by Joonas Ilmavirta, Keijo Mönkkönen
Abstract:
We show that the geodesic ray transform is injective on scalar functions on spherically symmetric reversible Finsler manifolds where the Finsler norm satisfies a Herglotz condition. We use angular Fourier series to reduce the injectivity problem to the invertibility of generalized Abel transforms and by Taylor expansions of geodesics we show that these Abel transforms are injective. Our result has applications in linearized boundary rigidity problem on Finsler manifolds and especially in linearized elastic travel time tomography.
Reference:
The geodesic ray transform on spherically symmetric reversible Finsler manifolds (Joonas Ilmavirta, Keijo Mönkkönen), The Journal of Geometric Analysis, volume 33, 2023.
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We show that the geodesic ray transform is injective on scalar functions on spherically symmetric reversible Finsler manifolds where the Finsler norm satisfies a Herglotz condition. We use angular Fourier series to reduce the injectivity problem to the invertibility of generalized Abel transforms and by Taylor expansions of geodesics we show that these Abel transforms are injective. Our result has applications in linearized boundary rigidity problem on Finsler manifolds and especially in linearized elastic travel time tomography.
[arXiv]
Bibtex Entry:
@article{xrt-spherical-finsler,
author = {Joonas Ilmavirta and Keijo M{\"o}nkk{\"o}nen},
title = {{The geodesic ray transform on spherically symmetric reversible Finsler manifolds}},
journal = {The Journal of Geometric Analysis},
volume = 33,
issue = 4,
month = feb,
year = {2023},
abstract = {We show that the geodesic ray transform is injective on scalar functions on spherically symmetric reversible Finsler manifolds where the Finsler norm satisfies a Herglotz condition. We use angular Fourier series to reduce the injectivity problem to the invertibility of generalized Abel transforms and by Taylor expansions of geodesics we show that these Abel transforms are injective. Our result has applications in linearized boundary rigidity problem on Finsler manifolds and especially in linearized elastic travel time tomography.},
url={http://users.jyu.fi/~jojapeil/pub/xrt-spherical-finsler.pdf},
arxiv = {2203.16886},
doi = {10.1007/s12220-022-01182-w}
}