# Global solution of second order ODE defined on riemannian manifold

Consider the differential equation $$\nabla \dot X + \frac{3}{t} \dot X + gradf(X) =0$$, defined on a riemannian manifold $$(M,g)$$ ($$\nabla$$ is the Levi-Civita connection and $$gradf(X)$$ is the riemannian gradient). Suppose that our manifold is geodesically convex in the sense that $$\exp$$ and $$\exp^{-1}$$ are well defined for the whole manifold and not just locally. I would like to prove that the equation has a unique global solution under some suitable assumptions, given an initial condition for $$X$$ and $$\dot X$$ at $$0$$. The procedure I have in mind is firstly modifying the initial equation making it $$\nabla \dot X+\frac{3}{\max(\delta,t)} \dot X + gradf(X)=0$$, in order to use Arzela-Ascoli theorem at the end, taking $$\delta \rightarrow 0$$. Writing each of the modified ODEs in local coordinates, we get the system $$\ddot c^k + \sum_{i,j=1}^m \Gamma_{ij}^k(c)\dot c^i\dot c^j+\frac{3}{\max(\delta,t)} \dot c^k+ \sum_{i=1}^m g^{ik}\frac{\partial(f o \psi)}{\partial x^i}(c)=0$$, for $$k=1,...,m$$, where $$\psi=\phi^{-1}$$ is a parametrization and $$c=\phi o X$$. By Picard-Lindelof theorem this equation has a local solution around $$0$$, but I need a global one. Suppose that everything in the equation in local coordinates is Lipschitz continuous. Intuitively there is no global solution in general, because geodesics, which are defined by a similar equation, are not defined globally in general. Do you have any idea about an extra assumtion on the manifold $$M$$ which could provide a global solution? I believe that completeness could work, but I do not know how.