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On mixed and transverse ray transforms on orientable surfaces

Joonas Ilmavirta, Keijo Mönkkönen, Jesse Railo

Abstract

The geodesic ray transform, the mixed ray transform and the transverse ray transform of a tensor field on a manifold can all be seen as what we call mixing ray transforms, compositions of the geodesic ray transform and an invertible linear map on tensor fields. We show that the characterization of the kernel and the stability of a mixing ray transform can be reduced to the same properties of any other mixing ray transform. Our approach applies to various geometries and ray transforms, including the light ray transform. In particular, we extend studies in de Hoop--Saksala--Zhai (2019) from compact simple surfaces to orientable surfaces with solenoidally injective geodesic ray transform. Our proofs are based on algebraic arguments.

On mixed and transverse ray transforms on orientable surfaces

Abstract

The geodesic ray transform, the mixed ray transform and the transverse ray transform of a tensor field on a manifold can all be seen as what we call mixing ray transforms, compositions of the geodesic ray transform and an invertible linear map on tensor fields. We show that the characterization of the kernel and the stability of a mixing ray transform can be reduced to the same properties of any other mixing ray transform. Our approach applies to various geometries and ray transforms, including the light ray transform. In particular, we extend studies in de Hoop--Saksala--Zhai (2019) from compact simple surfaces to orientable surfaces with solenoidally injective geodesic ray transform. Our proofs are based on algebraic arguments.

Paper Structure

This paper contains 25 sections, 16 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Mixing ray transforms
  3. Related problems
  4. Preliminaries
  5. Notation
  6. Sobolev norms of tensor fields
  7. Hodge star on orientable Riemannian surfaces
  8. The geodesic ray transform
  9. The mixed and transverse ray transforms
  10. The algebraic structure of mixing ray transforms
  11. Decompositions of tensor fields
  12. The mixing ray transform
  13. Solenoidal injectivity
  14. General results
  15. Mixed ray transform on compact simple surfaces
  16. ...and 10 more sections

Key Result

Proposition 3.1

Suppose that $m \geq 2$ and let $M$ be a Riemannian (or pseudo-Riemannian) manifold. Let $x \in M$ and define the sets Then $V_1 = V_2 = V_3$, $W_1 = W_2 = W_3$, and $V_i \oplus W_j = (T_x^*M)^{\otimes m}$ for any $i, j =1,2,3$.

Theorems & Definitions (39)

  • Proposition 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • proof
  • Corollary 3.5
  • Corollary 3.6
  • proof
  • ...and 29 more