Diffeomorphism Group Actions on Para-Kähler Structures: Lifts and Induced Dynamics on 3-Webs

Let $M$ be a smooth manifold equipped with a para-Kähler structure. We define an action of the diffeomorphism group $\mathrm{Diff}(M)$ on the space of para-Kähler structures and prove that this action preserves the associated affine structures. Although this result is well known, we provide a proof based on this action, showing in particular that an isometry commutes with the Levi–Civita connection. We then construct several natural liftings of this action to the tangent bundle $TM$, the cotangent bundle $T^{\ast }M$, and the Whitney sum $E=TM\oplus T^{\ast }M$. In certain cases, these lifted actions induce natural dynamical systems on a distinguished class of Lagrangian $3$-webs.