Rotations with Constant Curl are Constant

TL;DR

This paper generalizes a classical rigidity result to matrix fields with constant curl, showing such fields must be constant, and explores regularity of measurable rotation fields based on their distributional curl.

Contribution

It extends classical rigidity results to matrix fields with non-zero curl and establishes regularity properties of measurable rotation fields based on their distributional curl.

Findings

Matrix fields with constant curl in SO(3) are necessarily constant.

Measurable rotation fields have regularity determined by their distributional curl.

Smoothness of curl implies smoothness of the rotation field.

Abstract

It is a classical result that if $u \in C^2(\mathbb{R}^n;\mathbb{R}^n)$ and $\nabla u \in SO(n)$ it follows that $u$ is rigid. In this article this result is generalized to matrix fields with non-vanishing curl. It is shown that every matrix field $R\in C^2(\Omega \subseteq \mathbb{R}^3;SO(3))$ such that $\operatorname{curl } R = constant$ is necessarily constant. Moreover, it is proved in arbitrary dimensions that a measurable rotation field is as regular as its distributional curl allows. In particular, a measurable matrix field $R: \Omega \to SO(n)$, whose curl in the sense of distributions is smooth, is also smooth.