I read about Laplace Operator here. As given in the link, given the metric, we can find the expression for Laplace operator. I am curious to know whether it is always possible to reduce the given variable coefficient Laplacian to the standard form by choosing the metric appropriately.
For example, given the variable coefficient Laplacian in standard Euclidean metric \begin{equation} L=\sum_{i=1}^{n}a_i(x)\partial_{x_i}^2 \end{equation} where $a_i(x)>\delta>0, \quad \forall x \in \mathbb{R}^n$. Let \begin{equation} L_g=\sum_{i=1}^{n}\partial_{y_i}^2 \end{equation} Is there a metric $g$ for which $L$ has the form $L_g$?