Rotations with Constant $$\mathbf {{\text {curl }}}$$ are Constant

Abstract We address a problem that extends fundamental classical result of continuum mechanics from the time its inception, as well answers question in recent, modern nonlinear elastic theory dislocations. Interestingly, implication our latter case is qualitatively different well-established analog linear It if $$u\in C^2({\mathbb {R}}^n;{\mathbb {R}}^n)$$ u ∈ C 2 ( R n ; ) and $$\nabla u \in SO(n)$$ ∇ S O , it follows rigid. In this article generalized to matrix fields with non-vanishing $${\text {curl }}$$ curl . shown every field $$R\in C^2(\varOmega ;SO(3))$$ Ω 3 such }}R = constant$$ = c o s t a necessarily constant. Moreover, proved arbitrary dimensions measurable rotation regular distributional allows. particular, $$R: \varOmega \rightarrow : → whose sense distributions smooth, also smooth.