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

Abstract

We consider an $\mathcal{MP}$-system, that is, a compact Riemannian manifold with boundary, endowed with a magnetic field and a potential. On simple $\mathcal{MP}$-systems, we study the $\mathcal{MP}$-ray transform in order to obtain new boundary rigidity results for $\mathcal{MP}$-systems. We show that there is an explicit relation between the $\mathcal{MP}$-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 $\mathcal{MP}$-systems, which is open and dense, such that any two $\mathcal{MP}$-systems close to an element in it and having the same boundary action function, must be $k$-gauge equivalent.