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

Abstract We consider an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">MP</mml:mi> </mml:mrow> </mml:mrow> </mml:math> -system, that is, a compact Riemannian manifold with boundary, endowed with a magnetic field and a potential. On simple <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">MP</mml:mi> </mml:mrow> </mml:mrow> </mml:math> -systems, we study the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">MP</mml:mi> </mml:mrow> </mml:mrow> </mml:math> -ray transform in order to obtain new boundary rigidity results for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">MP</mml:mi> </mml:mrow> </mml:mrow> </mml:math> -systems. We show that there is an explicit relation between the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">MP</mml:mi> </mml:mrow> </mml:mrow> </mml:math> -ray transform and the magnetic one, which allows us to apply results from Dairbekov et al (2007 Adv. Math. 216 535–609) to our case. Regarding rigidity, we show that there exists a generic set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">G</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> of simple <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">MP</mml:mi> </mml:mrow> </mml:mrow> </mml:math> -systems, which is open and dense, such that any two <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">MP</mml:mi> </mml:mrow> </mml:mrow> </mml:math> -systems close to an element in it and having the same boundary action function, must be k -gauge equivalent.