Before revision:  We show our methods have high-order convergence rates for the metric and other geometric quantities of the manifold , for the truncation errors of exterior calculus operators , and for the solution errors of the Stokes problem for hydrodynamic flows on curved surfaces.
After revision:  We show our methods have high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold , for the truncation errors of exterior calculus operators , and for the solution errors of the Stokes problem for hydrodynamic flows on curved surfaces.