# Beltrami fields with nonconstant proportionality factor

## TL;DR

This paper investigates which nonconstant functions can serve as proportionality factors for Beltrami fields in three-dimensional space, analyzing the influence of the geometry of level surfaces and classifying fields with symmetries.

## Contribution

It advances the understanding of Beltrami fields with nonconstant proportionality factors using Cartan's method and classifies fields with specific symmetries.

## Key findings

- Characterization of possible nonconstant proportionality functions
- Dependence of Beltrami field spaces on level surface geometry
- Complete classification of symmetric Beltrami fields

## Abstract

We consider the question raised by Enciso and Peralta-Salas in [4] (see arXiv:1402.6825): What nonconstant functions $f$ can occur as the proportionality factor for a Beltrami field $\mathbf{u}$ on an open subset $U \subset \mathbb{R}^3$? We also consider the related question: For any such $f$, how large is the space of associated Beltrami fields? By applying Cartan's method of moving frames and the theory of exterior differential systems, we are able to improve upon the results given in [4]. In particular, the answer to the second question depends crucially upon the geometry of the level surfaces of $f$. We conclude by giving a complete classification of Beltrami fields that possess either a translation symmetry or a rotation symmetry.