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November  2010, 4(4): 713-720. doi: 10.3934/ipi.2010.4.713

X-ray transform on Damek-Ricci spaces

François Rouvière 1,

1. 

Laboratoire J.A. Dieudonné, Université de Nice, Parc Valrose, 06108 Nice cedex 2, France

Received  December 2008 Revised  May 2009 Published  September 2010

Damek-Ricci spaces, also called harmonic $NA$ groups, make up a large class of harmonic Riemannian manifolds including all hyperbolic spaces. We prove here an inversion formula and a support theorem for the X-ray transform, i.e. integration along geodesics, on those spaces.
   Using suitably chosen totally geodesic submanifolds we reduce the problems to similar questions on low-dimensional hyperbolic spaces.
Keywords: hyperbolic space, totally geodesic submanifold., Damek-Ricci space, X-ray transform.
Mathematics Subject Classification: Primary: 44A12, 53C25; Secondary: 22E25, 22E30, 53C2.
Citation: François Rouvière. X-ray transform on Damek-Ricci spaces. Inverse Problems & Imaging, 2010, 4 (4) : 713-720. doi: 10.3934/ipi.2010.4.713
References:
[1]

M. Cowling, A. Dooley, A. Korányi and F. Ricci, $H$-type groups and Iwasawa decompositions, Adv. Math., 87 (1991), 1-41. doi: doi:10.1016/0001-8708(91)90060-K.  Google Scholar

[2]

M. Cowling, A. Dooley, A. Korányi and F. Ricci, An approach to symmetric spaces of rank one via groups of Heisenberg type, J. Geom. Anal., 8 (1998), 199-237.  Google Scholar

[3]

E. Damek and F. Ricci, A class of nonsymmetric harmonic Riemannian spaces, Bull. Amer. Math. Soc., 27 (1992), 139-142. doi: doi:10.1090/S0273-0979-1992-00293-8.  Google Scholar

[4]

E. Damek and F. Ricci, Harmonic analysis on solvable extensions of $H$-type groups, J. Geom. Anal., 2 (1992), 213-248.  Google Scholar

[5]

S. Helgason, "Differential Geometry, Lie Groups and Symmetric Spaces," Academic Press, 1978.  Google Scholar

[6]

S. Helgason, "The Radon Transform," second edition, Birkhäuser, 1999.  Google Scholar

[7]

S. Helgason, The Abel, Fourier and Radon transforms on symmetric spaces, Indag. Math., 16 (2005), 531-551, arXiv:math/0506049.  Google Scholar

[8]

F. Rouvière, Espaces de Damek-Ricci, géométrie et analyse, Sémin. Congr., 7, Soc. Math. France, 2003, 45-100.  Google Scholar

[9]

, revised version of [8] at, \url{http://math.unice.fr/ frou/recherche/Damek-Ricci.pdf}, ().   Google Scholar

[10]

F. Rouvière, Transformation aux rayons X sur un espace symétrique, C. R. Math. Acad. Sci. Paris, 342 (2006), 1-6.  Google Scholar

show all references

References:
[1]

M. Cowling, A. Dooley, A. Korányi and F. Ricci, $H$-type groups and Iwasawa decompositions, Adv. Math., 87 (1991), 1-41. doi: doi:10.1016/0001-8708(91)90060-K.  Google Scholar

[2]

M. Cowling, A. Dooley, A. Korányi and F. Ricci, An approach to symmetric spaces of rank one via groups of Heisenberg type, J. Geom. Anal., 8 (1998), 199-237.  Google Scholar

[3]

E. Damek and F. Ricci, A class of nonsymmetric harmonic Riemannian spaces, Bull. Amer. Math. Soc., 27 (1992), 139-142. doi: doi:10.1090/S0273-0979-1992-00293-8.  Google Scholar

[4]

E. Damek and F. Ricci, Harmonic analysis on solvable extensions of $H$-type groups, J. Geom. Anal., 2 (1992), 213-248.  Google Scholar

[5]

S. Helgason, "Differential Geometry, Lie Groups and Symmetric Spaces," Academic Press, 1978.  Google Scholar

[6]

S. Helgason, "The Radon Transform," second edition, Birkhäuser, 1999.  Google Scholar

[7]

S. Helgason, The Abel, Fourier and Radon transforms on symmetric spaces, Indag. Math., 16 (2005), 531-551, arXiv:math/0506049.  Google Scholar

[8]

F. Rouvière, Espaces de Damek-Ricci, géométrie et analyse, Sémin. Congr., 7, Soc. Math. France, 2003, 45-100.  Google Scholar

[9]

, revised version of [8] at, \url{http://math.unice.fr/ frou/recherche/Damek-Ricci.pdf}, ().   Google Scholar

[10]

F. Rouvière, Transformation aux rayons X sur un espace symétrique, C. R. Math. Acad. Sci. Paris, 342 (2006), 1-6.  Google Scholar

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