  In the planar setting the Rad\'o-Kneser-Choquet theorem states that a
harmonic map from the unit disk onto a Jordan domain bounded by a convex curve
is a diffeomorphism provided that the boundary mapping is a homeomorphism. We
prove the injectivity criterion of Rad\'o-Kneser-Choquet for $p$-harmonic
mappings between Riemannian surfaces.
  In our proof of the injecticity criterion we approximate the $p$-harmonic map
with auxiliary mappings that solve uniformly elliptic systems. We prove that
each auxiliary mapping has a positive Jacobian by a homotopy argument. We keep
the maps injective all the way through the homotopy with the help of the
minimum principle for a certain subharmonic expression that is related to the
Jacobian.
