Suppose functions $a_{ij}: \mathbb{R}^{n}\to\mathbb{R}$ are given. The task is to find a diffeomorphism $y:U(x^{0})\to \mathbb{R}^{n}$ such that $$\displaystyle \sum\limits_{i,j=1}^{n}a_{ij}(x(y))\dfrac{\partial y_{k}}{\partial x_{i}}\dfrac{\partial y_{l}}{\partial x_{j}} = 0$$ for every $y\in V(y^{0})$ -- some neighborhood of $y^{0}$ (in the standart topology of $\mathbb{R}^{n}$). The function $x(y)$ is the inverse of $y$. Let's also suppose that $a_{ij}(x)\equiv a_{ji}(x)$
The desired diffeomorphism is $y = (y_{1},\ldots,y_{n})$, so we want to find $n$ functions, however they must satisfy $(n^2-n)/2$ equiations. So our system is overdetermined. Why such system does not have solutions (provided $n>3$) in general? One could just construct $a_{ij}$ and work out the example, but is there more straightforward approach to show that such system is inconsistent in general?