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Tensor tomography on negatively curved manifolds of low regularity

Joonas Ilmavirta, Antti Kykkänen

Abstract

We prove solenoidal injectivity for the geodesic X-ray transform of tensor fields on simple Riemannian manifolds with $C^{1,1}$ metrics and non-positive sectional curvature. The proof of the result rests on Pestov energy estimates for a transport equation on the non-smooth unit sphere bundle of the manifold. Our low regularity setting requires keeping track of regularity and making use of many functions on the sphere bundle having more vertical than horizontal regularity. Some of the methods, such as boundary determination up to gauge and regularity estimates for the integral function, have to be changed substantially from the smooth proof. The natural differential operators such as covariant derivatives are not smooth.

Tensor tomography on negatively curved manifolds of low regularity

Abstract

We prove solenoidal injectivity for the geodesic X-ray transform of tensor fields on simple Riemannian manifolds with metrics and non-positive sectional curvature. The proof of the result rests on Pestov energy estimates for a transport equation on the non-smooth unit sphere bundle of the manifold. Our low regularity setting requires keeping track of regularity and making use of many functions on the sphere bundle having more vertical than horizontal regularity. Some of the methods, such as boundary determination up to gauge and regularity estimates for the integral function, have to be changed substantially from the smooth proof. The natural differential operators such as covariant derivatives are not smooth.
Paper Structure (34 sections, 22 theorems, 153 equations)

This paper contains 34 sections, 22 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Regularity discussion
  4. Related results
  5. Acknowledgements
  6. Proof of the main theorem
  7. Basic definitions and notation
  8. Bundles
  9. Horizontal--vertical decomposition
  10. Geodesic flow
  11. The X-ray transform
  12. Differentiability
  13. Curvature
  14. Sobolev spaces
  15. Spherical harmonics
  16. ...and 19 more sections

Key Result

Theorem 1

Let $(M,g)$ be a simple $C^{1,1}$ manifold (see section sec:definitions-notation) with almost everywhere non-positive sectional curvature. Let $m \ge 1$ be an integer.

Theorems & Definitions (47)

  • Theorem 1
  • Lemma 2: Boundary determination
  • Lemma 3: Regularity of spherical harmonic components
  • Lemma 4
  • Lemma 5
  • Lemma 6: Injectivity of $X_+$
  • Lemma 7
  • proof : Proof of theorem \ref{['thm:c11-s-injectivity']}
  • Remark 8
  • Remark 9
  • ...and 37 more