Generalized Riemann minimal surfaces examples in three-dimensional manifolds products

In this paper, we construct and classify minimal surfaces foliated by horizontal constant curvature curves in product manifolds $M\times \mathbb{R}$, where $M$ is the hyperbolic plane, the Euclidean plane or the two dimensional sphere. The main tool is the existence of a Jacobi field which characterize the property to be foliated in circles and geodesics in these product manifolds. It is related to harmonic maps.