This question is about how the principal part (or symbol) is defined on a manifold?-I assume that the answer is: As in $\mathbb{R}^n$ using local coordinates, i.e.

A differential operator $P=\sum_{|\alpha|\le m} a_{\alpha}(x)D^{\alpha}$ has by definition the symbol $P_m(x,\xi)=\sum_{|\alpha|=m} a_{\alpha}(x)\xi^{\alpha}.$

Now, I would like to understand this in the case of $\Delta_{\mathbb{S}^2}$ on the two-sphere. Recall that $$\Delta_{\mathbb{S}^2}=\left(\frac{\partial^{2}}{\partial\vartheta^{2}}+\frac{\cos\vartheta}{\sin\vartheta}\frac{\partial}{\partial\vartheta}+\frac{1}{\sin^{2}\vartheta}\frac{\partial^{2}}{\partial\varphi^{2}}\right)$$

so the first obvious thing would be to consider

$$\left(\xi_1^2+\frac{1}{\sin^{2}x_1}\xi_2^2\right)?$$ The problem seems to be that spherical coordinates do not form a chart, so troubles are at the poles. Could anybody here elaborate on that, please?

Edit: Since somebody mentioned spectral theory in the comments before, the symbol I am talking about here is the one used to define the characteristic set on the cotangent bundle.