensional submanifold in ${\mathbb R}^n$, where $n=k+\ell+1$. We also extend the Biot-Savart formula to higher dimensions.