To an equivariant noncommutative principal bundle one associates an Atiyah sequence of braided derivations whose splittings give connections on the bundle. There is an explicit action of vertical braided derivations as infinitesimal gauge transformations on connections. From the sequence one derives a Chern—Weil homomorphism and braided Chern—Simons terms.

On the principal bundle of orthonormal frames over the quantum sphere S^{2n}_\theta, the splitting of the sequence leads to a Levi-Civita connection on the corresponding module of braided derivations. The connection is torsion free and compatible with the `round' metric. We work out the corresponding Riemannian geometry.