On Radon transforms on compact Lie groups
HTML articles powered by AMS MathViewer
- by Joonas Ilmavirta PDF
- Proc. Amer. Math. Soc. 144 (2016), 681-691 Request permission
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