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.
KeywordsTensor Field Carleman Estimate Simple Surface Commutator Formula Canonical Line Bundle
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.
- 7.Ivanov, S.: Volume comparison via boundary distances. In: Proceedings of the International Congress of Mathematicians, New Delhi, vol. II, pp. 769–784 (2010) Google Scholar
- 12.Pestov, L.: Well-Posedness Questions of the Ray Tomography Problems. Siberian Science Press, Novosibirsk (2003) (Russian) Google Scholar
- 14.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) Google Scholar