On a Riemannian manifold which is globally diffeomorphic to $\Bbb R^n$, can we write the Laplace-Beltrami in the prescribed form on the whole space?

Question

I have this one confusion.

Let $G$ be a Riemannian manifold which is globally diffeomorphic to $\Bbb R^n$ . Then can we write its Laplace-Beltrami $L$ in the 'usual elliptic form' i.e. $$\sum_{i,j=1}^n a_{ij}(x) \partial_i \partial_j + \sum_{j=1}^n b_i(x)\partial_i +c(x)$$ where the matrices $\{a_{ij}(x)\}_{1 \le i,j \le n}$ are symmetric positive-definite, and the coefficients vary continuously and satisfy the ellipticity condition?

I know that all the coefficients can be suitably arranged for each chart as it is coming from the Riemannian metric. My main question is whether the global diffeomorphism takes care of such an expression globally, i.e. not in bits and pieces but on the whole space ( just like in the situation of $\Bbb R^n$) ?

Please help me clear my confusions.