  We introduce $q$-deformed connections on the quantum 2-sphere and 3-sphere,
satisfying a twisted Leibniz rule in analogy with $q$-deformed derivations. We
show that such connections always exist on projective modules. Furthermore, a
condition for metric compatibility is introduced, and an explicit formula is
given, parametrizing all metric connections on a free module. On the quantum
3-sphere, a q-deformed torsion freeness condition is introduced and we derive
explicit expressions for the Christoffel symbols of a Levi-Civita connection
for a general class of metrics. We also give metric connections on a class of
projective modules over the quantum 2-sphere. Finally, we outline a
generalization to any Hopf algebra with a (left) covariant calculus and
associated quantum tangent space.
