parabolic equation leads to the eikonal equation which is, hence, satisfied by $d^2$. Moreover we know by $V$ and the fact that the squareroot of $d^2$ is a metric. Hence $d(x,y)=0$ if and only if $x=y$, and the same holds for $d^2$. Hence, we conclude that the global solution $d^2$ of the system of equations , and is unique. Moreover, from the preceding theorem we can conclude that the function $(x,y)\rightarrow d^2(x,y)$ is also smooth with respect to both variables.

Further analysis of the equation for the squared metric $d^2$
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Next we observe that the local representation of the solution of the equations , with the boundary condition has a local representation which starts with the quadratic terms. This will be used in the construction of a global approximation. The analysis presented here gives us two other insights. First, a powere series ansatz leads atmost to local and not to global solutions. Even if there is a local power series representation of the solution at each point of the domain, we do not know how a global solution can be constructed from this information, because we do not know the location of the geodesic the length of which we 