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

Sebastián Muñoz-Thon

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.

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

Abstract

We consider an -system, that is, a compact Riemannian manifold with boundary, endowed with a magnetic field and a potential. On simple -systems, we study the -ray transform in order to obtain new boundary rigidity results for -systems. We show that there is an explicit relation between the -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 of simple -systems, which is open and dense, such that any two -systems close to an element in it and having the same boundary action function, must be -gauge equivalent.
Paper Structure (17 sections, 18 theorems, 99 equations)

This paper contains 17 sections, 18 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Previous results
  3. Structure of the paper
  4. Review of MP-systems
  5. The MP-ray transform
  6. Definitions
  7. Linearization
  8. Relation with the magnetic ray transform
  9. Potential triples
  10. The definition of potential for the MP-ray transform
  11. Relation between MP-potentials and magnetic ones
  12. s-injectivity of the ray transform for analytic systems
  13. The linear problem
  14. Generic results
  15. Generic s-injectivity
  16. ...and 2 more sections

Key Result

Lemma 2.3

Let $(g,\alpha,U)$ be $\mathcal{MP}$-system with energy $k$ an let $(G,\alpha):=(2(k-U)g,\alpha)$ be its reduction to a magnetic system of energy $\frac{1}{2}$.

Theorems & Definitions (46)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: az*Proposition 1, 2, 3
  • Definition 2.4
  • Lemma 2.5: mt23*Lemma 4.3
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 36 more