There seems to be something curious about the so-called Van-Vleck-Morette determinant, as I cannot find any source that properly defines it in terms of expressions previously defined in that source and that actually proves properties.

To my understanding, a possible definition is $$ \triangle(p, q) = (\det(g_{ij}(x(p)))\det(g_{ij}(y(q)))^{-1/2} \det \left(- \frac{\partial^2\sigma}{\partial x^i \partial x^j}\right),$$ where $\sigma = d(p, q)^2/2$ and the expression means the following: Choose charts $x$ and $y$ such that $p$ and $q$ are in the respective domains of these charts, then we denote by $g_{ij}(x)$ is the matrix of the metric in the chart $x$ at the point $p$ and $g_{ij}(y)$ the corresponding matrix with respect to $y$ at $q$. One can check easily be the usual transformation laws under coordinate changes that this expression is well-defined.

Define on the other hand $$ \mu(p, q) = \det_g (d \exp_p|_X),$$ where $X$ is such that $\exp_p(X) = q$. Now if $x$ is a geodesic chart about $p$, then $\mu(p, q) = \det(g_{ij}(x))^{1/2}$.

Now my question is: These two functions should be related, if one trusts the literature, if I am not mistaken, $\triangle = \mu^{-1}$. However, I cannot find a proof anywhere and I couldn't figure out a proof myself. This should be a straightforward calculation of Riemannian geometry; however, the definition of $\triangle$ might be wrong. Does anybody know how to show this?