  Although Regge finite element functions are not continuous, useful
generalizations of nonlinear derivatives like the curvature, can be defined
using them. This paper is devoted to studying the convergence of the finite
element lifting of a generalized (distributional) Gauss curvature defined using
a metric tensor approximation in the Regge finite element space. Specifically,
we investigate the interplay between the polynomial degree of the curvature
lifting by Lagrange elements and the degree of the metric tensor in the Regge
finite element space. Previously, a superconvergence result, where convergence
rate of one order higher than expected, was obtained when the approximate
metric is the canonical Regge interpolant of the exact metric. In this work, we
show that an even higher order can be obtained if the degree of the curvature
lifting is reduced by one polynomial degree and if at least linear Regge
elements are used. These improved convergence rates are confirmed by numerical
examples.
