Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Invertibility of local geodesic transverse and mixed ray transforms I: basic cases

Gunther Uhlmann, Jian Zhai

Abstract

Consider a compact Riemannian manifold in dimension $n\geq 3$ with strictly convex boundary. We show that the transverse ray transform of $1$ tensors and the mixed ray transform of $1+1$ tensors are invertible, up to natural obstructions, near a boundary point. When the manifold admits a strictly convex function, this local invertibility result leads to a global result by a layer stripping argument.

Invertibility of local geodesic transverse and mixed ray transforms I: basic cases

Abstract

Consider a compact Riemannian manifold in dimension with strictly convex boundary. We show that the transverse ray transform of tensors and the mixed ray transform of tensors are invertible, up to natural obstructions, near a boundary point. When the manifold admits a strictly convex function, this local invertibility result leads to a global result by a layer stripping argument.
Paper Structure (8 sections, 14 theorems, 250 equations)

This paper contains 8 sections, 14 theorems, 250 equations.

Table of Contents

  1. Introduction
  2. Setting up within the scattering algebra
  3. Redefinition of the mixed ray transforms
  4. Geodesics near the lifted diagonal
  5. Transverse ray transform of $1+0$ tensors
  6. Mixed ray transform of $1+1$ tensors
  7. Ellipticity
  8. The gauge condition and the proof of the main results

Key Result

Lemma 1

The operator $N_\mathsf{F}$ is elliptic at fiber infinity of $^{sc}T^*X$ in $\widetilde{M}$.

Theorems & Definitions (25)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • ...and 15 more