This is an exercise in AC da Silva's Lectures onn Symplectic Geometry; I am having trouble showing the following.
$(X,g)$ is a geodesically complete manifold, and $f: X \times X \to \mathbb{R}$ is given by $f(x,y) = - \frac{1}{2} d(x,y)^2$ where $d$ is the Riemannian distance function (i.e. the infimum of arc-lengths of piecewise smooth curves joining $x$ to $y$).
I need to show that under the identification of $TX$ and $T^*X$ by the metric $g$, the symplectomorphism generated by $f$ coincides with the map $TX \to TX$ given by $(x,v) \to (\exp(x,v)(1), \frac{d}{dt} \exp(x,v) (1))$.
This amounts to solving the equations $g_x(v, \cdot ) = d_x f (\cdot )$ and $g_y (w, \cdot) = -d_y f(\cdot)$ for $(y, w)$ with fixed $(x,v)$, where $d_x f, d_y f$ are components of $df_{(x,y)}$ where $T^*_{(x,y)} (X \times X) \simeq T_x^* X \times T_y ^* X$.
I am having trouble in trying to solve the first equation, $g_x (v, \cdot) = d_x f(\cdot)$ for $y$, where both are members of $T_x^* X$. (The answer should be $y = \exp(x,v)=1$.) In particular, when $g$ (and hence $d$) is not concretely given, how should I go about finding $df$ at a fixed point $(x,y)$? Thank you for any help.