Loxodromes on invariant surface in three-manifolds

In this paper, we prove important results concerning the loxodromes on an invariant surface in a three-dimensional Riemannian manifold, some of which generalize classical results about loxodromes on rotational surfaces in ${\mathbb{R}}^3$. In particular, we show how to parametrize a loxodrome on an invariant surface of ${\mathbb{H}}^2\times \mathbb{R}$ and ${\mathbb{H}}_3$ and we exhibit the loxodromes of some remarkable minimal invariant surfaces of these spaces. In addition, we give an explicit description of the loxodromes on an invariant surface with constant Gauss curvature.