I would like to better understand what it means for the leading symbol of a differential operator to be scalar.
Concretely, I am currently looking at the Laplace - Beltrami operator on an n-dimensional Riemannian manifold $(M,g)$.
Locally, \begin{equation} \triangle_g = \frac{1}{\sqrt{|g|}} \sum^n_{i,j = 1} \partial_i g^{ij} \sqrt{|g|} \partial_j \end{equation} where $|g|$ denotes the determinant and $(g^{ij})$ the inverse matrix to $(g_{ij})$.
Expanding the above expression we obtain \begin{equation} \triangle_g = \sum_{i,j = 1}^n g^{ij} \partial_i \partial_j + \sum_{i,j = 1}^n (g^{ij} \frac{\partial_i \sqrt{|g|}}{\sqrt{|g|}} + \partial_ig^{ij})\partial_j \end{equation}
and so the leading symbol is given by \begin{equation} p_2(x,\xi) = \sum_{i,j = 1}^n g^{ij} \xi_i \xi_j \end{equation}
(hope I am correct so far)
Now, there are two questions that I am trying to look up but cannot find an answer to that helps me fill in all the gaps that I currently have in my knowledge:
(1) what does it mean when it is pointed out that the leading symbol is scalar ? Does it mean that I have $g^{ij} = \lambda \delta_{ij}$ for some number $\lambda$ ?
(2) since $\triangle_g$ is elliptic, does this mean that the leading symbol is scalar ? I don't think that would be true, but can I diagonalize the matrix $g$ so that it the leading symbol becomes scalar ? But what does it mean for a non-constant matrix $g$ to be diagonalized ?
I hope the questions are not too confused, please let me know in case more clarification is necessary.
Is there a book that you'd recommend me looking at, given my above questions ?
Many thanks!
