  This article presents a novel resolution to the problem of spline
interpolation versus least-squares fitting on smooth Riemannian manifolds
utilizing the method of gradient flows of networks. This approach represents a
contribution to both geometric control theory and statistical shape data
analysis. Our work encompasses a rigorous proof for the existence of global
solutions in H\"{o}lder spaces for the gradient flow. The asymptotic limits of
these solutions establish the existence of the spline interpolation versus
least-squares fitting problem on smooth Riemannian manifolds, offering a
comprehensive solution. Notably, the constructive nature of the proof suggests
potential numerical schemes for finding solutions.
