as the second order terms of local representations around $y$. Having obtained this the representation is just a multivariate version of Taylor’s theorem. Note, however, that we do not need to invoke the uniqueness of theorem 2.3. but just consider a recursion obtained from a power series ansatz starting with second order terms. However, this would complicate the matter a bit so we take advantage that we know the second order terms of a local representation by the preceding argument. Finally we have to establish the recursion in . The recursion shows directly that the higher order coefficients $d^2_{\beta}(y)$ for $|\beta|\geq 3$ are uniquely determined. Moreover, it is clear from that in general the convergence radius of the full power series is small (if not zero). Hence in general there is no globally analytic solution the function $d^ 2:\Omega\times \Omega \rightarrow {\mathbb R}$ globally analytic if for each $y\in \mathbb{R}^n$ the Taylor expansion of $d^2$ at $y\in \mathbb{R}^n$ and $x\in \mathbb{R}^n$ equals $d^2$ globally, i.e. $$\label{geolgth}
d^ 2(x,y)=\sum_{\alpha}\frac{\partial_{\alpha}d^2(y)}{\alpha !}(x-y)^ {\alpha}~~\mbox{forall}~x,y\in \mathbb{R}^n.$$ Invoking the implicit function theorem equation is equivalent to $$\label{eik2}
d^2=\frac{1}{4}\sum_{i}\lambda_{i}(x)d^2_{x_i}d^2_{x_i},$$ where $\lambda_i(x), 1\leq i\leq n$ is the spectrum of the positive $(a_{ij}(x))$. Since $d^2_{x_i}=2d d_{x_i}$ this is equivalent 