of the solution in the next Section. We consider Riemannian manifolds where any two points can be connected by a minimal geodesic. For our purposes it is sufficient to consider manifolds which are geodesically complete. Recall that a Riemannian manifold $M$ is geodesically complete if for all $p\in M$ the exponential map $\exp_p: T_pM\rightarrow M$ is defined globally on $T_pM$. Here, $T_pM$ denotes the tangential space of the manifold $M$ at $p\in M$. The Hopf-Rinow theorem provides conditions for Riemannian manifolds to be geodesically complete. Especially we have

For a Riemannian manifold $M$ the following statements are equivalent:

-   $M$ is complete as a metric space.

-   The closed and bounded sets of $M$ are compact.

-   $M$ is geodesically complete.

Each of these equivalent statements implies that geodesics are curves of shortest length. Moreover, if $M$ is geodesically complete, then any two points of $M$ can be joined by a minimal geodesic.

The connection between the arclength and equation can be established as follows. First equations for minimal geodesics are obtained from variation of the length functional. Second Hamilton-Jacobi calculus shows that the length functional satisfies the eikonal equation . Since this is known we only sketch the main 