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 $\mathrm{curl}R=constant$ 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.

16 pages, 1 figure

References in corpus (3)

• Finite Element Approximation of Finite Deformation Dislocation Mechanics
• An Energy Estimate for Dislocation Configurations and the Emergence of Cosserat-Type Structures in Metal Plasticity
• Geometric Rigidity Estimates for Incompatible Fields in Dimension $\ge$ 3