Let $M$ be a closed riemannian manifold and $\phi: \begin{array}{ccc} TM&\to &M\times M \\ (x,v)&\mapsto & (\exp_x(v),\exp_x(-v)) \end{array}$
I need an asymptotic expression for the jacobian of $\phi^{-1}$ as $\|v\|\to 0$, but I'm unable to compute it ... I guess the order $0$ term is $2$, but even this I'm unable to compute correctly... and I'm also interested in the quadratic term (the one of order $\|v\|^2$), wish I guess depends on curvatures at $x$. I guess it's someway linked to Jacobi field... If that simplify thing, the manifold $M$ has dimension $3$ in my problem.
I'm interested as much in the result itsef as in the way to obtain (and understand) it.
Thanks a lot, have a good day