Generalization of Newton's minimal resistance problem to Riemannian surfaces

We extend Newton's problem of minimal resistance to Riemannian surfaces endowed with a geodesic coordinate system, which includes the two-dimensional space forms such as the sphere and the hyperbolic plane. Assuming that the fluid particles flow along radial geodesics, we derive the resistance functional and prove that its smooth extremals are the loxodromes of the surface. Furthermore, we analyze the constrained minimization problem, establishing the absence of strong local minima for smooth extremals, and characterizing their global minimizers.