by Joonas Ilmavirta
Abstract:
We show that the Radon transform related to closed geodesics is injective on a Lie group if and only if the connected components are not homeomorphic to $S^1$ nor to $S^3$.
This is true for both smooth functions and distributions.
The key ingredients of the proof are finding totally geodesic tori and realizing the Radon transform as a family of symmetric operators indexed by nontrivial homomorphisms from $S^1$.
Reference:
On Radon transforms on compact Lie groups (Joonas Ilmavirta), Proceedings of the American Mathematical Society, volume 144, number 2, pp. 681–691, 2016.
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We show that the Radon transform related to closed geodesics is injective on a Lie group if and only if the connected components are not homeomorphic to $S^1$ nor to $S^3$.
This is true for both smooth functions and distributions.
The key ingredients of the proof are finding totally geodesic tori and realizing the Radon transform as a family of symmetric operators indexed by nontrivial homomorphisms from $S^1$.
[arXiv]
Bibtex Entry:
@article{lie-radon,
author = {Joonas Ilmavirta},
title = {{On Radon transforms on compact Lie groups}},
journal = {Proceedings of the American Mathematical Society},
volume={144},
number={2},
pages={681--691},
month = feb,
year = {2016},
doi = {10.1090/proc12732},
arxiv = {1410.2114},
gsid = {5114831216001904216},
url={http://users.jyu.fi/~jojapeil/pub/lie-radon-arxiv.pdf},
abstract = {We show that the Radon transform related to closed geodesics is injective on a Lie group if and only if the connected components are not homeomorphic to $S^1$ nor to $S^3$.
This is true for both smooth functions and distributions.
The key ingredients of the proof are finding totally geodesic tori and realizing the Radon transform as a family of symmetric operators indexed by nontrivial homomorphisms from $S^1$.}
}