The linearization of the boundary rigidity problem for MP-systems and generic local boundary rigidity

We consider an $\mathcal{M}\mathcal{P}$-system, that is, a compact Riemannian manifold with boundary, endowed with a magnetic field and a potential. On simple $\mathcal{M}\mathcal{P}$-systems, we study the $\mathcal{M}\mathcal{P}$-ray transform in order to obtain new boundary rigidity results for $\mathcal{M}\mathcal{P}$-systems. We show that there is an explicit relation between the $\mathcal{M}\mathcal{P}$-ray transform and the magnetic one, which allow us to apply results from [DPSU07] to our case. Regarding rigidity, we show that there exists a generic set ${\mathcal{G}}^m$ of simple $\mathcal{M}\mathcal{P}$-systems, which is open and dense, such that any two $\mathcal{M}\mathcal{P}$-systems close to an element in it and having the same boundary action function, must be $k$-gauge equivalent.