@PierrePC : Thank you for the reference to subriemannian manifolds: I haven't seen them before. I am ultimately looking not just into these "generalized Euclidean spaces", but also into non-expanding linear maps between them. The ultimate motivation is Arakelov theory, so I also have lattices inside these spaces.

Thank you. Yes, this is indeed my convention. I realized that I actually need an extra condition: all 5 points with ramification index 3 being mapped to the same point in P^1. My motivation: a possible counterexample to the Jacobian Conjecture.