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Associated with an equivariant noncommutative principal bundle we give anAtiyah sequence of braided derivations whose splittings give connections on thebundle. Vertical braided derivations act as infinitesimal gauge transformationson connections. For the $SU(2)$-principal bundle over the sphere $S^{4}_\theta$an equivariant splitting of the Atiyah sequence recovers the instantonconnection. An infinitesimal action of the braided conformal Lie algebra$so_\theta(5,1)$ yields a five parameter family of splittings. On the principal$SO_\theta(2n,\mathbb{R})$-bundle of orthonormal frames over the sphere$S^{2n}_\theta$, the splitting of the sequence leads to the Levi-Civitaconnection for the `round' metric on the $S^{2n}_\theta$. The correspondingRiemannian geometry of $S^{2n}_\theta$ is worked out.

Atiyah sequences of braided Lie algebras and their splittings