Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure

We describe in detail a construction of weakly mixing $C^\infty$ diffeomorphisms preserving a smooth measure and a measurable Riemannian metric as well as ${\mathbb} Z^k$ actions with similar properties. We construct those as a perturbation of elements of a nontrivial non-transitive circle action. Our construction works on all compact manifolds admitting a nontrivial circle action. It is shown in the appendix that a Riemannian metric preserved by a weakly mixing diffeomorphism can not be square integrable.