Global Gronwall Estimates for Integral Curves on Riemannian Manifolds
  We prove Gronwall-type estimates for the distance of integral curves of smooth vector fields on a Riemannian manifold. Such estimates are of central importance for all methods of solving ODEs in a verified way, i.e., with full control of roundoff errors. Our results may therefore be seen as a prerequisite for the generalization of such methods to the setting of Riemannian manifolds. 
Michael Kunzinger, Hermann Schichl, Roland Steinbauer, James A.   Vickers