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Invertibility of local geodesic transverse and mixed ray transforms II: higher order tensors

Gunther Uhlmann, Jian Zhai

Abstract

Consider a compact Riemannian manifold in dimension $n$ with strictly convex boundary. We show the local invertibility near a boundary point of the transverse ray transform of $2$ tensors for $n\geq 3$ and the mixed ray transform of $2+2$ tensors for $n=3$. When the manifold admits a strictly convex function, this local invertibility result leads to global invertibility.

Invertibility of local geodesic transverse and mixed ray transforms II: higher order tensors

Abstract

Consider a compact Riemannian manifold in dimension with strictly convex boundary. We show the local invertibility near a boundary point of the transverse ray transform of tensors for and the mixed ray transform of tensors for . When the manifold admits a strictly convex function, this local invertibility result leads to global invertibility.
Paper Structure (5 sections, 9 theorems, 255 equations)

This paper contains 5 sections, 9 theorems, 255 equations.

Table of Contents

  1. Introduction
  2. Transverse ray transform of $2+0$ tensors
  3. Mixed ray transform of $2+2$ tensors
  4. Ellipticity
  5. The gauge condition and the proof of the main results

Key Result

Theorem 1

Assume $\partial M$ is strictly convex at $p\in\partial M$. There exists a function $\tilde{x}\in C^\infty(\widetilde{M})$ with $O_p=\{\tilde{x}>-c\}\cap M$ for sufficiently small $c>0$, such that a symmetric $(2,0)$-tensor $f$ can be uniquely determined by $T_2f$ restricted to $O_p$-local geodesics

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 1
  • proof
  • Remark 1
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 4 more