On Radon transforms on compact Lie groups (bibtex)

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 $S^1$.} }

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