Umbilical routes along geodesics and hypercycles in the hyperbolic space

Given a geodesic line $\gamma$ the hyperbolic space $\mathbb H^n$ we formulate a necessary and sufficient condition for a function along this geodesic which measure the mean curvature of totally umbilical leaves of a foliation orthogonal to $\gamma$. Then we extend the result to $\gamma$ being a hypercycle i.e. a geodesic on a hypersurface equidistant from the totally geodesic one.