I've seen the following defined as the "volume density function" of a Riemannian manifold: $$ \theta_p(q) = \sqrt{ |\det g(\partial_i|_q,\partial_j|_q)| } $$ where we are in a normal coordinate chart around $p$.
I know that $v(x)=\sqrt{|\det g(x)|}$ intuitively measures the local spatial scaling of the manifold wrt Euclidean space. It would seem that this is an invariant scalar field on the manifold. It's not clear to me what it has to do with $\theta_p(q)$ or why the coordinates matter here. In other words, I'm not sure why $p$ should matter; i.e. how it's different from $v(q)$. I'm sure something is terribly wrong with my understanding (especially of smooth manifold notation).
Questions:
- What is the intuitive meaning of this function?
- Apparently it's only defined when $d(p,q)$ is less than the injectivity radius. Why?
- How to explicitly compute this in a simple case (e.g. for a 2D surface like $z(x,y)$)?