Let $(M,g)$ be a compact Riemannian Manifold. Where possible, define the following function
$$\Theta(x,y) := |\text{det}_g \ D_{\exp_x^{-1}(y)} \exp_x|$$
What is the meaning of this function and in particular $\text{det}_g$? I think the author says that it is the Riemannian volume density at $x$ in normal coordinates at $y$ but I'm not sure why.