A Riemannian version of Korn's inequality

We prove a Korn type inequality for vector fields on a Riemann manifold. This inequality includes the special cases proved in the literature for domains in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^3$\end{document}. If the domain is convex, we can considerably weaken the needed assumption on the boundary values.