Abstract
We show that on simple surfaces the geodesic ray transform acting on solenoidal symmetric tensor fields of arbitrary order is injective. This solves a long standing inverse problem in the two-dimensional case.
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Anikonov, Yu., Romanov, V.: On uniqueness of determination of a form of first degree by its integrals along geodesics. J. Inverse Ill-Posed Probl. 5, 467–480 (1997)
Dairbekov, N.S.: Integral geometry problem for nontrapping manifolds. Inverse Probl. 22, 431–445 (2006)
Dos Santos Ferreira, D., Kenig, C.E., Salo, M., Uhlmann, G.: Limiting Carleman weights and anisotropic inverse problems. Invent. Math. 178, 119–171 (2009)
Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley Classics Library. Wiley, New York (1994). Reprint of the 1978 original
Guillemin, V., Kazhdan, D.: Some inverse spectral results for negatively curved 2-manifolds. Topology 19, 301–312 (1980)
Hörmander, L.: The Analysis of Linear Partial Differential Operators. IV. Fourier Integral Operators. Classics in Mathematics. Springer, Berlin (2009). Reprint of the 1994 edition
Ivanov, S.: Volume comparison via boundary distances. In: Proceedings of the International Congress of Mathematicians, New Delhi, vol. II, pp. 769–784 (2010)
Michel, R.: Sur la rigidité imposée par la longueur des géodésiques. Invent. Math. 65, 71–83 (1981)
Mukhometov, R.G.: The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry. Dokl. Akad. Nauk SSSR 232(1), 32–35 (1977) (Russian)
Paternain, G.P.: Transparent connections over negatively curved surfaces. J. Mod. Dyn. 3, 311–333 (2009)
Paternain, G.P., Salo, M., Uhlmann, G.: The attenuated ray transform for connections and Higgs fields. Geom. Funct. Anal. 22, 1460–1489 (2012)
Pestov, L.: Well-Posedness Questions of the Ray Tomography Problems. Siberian Science Press, Novosibirsk (2003) (Russian)
Pestov, L., Sharafutdinov, V.A.: Integral geometry of tensor fields on a manifold of negative curvature. Sib. Math. J. 29, 427–441 (1988)
Pestov, L., Uhlmann, G.: On characterization of the range and inversion formulas for the geodesic X-ray transform. Int. Math. Res. Not. 4331–4347 (2004)
Pestov, L., Uhlmann, G.: Two dimensional compact simple Riemannian manifolds are boundary distance rigid. Ann. Math. 161, 1089–1106 (2005)
Salo, M., Uhlmann, G.: The attenuated ray transform on simple surfaces. J. Differ. Geom. 88, 161–187 (2011)
Sharafutdinov, V.A.: Integral Geometry of Tensor Fields. Inverse and Ill-Posed Problems Series. VSP, Utrecht (1994)
Sharafutdinov, V.A.: Integral geometry of a tensor field on a surface of revolution. Sib. Math. J. 38, 603–620 (1997)
Sharafutdinov, V.A.: A problem in integral geometry in a nonconvex domain. Sib. Math. J. 43, 1159–1168 (2002)
Sharafutdinov, V.A.: Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds. J. Geom. Anal. 17, 147–187 (2007)
Sharafutdinov, V.A., Skokan, M., Uhlmann, G.: Regularity of ghosts in tensor tomography. J. Geom. Anal. 15, 517–560 (2005)
Singer, I.M., Thorpe, J.A.: Lecture Notes on Elementary Topology and Geometry. Undergraduate Texts in Mathematics. Springer, New York (1976). Reprint of the 1967 edition
Stefanov, P., Uhlmann, G.: Stability estimates for the X-ray transform of tensor fields and boundary rigidity. Duke Math. J. 123, 445–467 (2004)
Stefanov, P., Uhlmann, G.: Boundary and lens rigidity, tensor tomography and analytic microlocal analysis. In: Aoki, T., Majima, H., Katei, Y., Tose, N. (eds.) Algebraic Analysis of Differential Equations, Festschrift in Honor of Takahiro Kawai, pp. 275–293 (2008)
Stefanov, P., Uhlmann, G.: Linearizing non-linear inverse problems and its applications to inverse backscattering. J. Funct. Anal. 256, 2842–2866 (2009)
Thorbergsson, G.: Closed geodesics on non-compact Riemannian manifolds. Math. Z. 159, 249–258 (1978)
Acknowledgements
M.S. was supported in part by the Academy of Finland, and G.U. was partly supported by NSF and a Rothschild Distinguished Visiting Fellowship at the Isaac Newton Institute. The authors would like to express their gratitude to the Newton Institute and the organizers of the program on Inverse Problems in 2011 where this work was carried out. They would also like to thank the referees for their constructive and useful comments.
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Paternain, G.P., Salo, M. & Uhlmann, G. Tensor tomography on surfaces. Invent. math. 193, 229–247 (2013). https://doi.org/10.1007/s00222-012-0432-1
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DOI: https://doi.org/10.1007/s00222-012-0432-1