## On Radon transforms on compact Lie groups

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## 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 $S^1$.## References

- Ahmed Abouelaz and François Rouvière,
*Radon transform on the torus*, Mediterr. J. Math.**8**(2011), no. 4, 463–471. MR**2860679**, DOI 10.1007/s00009-010-0111-7 - Swanhild Bernstein, Svend Ebert, and Isaac Z. Pesenson,
*Generalized splines for Radon transform on compact Lie groups with applications to crystallography*, J. Fourier Anal. Appl.**19**(2013), no. 1, 140–166. MR**3019773**, DOI 10.1007/s00041-012-9241-6 - Christopher Croke,
*Boundary and lens rigidity of finite quotients*, Proc. Amer. Math. Soc.**133**(2005), no. 12, 3663–3668. MR**2163605**, DOI 10.1090/S0002-9939-05-07927-X - Christopher B. Croke and Vladimir A. Sharafutdinov,
*Spectral rigidity of a compact negatively curved manifold*, Topology**37**(1998), no. 6, 1265–1273. MR**1632920**, DOI 10.1016/S0040-9383(97)00086-4 - Nurlan S. Dairbekov and Vladimir A. Sharafutdinov,
*Some problems of integral geometry on Anosov manifolds*, Ergodic Theory Dynam. Systems**23**(2003), no. 1, 59–74. MR**1971196**, DOI 10.1017/S0143385702000822 - P. Funk,
*Über Flächen mit lauter geschlossenen geodätischen Linien*, Math. Ann.**74**(1913), no. 2, 278–300 (German). MR**1511763**, DOI 10.1007/BF01456044 - Eric L. Grinberg,
*Spherical harmonics and integral geometry on projective spaces*, Trans. Amer. Math. Soc.**279**(1983), no. 1, 187–203. MR**704609**, DOI 10.1090/S0002-9947-1983-0704609-1 - Eric L. Grinberg,
*Integration over minimal spheres in Lie groups and symmetric spaces of compact type*, 75 years of Radon transform (Vienna, 1992) Conf. Proc. Lecture Notes Math. Phys., IV, Int. Press, Cambridge, MA, 1994, pp. 167–174. MR**1313932** - Eric L. Grinberg and Steven Glenn Jackson,
*On the kernel of the maximal flat Radon transform on symmetric spaces of compact type*, (2014), arXiv:1406.7219. - V. Guillemin and D. Kazhdan,
*Some inverse spectral results for negatively curved $2$-manifolds*, Topology**19**(1980), no. 3, 301–312. MR**579579**, DOI 10.1016/0040-9383(80)90015-4 - Ralf Hielscher,
*The Inversion of the Radon Transform on the Rotational Group and Its Application to Texture Analysis*, Ph.D. thesis, Technischen Universität Bergakademie Freiberg, 2006. - Joonas Ilmavirta,
*On Radon transforms on tori*, Fourier Analysis and Applications (2014), To appear, arXiv:1402.6209. - K. Modin, M. Perlmutter, S. Marsland, and R. McLachlan,
*Geodesics on Lie groups: Euler equations and totally geodesic subgroups*, Research Letters in the Information and Mathematical Sciences**14**(2010), 79–106. - Gabriel P. Paternain, Mikko Salo, and Gunther Uhlmann,
*Invariant distributions, Beurling transform and tensor tomography in higher dimensions*, (2014), arXiv:1404.7009. - Gabriel P. Paternain, Mikko Salo, and Gunther Uhlmann,
*Spectral rigidity and invariant distributions on Anosov surfaces*, J. Differential Geom.**98**(2014), no. 1, 147–181. MR**3263517** - Michael Ruzhansky and Ville Turunen,
*Pseudo-differential operators and symmetries*, Pseudo-Differential Operators. Theory and Applications, vol. 2, Birkhäuser Verlag, Basel, 2010. Background analysis and advanced topics. MR**2567604**, DOI 10.1007/978-3-7643-8514-9 - Robert S. Strichartz,
*Radon inversion—variations on a theme*, Amer. Math. Monthly**89**(1982), no. 6, 377–384, 420–423. MR**660917**, DOI 10.2307/2321649

## Additional Information

**Joonas Ilmavirta**- Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD) FI-40014 University of Jyväskylä, Finland
- MR Author ID: 1014879
- Email: joonas.ilmavirta@jyu.fi
- Received by editor(s): October 8, 2014
- Received by editor(s) in revised form: January 21, 2015
- Published electronically: July 24, 2015
- Communicated by: Michael Hitrik
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 681-691 - MSC (2010): Primary 46F12, 44A12, 22C05, 22E30
- DOI: https://doi.org/10.1090/proc12732
- MathSciNet review: 3430844