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Generic injectivity of the X-ray transform

Mihajlo Cekić, Thibault Lefeuvre

Abstract

In dimensions $\geq 3$, we prove that the X-ray transform of symmetric tensors of arbitrary degree is generically injective with respect to the metric on closed Anosov manifolds and on manifolds with spherical strictly convex boundary, no conjugate points and a hyperbolic trapped set. This has two immediate corollaries: local spectral rigidity, and local marked length spectrum rigidity (building on earlier work by Guillarmou, Knieper and the second author [arXiv:1806.04218], [arXiv:1909.08666]), in a neighbourhood of a generic Anosov metric. In both cases, this is the first work going beyond the negatively curved assumption or dimension $2$. Our method, initiated in [arXiv:2008.09191] and fully developed in the present paper, is based on a perturbative argument of the $0$-eigenvalue of elliptic operators via microlocal analysis which turn the analytic problem of injectivity into an algebraic problem of representation theory. When the manifold is equipped with a Hermitian vector bundle together with a unitary connection, we also show that the twisted X-ray transform of symmetric tensors (with values in that bundle) is generically injective with respect to the connection. This property turns out to be crucial when solving the $\textit{holonomy inverse problem}$, as studied in a subsequent article [arXiv:2105.06376].

Generic injectivity of the X-ray transform

Abstract

In dimensions , we prove that the X-ray transform of symmetric tensors of arbitrary degree is generically injective with respect to the metric on closed Anosov manifolds and on manifolds with spherical strictly convex boundary, no conjugate points and a hyperbolic trapped set. This has two immediate corollaries: local spectral rigidity, and local marked length spectrum rigidity (building on earlier work by Guillarmou, Knieper and the second author [arXiv:1806.04218], [arXiv:1909.08666]), in a neighbourhood of a generic Anosov metric. In both cases, this is the first work going beyond the negatively curved assumption or dimension . Our method, initiated in [arXiv:2008.09191] and fully developed in the present paper, is based on a perturbative argument of the -eigenvalue of elliptic operators via microlocal analysis which turn the analytic problem of injectivity into an algebraic problem of representation theory. When the manifold is equipped with a Hermitian vector bundle together with a unitary connection, we also show that the twisted X-ray transform of symmetric tensors (with values in that bundle) is generically injective with respect to the connection. This property turns out to be crucial when solving the , as studied in a subsequent article [arXiv:2105.06376].

Paper Structure

This paper contains 45 sections, 40 theorems, 281 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Generic injectivity of the X-ray transform with respect to the metric: closed case
  3. Application to rigidity problems
  4. The marked length spectrum rigidity conjecture
  5. Spectral rigidity
  6. Generic injectivity of the X-ray transform with respect to the connection
  7. Strategy of proof, organization of the paper
  8. Technical preliminaries
  9. Elementary Riemannian geometry
  10. Symmetric tensors
  11. Symmetric tensors in Euclidean space
  12. Symmetric tensors on Riemannian manifolds
  13. Twisted symmetric tensors
  14. Pollicott-Ruelle theory
  15. Meromorphic extension of the resolvents
  16. ...and 30 more sections

Key Result

Theorem 1.1

There exists an integer $k_0 \gg 1$ such that the following holds. Let $M$ be a smooth closed manifold of dimension $\geq 3$ carrying Anosov metrics. For all $m \in \mathbb{Z}_{\geq 0}$,For $m=0,1$, the s-injectivity is already established Dairbekov-Sharafutdinov-03. there exists an open dense set $

Figures (1)

  • Figure 1: The function $R$ is constant along the red orbits which correspond to the rotation around the $v_0$-axis.

Theorems & Definitions (82)

  • Theorem 1.1
  • Conjecture 1.2
  • Corollary 1.3: of Theorem \ref{['theorem:genericity-metric']} and Guillarmou-Lefeuvre-18Guillarmou-Knieper-Lefeuvre-19
  • Corollary 1.4: of Theorem \ref{['theorem:genericity-metric']}
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • ...and 72 more