The Geodesic Ray Transform on Spherically Symmetric Reversible Finsler Manifolds

Ilmavirta, Joonas; Mönkkönen, Keijo

doi:10.1007/s12220-022-01182-w

Cited by 3 publications

(3 citation statements)

References 52 publications

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“…The Pestov identity approach has been studied in more general geometries than Riemannian. For results in Finsler geometry, see [3,19] and for pseudo-Riemannian geometry, [17].…”

Section: Related Resultsmentioning

confidence: 99%

Tensor Tomography on Negatively Curved Manifolds of Low Regularity

Ilmavirta,

Kykkänen

2024

J Geom Anal

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We prove solenoidal injectivity for the geodesic X-ray transform of tensor fields on simple Riemannian manifolds with $$C^{1,1}$$ 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.

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“…The Pestov identity approach has been studied in more general geometries than Riemannian. For results in Finsler geometry, see [3,19] and for pseudo-Riemannian geometry, [17].…”

Section: Related Resultsmentioning

confidence: 99%

Tensor Tomography on Negatively Curved Manifolds of Low Regularity

Ilmavirta,

Kykkänen

2024

J Geom Anal

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“…This result was later generalized to higher dimensions and tensor fields of any order in [37]. In the case of rotational (or spherical) symmetry, one may sometimes solve these and related problems using local results and data avoiding the obstacle when the manifold satisfies the Herglotz condition [12,36,64]. Broken lens rigidity was studied recently in [13], and a broken non-Abelian ray transform in Minkowski space in [63].…”

Section: Introductionmentioning

confidence: 99%

Broken Ray Transform for Twisted Geodesics on Surfaces with a Reflecting Obstacle

Jathar,

Kar,

Railo

2024

J Geom Anal

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We prove a uniqueness result for the broken ray transform acting on the sums of functions and 1-forms on surfaces in the presence of an external force and a reflecting obstacle. We assume that the considered twisted geodesic flows have nonpositive curvature. The broken rays are generated from the twisted geodesic flows by the law of reflection on the boundary of a suitably convex obstacle. Our work generalizes recent results for the broken geodesic ray transform on surfaces to more general families of curves including the magnetic flows and Gaussian thermostats.

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“…The Pestov identity approach has been studied in more general geometries than Riemannian. For results in Finsler geometry see [AD18,IM22] and for pseudo-Riemannian geometry [Ilm16].…”

Section: Introductionmentioning

confidence: 99%

Tensor tomography on negatively curved manifolds of low regularity

Ilmavirta¹,

Kykkänen²

2023

Preprint

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We prove solenoidal injectivity for the geodesic X-ray transform of tensor fields on simple Riemannian manifolds with C 1,1 metrics and nonpositive 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.

show abstract

The Geodesic Ray Transform on Spherically Symmetric Reversible Finsler Manifolds

Cited by 3 publications

References 52 publications

Tensor Tomography on Negatively Curved Manifolds of Low Regularity

Tensor Tomography on Negatively Curved Manifolds of Low Regularity

Broken Ray Transform for Twisted Geodesics on Surfaces with a Reflecting Obstacle

Tensor tomography on negatively curved manifolds of low regularity

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