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A support theorem for the X-ray transform on manifolds with plane covers

Part of: Lie groups Integral transforms, operational calculus Global differential geometry

Published online by Cambridge University Press:  25 April 2019

NORBERT PEYERIMHOFF  and
EVANGELIA SAMIOU
NORBERT PEYERIMHOFF
Affiliation:
Department of Mathematical Sciences, Durham University, Science Laboratories South Road, Durham, DH1 3LE. e-mail: norbert.peyerimhoff@durham.ac.uk
EVANGELIA SAMIOU
Affiliation:
Department of Mathematics and Statistics, University of Cyprus P.O. Box 20537, 1678 Nicosia, Cyprus. e-mail: samiou@ucy.ac.cy
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Abstract

This paper is concerned with support theorems of the X-ray transform on non-compact manifolds with conjugate points. In particular, we prove that all simply connected 2-step nilpotent Lie groups have a support theorem. Important ingredients of the proof are the concept of plane covers and a support theorem for simple manifolds by Krishnan. We also provide examples of non-homogeneous 3-dimensional simply connected manifolds with conjugate points which have support theorems.

MSC classification

Primary: 53C65: Integral geometry 44A12: Radon transform
Secondary: 22E25: Nilpotent and solvable Lie groups
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 1 , July 2020 , pp. 149 - 158
Copyright
© Cambridge Philosophical Society 2019

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References

REFERENCES

Berndt, J., Tricerri, F. and Vanhecke, L.. Generalised Heisenberg groups and Damek–Ricci harmonic spaces. Lecture Notes in Mathematics 1598. (Springer, 1995).CrossRefGoogle Scholar
Epstein, C. L.. Introduction to the mathematics of medical imaging, Second Edition. Society for Industrial and Applied Mathematics (SIAM) (Philadelphia, PA, 2008).Google Scholar
Feeman, T. G.. The Mathematics of Medical Imaging, A Beginner’s Guide. Second Edition (Springer Undergraduate Texts in Mathematics and Technology, Springer, Cham, 2015).Google Scholar
Grinberg, E. L. and Jackson, S. G.. On the kernel of the maximal flat Radon transform on symmetric spaces of compact type. J. Lie Theory 27(3) (2017), 623–636.Google Scholar
Helgason, S.. A duality in integral geometry; some generalizations of the Radon transform. Bull. Amer. Math. Soc. 70 (1964), 435–446.CrossRefGoogle Scholar
Helgason, S.. The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds. Acta Math. 113 (1965), 153–180.CrossRefGoogle Scholar
Helgason, S.. Support of Radon transforms, Adv. in Math. 38(1) (1980), 91–100.CrossRefGoogle Scholar
Helgason, S.. The X-ray transform on a symmetric space, in Global differential geometry and global analysis (Berlin, 1979). Lecture Notes in Math. 838, 145–148. (Springer, Berlin-New York. 1981).Google Scholar
Helgason, S.. Integral Geometry and Radon Transforms. (Springer, New York, 2011).CrossRefGoogle Scholar
Ilmavirta, J.. On Radon transforms on compact Lie groups. Proc. Amer. Math. Soc. 144(2) (2016), 681–691.CrossRefGoogle Scholar
Ilmavirta, J. and Monard, F.. Integral geometry on manifolds with boundary and applications. arXiv:1806.06088.Google Scholar
Ilmavirta, J. and Uhlmann, G.. Tensor tomography in periodic slabs. J. Funct. Anal. 275(2) (2018), 288–299.CrossRefGoogle Scholar
Kerr, M. M. and Payne, T. L.. The geometry of filiform nilpotent Lie groups. Rocky Mountain Journal of Mathematics 40(5) (2010), 1587–1610.CrossRefGoogle Scholar
Krishnan, V. P.. A support theorem for the geodesic ray transform on functions. Fourier Anal. Appl. 15 (2009), 515–520.CrossRefGoogle Scholar
Kuchment, P.. The Radon transform and medical imaging. CBMS-NSF Regional Conference Series in Applied Mathematics 85. Society for Industrial and Applied Mathematics (SIAM). (Philadelphia, PA, 2014).Google Scholar
Lehtonen, J.. The geodesic ray transform on two-dimensional Cartan–Hadamard manifolds. arXiv:1612.04800.Google Scholar
Lehtonen, J., Railo, J.. and Salo, M.. Tensor tomography on Cartan–Hadamard manifolds. Inverse Problems 34(4) (2018), Article ID 044004, 27pp.CrossRefGoogle Scholar
Paternain, G. P., Salo, M.. and Uhlmann, G.. Tensor tomography on surfaces. Invent. Math. 193(1) (2103), 229–247.CrossRefGoogle Scholar
Paternain, G. P., Salo, M.. and Uhlmann, G.. Tensor tomography: progress and challenges. Chin. Ann. Math. Ser. B 35(3) (2014), 399–428.CrossRefGoogle Scholar
Pestov, L. N. and Sharafutdinov, V. A.. Integral geometry of tensor fields on a manifold of negative curvature. Siberian Math. J. 29(3) (1988), 427–441.CrossRefGoogle Scholar
Peyerimhoff, N. and Samiou, E.. The X-ray transform on 2-step nilpotent Lie groups of higher rank, arXiv:1601.04614. To appear in Rendiconti Seminario Matematico Univ. Pol. Torino, Workshop for Sergio Console, Vol. 74, 1 (2016), 297–305.Google Scholar
Radon, J.. Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber. Verh. Sächs. Akad. Wiss. Leipzig Math. Nat. Kl. 69 (1917), 262–277.Google Scholar
Rouvière, F.. Transformation aux rayons X sur un espace symétrique. C. R. Math. Acad. Sci. Paris 342(1) (2006), 1–6.CrossRefGoogle Scholar
Rouvière, F.. X-ray transform on Damek–Ricci spaces. Inverse Probl. Imaging 4(4) (2010), 713–720.CrossRefGoogle Scholar
Salo, M.. The Calderón problem and normal forms. arXiv:1702.02136.Google Scholar
Samiou, E.. 2-step nilpotent Lie groups of higher rank. Manuscripta Math. 107(1) (2002), 101–110.CrossRefGoogle Scholar
Sharafutdinov, V. A.. Integral geometry of tensor fields. Inverse and Ill-posed Problems Series, (VSP, Utrecht, 1994).Google Scholar
Strichartz, R.. L p-harmonic analysis and Radon transforms on the Heisenberg group. J. Funct. Anal. 96 (1991), no. 2, 350–406.CrossRefGoogle Scholar
Uhlmann, G. and Vasy, A.. The inverse problem for the local geodesic ray transform. Invent. Math. 205(1) (2016), 83–120.CrossRefGoogle Scholar
Wojtkowski, M.. Geodesics on open surfaces containing horns. Studia Math. 73(2) (1982), 115–151.CrossRefGoogle Scholar