First variation of flat traces on negatively curved surfaces
For a closed negatively curved surface $(X,g)$ the flat trace of the geodesic Koopman operators $V_g^\tau f=f\circ G_g^\tau$ is the periodic orbit distribution \[ \mathrm{Tr}^{\flat} V_{g}(\tau)=\sum_{\gamma}\frac{L_\gamma^{\#}}{\lvert\det(I-P_\gamma)\rvert}\,\delta(\tau-L_\gamma), \qquad \tau>0, \] supported on the length spectrum and weighted by the linearized Poincar\'e maps $P_\gamma$. For a smooth family of negatively curved metrics $g_t$ we compute the first variation $\partial_t\vert_{0}\,\mathrm{Tr}^{\flat} V_{g_t}$ as a distribution. At an isolated length $\ell$ the leading singularity is a multiple of $\delta'(\tau-\ell)$, and its coefficient is an explicit linear functional of the length variations $\dot L_{\gamma^m}$ of the closed geodesics with $L_{\gamma^m}=\ell$. This transport coefficient forces the marked lengths to be locally constant along any deformation with constant flat trace. As an application, if $\mathrm{Tr}^{\flat} V_{g_t}=\mathrm{Tr}^{\flat} V_{g_0}$ for all $t$ then $g_t$ is isometric to $g_0$ for all $t$. Together with Sunada-type constructions of non isometric pairs with equal flat traces, this shows that the flat trace is globally non-unique yet locally complete along smooth families.