May  2015, 9(2): 317-335. doi: 10.3934/ipi.2015.9.317

On the range of the attenuated magnetic ray transform for connections and Higgs fields

1. 

Trinity College, Cambridge, CB2 1TQ, United Kingdom

2. 

Department of Mathematics, University of Washington, Seattle, WA 98195-4350, United States

Received  November 2013 Revised  July 2014 Published  March 2015

For a two-dimensional simple magnetic system, we study the attenuated magnetic ray transform $I_{A,\Phi}$, with attenuation given by a unitary connection $A$ and a skew-Hermitian Higgs field $\Phi$. We give a description for the range of $I_{A,\Phi}$ acting on $\mathbb{C}^n$-valued tensor fields.
Citation: Gareth Ainsworth, Yernat M. Assylbekov. On the range of the attenuated magnetic ray transform for connections and Higgs fields. Inverse Problems & Imaging, 2015, 9 (2) : 317-335. doi: 10.3934/ipi.2015.9.317
References:
[1]

G. Ainsworth, The attenuated magnetic ray transform on surfaces,, Inverse Probl. Imaging, 7 (2013), 27. doi: 10.3934/ipi.2013.7.27. Google Scholar

[2]

D. V. Anosov and Y. G. Sinai, Certain smooth ergodic systems [Russian],, Uspekhi Mat. Nauk., 22 (1967), 107. Google Scholar

[3]

V. I. Arnold, Some remarks on flows of line elements and frames,, Dokl. Akad. Nauk SSSR, 138 (1961), 255. Google Scholar

[4]

V. I. Arnold and A. B. Givental, Symplectic geometry,, in Dynamical Systems IV, (1990), 1. doi: 10.1007/978-3-662-06793-2. Google Scholar

[5]

N. Bourbaki, Topological Vector Spaces,, Springer-Verlag, (1987). doi: 10.1007/978-3-642-61715-7. Google Scholar

[6]

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. doi: 10.1016/j.aim.2007.05.014. Google Scholar

[7]

N. Dairbekov and G. Uhlmann, Reconstructing the metric and magnetic field from the scattering relation,, Inverse Probl. Imaging, 4 (2010), 397. doi: 10.3934/ipi.2010.4.397. Google Scholar

[8]

M. Dunajski, Solitons, Instantons, and Twistors,, Oxford Graduate Texts in Mathematics, (2010). Google Scholar

[9]

N. J. Hitchin, G. B. Segal and R. S. Ward, Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces,, Oxford Graduate Texts in Mathematics, (1997). Google Scholar

[10]

V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved 2-manifolds,, Topology, 19 (1980), 301. doi: 10.1016/0040-9383(80)90015-4. Google Scholar

[11]

S. Kobayashi, Differential Geometry of Complex Vector Bundles,, Publications of the Mathematical Society of Japan 15, (1987). doi: 10.1515/9781400858682. Google Scholar

[12]

V. V. Kozlov, Calculus of variations in the large and classical mechanics,, Uspekhi Mat. Nauk, 40 (1985), 33. Google Scholar

[13]

N. Manton and P. Sutcliffe, Topological Solitons,, Cambridge Monographs on Mathematical Physics, (2004). doi: 10.1017/CBO9780511617034. Google Scholar

[14]

L. J. Mason and N. M. J. Woodhouse, Integrability, Self-duality, and Twistor Theory,, London Mathematical Society Monographs, (1996). Google Scholar

[15]

R. Michel, Sur la rigidité imposée par la longueur des géodésiques,, Invent. Math., 65 (1981), 71. doi: 10.1007/BF01389295. Google Scholar

[16]

S. P. Novikov, Variational methods and periodic solutions of equations of Kirchhoff type. II,, (Russian) Funktsional. Anal. i Prilozhen, 15 (1981), 37. Google Scholar

[17]

S. P. Novikov, Hamiltonian formalism and a multivalued analogue of Morse theory,, (Russian) Uspekhi Mat. Nauk, 37 (1982), 3. Google Scholar

[18]

S. P. Novikov and I. Shmel'tser, Periodic solutions of the Kirchhoff equations for the free motion of a rigid body in a liquid, and the extended Lyusternik-Schnirelmann-Morse theory. I,, (Russian) Funktsional. Anal. i Prilozhen, 15 (1981), 54. Google Scholar

[19]

G. P. Paternain, Transparent connections over negatively curved surfaces,, J. Mod. Dyn., 3 (2009), 311. doi: 10.3934/jmd.2009.3.311. Google Scholar

[20]

G. P. Paternain and M. Paternain, Anosov geodesic flows and twisted symplectic structures,, in International Congress on Dynamical Systems in Montevideo (A Tribute to Ricardo Mañé) (eds. F. Ledrappier, (1996), 132. Google Scholar

[21]

G. P. Paternain, M. Salo and G. Uhlmann, Spectral rigidity and invariant distributions on Anosov surfaces,, J. Diff. Geom., 98 (2014), 147. Google Scholar

[22]

G. P. Paternain, M. Salo, G. Uhlmann, On the range of the attenuated ray transform for unitary connections,, Int. Math. Res. Not., (2015), 873. doi: 10.1093/imrn/rnt228. Google Scholar

[23]

G. P. Paternain, M. Salo and G. Uhlmann, Tensor tomography on surfaces,, Invent. Math., 193 (2013), 229. doi: 10.1007/s00222-012-0432-1. Google Scholar

[24]

G. P. Paternain, M. Salo and G. Uhlmann, The attenuated ray transform for connections and Higgs fields,, Geom. Funct. Anal., 22 (2012), 1460. doi: 10.1007/s00039-012-0183-6. Google Scholar

[25]

L. Pestov and G. Uhlmann, On the characterization of the range and inversion formulas for the geodesic X-ray transform,, Int. Math. Res. Not., (2004), 4331. doi: 10.1155/S1073792804142116. Google Scholar

[26]

L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid,, Ann. of Math., 161 (2005), 1093. doi: 10.4007/annals.2005.161.1093. Google Scholar

[27]

E. Powell, Boundary Rigidity,, unpublished draft, (2014). Google Scholar

[28]

M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces,, J. Diff. Geom., 88 (2011), 161. Google Scholar

[29]

P. Stefanov, Personal Communication,, 12/02/2014., (). Google Scholar

[30]

M. E. Taylor, Partial Differential Equations I. Basic Theory,, Second edition, (2011). doi: 10.1007/978-1-4419-7055-8. Google Scholar

[31]

F. Treves, Topological Vector Spaces, Distributions and Kernels,, Academic Press, (1967). Google Scholar

show all references

References:
[1]

G. Ainsworth, The attenuated magnetic ray transform on surfaces,, Inverse Probl. Imaging, 7 (2013), 27. doi: 10.3934/ipi.2013.7.27. Google Scholar

[2]

D. V. Anosov and Y. G. Sinai, Certain smooth ergodic systems [Russian],, Uspekhi Mat. Nauk., 22 (1967), 107. Google Scholar

[3]

V. I. Arnold, Some remarks on flows of line elements and frames,, Dokl. Akad. Nauk SSSR, 138 (1961), 255. Google Scholar

[4]

V. I. Arnold and A. B. Givental, Symplectic geometry,, in Dynamical Systems IV, (1990), 1. doi: 10.1007/978-3-662-06793-2. Google Scholar

[5]

N. Bourbaki, Topological Vector Spaces,, Springer-Verlag, (1987). doi: 10.1007/978-3-642-61715-7. Google Scholar

[6]

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. doi: 10.1016/j.aim.2007.05.014. Google Scholar

[7]

N. Dairbekov and G. Uhlmann, Reconstructing the metric and magnetic field from the scattering relation,, Inverse Probl. Imaging, 4 (2010), 397. doi: 10.3934/ipi.2010.4.397. Google Scholar

[8]

M. Dunajski, Solitons, Instantons, and Twistors,, Oxford Graduate Texts in Mathematics, (2010). Google Scholar

[9]

N. J. Hitchin, G. B. Segal and R. S. Ward, Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces,, Oxford Graduate Texts in Mathematics, (1997). Google Scholar

[10]

V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved 2-manifolds,, Topology, 19 (1980), 301. doi: 10.1016/0040-9383(80)90015-4. Google Scholar

[11]

S. Kobayashi, Differential Geometry of Complex Vector Bundles,, Publications of the Mathematical Society of Japan 15, (1987). doi: 10.1515/9781400858682. Google Scholar

[12]

V. V. Kozlov, Calculus of variations in the large and classical mechanics,, Uspekhi Mat. Nauk, 40 (1985), 33. Google Scholar

[13]

N. Manton and P. Sutcliffe, Topological Solitons,, Cambridge Monographs on Mathematical Physics, (2004). doi: 10.1017/CBO9780511617034. Google Scholar

[14]

L. J. Mason and N. M. J. Woodhouse, Integrability, Self-duality, and Twistor Theory,, London Mathematical Society Monographs, (1996). Google Scholar

[15]

R. Michel, Sur la rigidité imposée par la longueur des géodésiques,, Invent. Math., 65 (1981), 71. doi: 10.1007/BF01389295. Google Scholar

[16]

S. P. Novikov, Variational methods and periodic solutions of equations of Kirchhoff type. II,, (Russian) Funktsional. Anal. i Prilozhen, 15 (1981), 37. Google Scholar

[17]

S. P. Novikov, Hamiltonian formalism and a multivalued analogue of Morse theory,, (Russian) Uspekhi Mat. Nauk, 37 (1982), 3. Google Scholar

[18]

S. P. Novikov and I. Shmel'tser, Periodic solutions of the Kirchhoff equations for the free motion of a rigid body in a liquid, and the extended Lyusternik-Schnirelmann-Morse theory. I,, (Russian) Funktsional. Anal. i Prilozhen, 15 (1981), 54. Google Scholar

[19]

G. P. Paternain, Transparent connections over negatively curved surfaces,, J. Mod. Dyn., 3 (2009), 311. doi: 10.3934/jmd.2009.3.311. Google Scholar

[20]

G. P. Paternain and M. Paternain, Anosov geodesic flows and twisted symplectic structures,, in International Congress on Dynamical Systems in Montevideo (A Tribute to Ricardo Mañé) (eds. F. Ledrappier, (1996), 132. Google Scholar

[21]

G. P. Paternain, M. Salo and G. Uhlmann, Spectral rigidity and invariant distributions on Anosov surfaces,, J. Diff. Geom., 98 (2014), 147. Google Scholar

[22]

G. P. Paternain, M. Salo, G. Uhlmann, On the range of the attenuated ray transform for unitary connections,, Int. Math. Res. Not., (2015), 873. doi: 10.1093/imrn/rnt228. Google Scholar

[23]

G. P. Paternain, M. Salo and G. Uhlmann, Tensor tomography on surfaces,, Invent. Math., 193 (2013), 229. doi: 10.1007/s00222-012-0432-1. Google Scholar

[24]

G. P. Paternain, M. Salo and G. Uhlmann, The attenuated ray transform for connections and Higgs fields,, Geom. Funct. Anal., 22 (2012), 1460. doi: 10.1007/s00039-012-0183-6. Google Scholar

[25]

L. Pestov and G. Uhlmann, On the characterization of the range and inversion formulas for the geodesic X-ray transform,, Int. Math. Res. Not., (2004), 4331. doi: 10.1155/S1073792804142116. Google Scholar

[26]

L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid,, Ann. of Math., 161 (2005), 1093. doi: 10.4007/annals.2005.161.1093. Google Scholar

[27]

E. Powell, Boundary Rigidity,, unpublished draft, (2014). Google Scholar

[28]

M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces,, J. Diff. Geom., 88 (2011), 161. Google Scholar

[29]

P. Stefanov, Personal Communication,, 12/02/2014., (). Google Scholar

[30]

M. E. Taylor, Partial Differential Equations I. Basic Theory,, Second edition, (2011). doi: 10.1007/978-1-4419-7055-8. Google Scholar

[31]

F. Treves, Topological Vector Spaces, Distributions and Kernels,, Academic Press, (1967). Google Scholar

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