by Joonas Ilmavirta and Gabriel P. Paternain
Abstract:
We show that the existence of a function in with constant geodesic X-ray transform imposes geometrical restrictions on the manifold. The boundary of the manifold has to be umbilical and in the case of a strictly convex Euclidean domain, it must be a ball. Functions of constant geodesic X-ray transform always exist on manifolds with rotational symmetry.
Reference:
Functions of constant geodesic X-ray transform (Joonas Ilmavirta and Gabriel P. Paternain), Inverse Problems, volume 35, number 6, 2019.
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We show that the existence of a function in with constant geodesic X-ray transform imposes geometrical restrictions on the manifold. The boundary of the manifold has to be umbilical and in the case of a strictly convex Euclidean domain, it must be a ball. Functions of constant geodesic X-ray transform always exist on manifolds with rotational symmetry.
[arXiv]
Bibtex Entry:
@article{constant-xrt,
author = {Joonas Ilmavirta and Gabriel P. Paternain},
title = {{Functions of constant geodesic X-ray transform}},
journal = {Inverse Problems},
volume = {35},
number = {6},
month = may,
year = {2019},
arxiv = {1812.03515},
url={http://users.jyu.fi/~jojapeil/pub/constant-xrt-v10.pdf},
doi = {10.1088/1361-6420/ab0b6f},
gsid = {12361788843628514661},
abstract = {We show that the existence of a function in $L^{1}$ with constant geodesic X-ray transform imposes geometrical restrictions on the manifold. The boundary of the manifold has to be umbilical and in the case of a strictly convex Euclidean domain, it must be a ball. Functions of constant geodesic X-ray transform always exist on manifolds with rotational symmetry.}
}