This paper presents a novel on-line path planning method that enables aerial robots to interact with surfaces. We present a solution to the problem of finding trajectories that drive a robot towards a surface and move along it. Triangular meshes are used as a surface map representation that is free of fixed discretization and allows for very large workspaces. We propose to leverage planar parametrization methods to obtain a lower-dimensional topologically equivalent representation of the original surface. Furthermore, we interpret the original surface and its lower-dimensional representation as manifold approximations that allow the use of Riemannian Motion Policies (RMPs), resulting in an efficient, versatile, and elegant motion generation framework. We compare against several Rapidly-exploring Random Tree (RRT) planners, a customized CHOMP variant, and the discrete geodesic algorithm. Using extensive simulations on real-world data we show that the proposed planner can reliably plan high-quality near-optimal trajectories at minimal computational cost. The accompanying multimedia attachment demonstrates feasibility on a real OMAV. The obtained paths show less than 10% deviation from the theoretical optimum while facilitating reactive re-planning at kHz refresh rates, enabling flying robots to perform motion planning for interaction with complex surfaces.