Rotations with Constant Curl are Constant

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.