  We construct a differential calculus over the noncommutative 4-sphere in the
framework of pseudo-Riemannian calculi, and show that for every metric in a
conformal class of perturbations of the round metric, there exists a unique
metric and torsion-free connection. Furthermore, we find a localization of the
projective module corresponding to the space of vector fields, which allows us
to formulate a Gauss-Chern-Bonnet type theorem for the noncommutative 4-sphere.
