# American Institute of Mathematical Sciences

February  2013, 7(1): 27-46. doi: 10.3934/ipi.2013.7.27

## The attenuated magnetic ray transform on surfaces

 1 Trinity College, Cambridge, CB2 1TQ, United Kingdom

Received  June 2012 Revised  October 2012 Published  February 2013

It has been shown in [10] that on a simple, compact Riemannian 2-manifold the attenuated geodesic ray transform, with attenuation given by a connection and Higgs field, is injective on functions and 1-forms modulo the natural obstruction. Furthermore, the scattering relation determines the connection and Higgs field modulo a gauge transformation. We extend the results obtained therein to the case of magnetic geodesics. In addition, we provide an application to tensor tomography in the magnetic setting, along the lines of [11].
Citation: Gareth Ainsworth. The attenuated magnetic ray transform on surfaces. Inverse Problems and Imaging, 2013, 7 (1) : 27-46. doi: 10.3934/ipi.2013.7.27
##### References:
 [1] D. V. Anosov and Ya. G. Sinai, Certain smooth ergodic systems (Russian), Uspekhi Mat. Nauk, 22 (1967), 107-172. [2] V. I. Arnol'd, Some remarks on flows of line elements and frames, Sov. Math. Dokl., 2 (1961), 562-564. [3] K. Burns and G. P. Paternain, Anosov magnetic flows, critical values and topological entropy, Nonlinearity, 15 (2002), 281-314. doi: 10.1088/0951-7715/15/2/305. [4] N. S. Dairbekov, G. P. Paternain, P. Stefanov and G. Uhlmann, The boundary rigidity problem in the presence of a magnetic field, Adv. Math., 216 (2007), 535-609. doi: 10.1016/j.aim.2007.05.014. [5] M. Dunajski, "Solitons, Instantons, and Twistors," Oxford Graduate Texts in Mathematics, 19, Oxford Univ. Press, Oxford, 2010. [6] P. Herreros, Scattering boundary rigidity in the presence of a magnetic field, Comm. Anal. Geom., 20 (2012), 501-528. [7] P. Herreros and J. Vargo, Scattering rigidity for analytic riemannian manifolds with a possible magnetic field, J. Geom. Anal., 21 (2011), 641-664. doi: 10.1007/s12220-010-9162-z. [8] V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved 2-manifolds, Topology, 19 (1980), 301-312. doi: 10.1016/0040-9383(80)90015-4. [9] R. Michel, Sur la rigidité imposée par la longeur des géodésiques,, Invent. Math., 65 (): 71.  doi: 10.1007/BF01389295. [10] G. P. Paternain, M. Salo and G. Uhlmann, The attenuated ray transform for connections and Higgs fields, Geom. Funct. Anal., 22 (2012), 1460-1489. doi: 10.1007/s00039-012-0183-6. [11] G. P. Paternain, M. Salo and G. Uhlmann, Tensor tomography on surfaces, preprint,, , (). [12] L. Pestov and G. Uhlmann, On characterization of the range and inversion formulas for the geodesic X-ray transform, Int. Math. Res. Not., 80 (2004), 4331-4347. doi: 10.1155/S1073792804142116. [13] L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math. (2), 161 (2005), 1093-1110. doi: 10.4007/annals.2005.161.1093. [14] M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces, J. Diff. Geom., 88 (2011), 161-187. [15] V. A. Sharafutdinov, "Integral Geometry of Tensor Fields," Inverse and Ill-posed Problem Series, VSP, Utrecht, 1994. doi: 10.1515/9783110900095.

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##### References:
 [1] D. V. Anosov and Ya. G. Sinai, Certain smooth ergodic systems (Russian), Uspekhi Mat. Nauk, 22 (1967), 107-172. [2] V. I. Arnol'd, Some remarks on flows of line elements and frames, Sov. Math. Dokl., 2 (1961), 562-564. [3] K. Burns and G. P. Paternain, Anosov magnetic flows, critical values and topological entropy, Nonlinearity, 15 (2002), 281-314. doi: 10.1088/0951-7715/15/2/305. [4] N. S. Dairbekov, G. P. Paternain, P. Stefanov and G. Uhlmann, The boundary rigidity problem in the presence of a magnetic field, Adv. Math., 216 (2007), 535-609. doi: 10.1016/j.aim.2007.05.014. [5] M. Dunajski, "Solitons, Instantons, and Twistors," Oxford Graduate Texts in Mathematics, 19, Oxford Univ. Press, Oxford, 2010. [6] P. Herreros, Scattering boundary rigidity in the presence of a magnetic field, Comm. Anal. Geom., 20 (2012), 501-528. [7] P. Herreros and J. Vargo, Scattering rigidity for analytic riemannian manifolds with a possible magnetic field, J. Geom. Anal., 21 (2011), 641-664. doi: 10.1007/s12220-010-9162-z. [8] V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved 2-manifolds, Topology, 19 (1980), 301-312. doi: 10.1016/0040-9383(80)90015-4. [9] R. Michel, Sur la rigidité imposée par la longeur des géodésiques,, Invent. Math., 65 (): 71.  doi: 10.1007/BF01389295. [10] G. P. Paternain, M. Salo and G. Uhlmann, The attenuated ray transform for connections and Higgs fields, Geom. Funct. Anal., 22 (2012), 1460-1489. doi: 10.1007/s00039-012-0183-6. [11] G. P. Paternain, M. Salo and G. Uhlmann, Tensor tomography on surfaces, preprint,, , (). [12] L. Pestov and G. Uhlmann, On characterization of the range and inversion formulas for the geodesic X-ray transform, Int. Math. Res. Not., 80 (2004), 4331-4347. doi: 10.1155/S1073792804142116. [13] L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math. (2), 161 (2005), 1093-1110. doi: 10.4007/annals.2005.161.1093. [14] M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces, J. Diff. Geom., 88 (2011), 161-187. [15] V. A. Sharafutdinov, "Integral Geometry of Tensor Fields," Inverse and Ill-posed Problem Series, VSP, Utrecht, 1994. doi: 10.1515/9783110900095.
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