## Jacobian between $TM$ and $M\times M$

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