  This paper investigates gradient recovery schemes for data defined on
discretized manifolds. The proposed method, parametric polynomial preserving
recovery (PPPR), does not require the tangent spaces of the exact manifolds,
and they have been assumed for some significant gradient recovery methods in
the literature. Another advantage of PPPR is that superconvergence is
guaranteed without the symmetric condition which has been asked in the existing
techniques. There is also numerical evidence that the superconvergence by PPPR
is high curvature stable, which distinguishes itself from the others. As an
application, we show its capability of constructing an asymptotically exact
\textit{a posteriori} error estimator. Several numerical examples on
two-dimensional surfaces are presented to support the theoretical results and
comparisons with existing methods are documented.
