$$\label{lf}
d^2=\frac{1}{4}\sum_{ij}d_{x_i}^2a_{ij}d_{x_j}^2,$$ $$\label{lg}
d^2=\frac{1}{4}\sum_{ij}d_{y_i}^2a_{ij}d_{y_j}^2$$ for all $x,y\in {\mathbb R}^n$ and with the boundary condition $$\label{lh}
d(x,y)=0 \mbox{ iff  $x=y$ for all $x,y\in {\mathbb R}^n$.}$$ Moreover, the squareroot $d$ is the Riemannian distance induced by $$\begin{array}{ll}
d(x,y):=\inf{\Bigg\{}\int_a^b&\sqrt{a^{ij}(\gamma)\stackrel{.}{\gamma}^i\stackrel{.}{\gamma}^j}dt|\gamma:[a,b]\rightarrow {\mathbb R}^n \mbox{ is piecewise } \\
&\mbox{ smooth with $\gamma(a)=x$ and $\gamma(b)=y$}{\Bigg \}}.
\end{array}$$ The function $d^2$ is a $C^{\infty}$-function with respect to both variables.

The variation of the length functional leads to the geodesic equation. On the other hand, Hamilton-Jacobi calculus leads us to the fact that the squared length functional $d^2$ satisfies the equation . It is clear that the squared length functional satisfies both equations and below. Moreover, it is clear that the squared length functional satisfies the initial condition . Uniqueness is a bit more subtle. In [@V] Varadhan showed that $$\label{V}
d^2(x,y)=\lim_{t\downarrow 0}2t\ln p(t,x,y),$$ where $p$ is the fundamental solution of a scalar parabolic equation with diffusion coefficient function $x\rightarrow a_{ij}(x)$. Since $p$ is unique for a strictly parabolic equation $d^2$ is uniquely determined by the equation . On the other hand one knows that for small $t>0$ $\ln p$ has for $C^{\infty}$ coefficients a representation of type is valid (cf. [@Gi; @Ka]). Plugging this into the correspondend 