  The metric tensor of a Riemannian manifold can be approximated using Regge
finite elements and such approximations can be used to compute approximations
to the Gauss curvature and the Levi-Civita connection of the manifold. It is
shown that certain Regge approximations yield curvature and connection
approximations that converge at a higher rate than previously known. The
analysis is based on covariant (distributional) curl and incompatibility
operators which can be applied to piecewise smooth matrix fields whose
tangential-tangential component is continuous across element interfaces. Using
the properties of the canonical interpolant of the Regge space, we obtain
superconvergence of approximations of these covariant operators. Numerical
experiments further illustrate the results from the error analysis.
