You can use the general local formula for the Laplace-Beltrami operator in terms of any local orthonormal frame:

$$\Delta = \sum_{i=1}^n X_i^2 +\mathrm{div}(X_i)X_i$$

where the $X_i$'s are seen as derivations on functions.

Thus, there are no local obstructions. You can always find a local frame of divergence-free vector fields $X_1,\ldots,X_n$, and in terms of this frame the Laplacian is just a "sum of squares".