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 and 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), 27-46. doi: 10.3934/ipi.2013.7.27.

[2]

D. V. Anosov and Y. G. Sinai, Certain smooth ergodic systems [Russian], Uspekhi Mat. Nauk., 22 (1967), 107-172.

[3]

V. I. Arnold, Some remarks on flows of line elements and frames, Dokl. Akad. Nauk SSSR, 138 (1961), 255-257.

[4]

V. I. Arnold and A. B. Givental, Symplectic geometry, in Dynamical Systems IV, Encyclopaedia of Mathematical Sciences, 4, Springer Verlag, Berlin, 1990, 1-136. doi: 10.1007/978-3-662-06793-2.

[5]

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

[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), 535-609. doi: 10.1016/j.aim.2007.05.014.

[7]

N. Dairbekov and G. Uhlmann, Reconstructing the metric and magnetic field from the scattering relation, Inverse Probl. Imaging, 4 (2010), 397-409. doi: 10.3934/ipi.2010.4.397.

[8]

M. Dunajski, Solitons, Instantons, and Twistors, Oxford Graduate Texts in Mathematics, 19, Oxford University Press, Oxford, 2010.

[9]

N. J. Hitchin, G. B. Segal and R. S. Ward, Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces, Oxford Graduate Texts in Mathematics, 4, The Clarendon Press, Oxford University Press, New York, 1997.

[10]

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.

[11]

S. Kobayashi, Differential Geometry of Complex Vector Bundles, Publications of the Mathematical Society of Japan 15, Kanô Memorial Lectures 5, Princeton University Press, Princeton, NJ; Iwanami Shoten, Tokyo, 1987. doi: 10.1515/9781400858682.

[12]

V. V. Kozlov, Calculus of variations in the large and classical mechanics, Uspekhi Mat. Nauk, 40 (1985), 33-60, 237.

[13]

N. Manton and P. Sutcliffe, Topological Solitons, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511617034.

[14]

L. J. Mason and N. M. J. Woodhouse, Integrability, Self-duality, and Twistor Theory, London Mathematical Society Monographs, New Series, 15, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996.

[15]

R. Michel, Sur la rigidité imposée par la longueur des géodésiques, Invent. Math., 65 (1981), 71-83. doi: 10.1007/BF01389295.

[16]

S. P. Novikov, Variational methods and periodic solutions of equations of Kirchhoff type. II, (Russian) Funktsional. Anal. i Prilozhen, 15 (1981), 37-52, 96.

[17]

S. P. Novikov, Hamiltonian formalism and a multivalued analogue of Morse theory, (Russian) Uspekhi Mat. Nauk, 37 (1982), 3-49, 248.

[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), 54-66.

[19]

G. P. Paternain, Transparent connections over negatively curved surfaces, J. Mod. Dyn., 3 (2009), 311-333. doi: 10.3934/jmd.2009.3.311.

[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, J. Lewowicz, S. Newhouse), Pitman Research Notes in Math., 362, Longman, Harlow, 1996, 132-145.

[21]

G. P. Paternain, M. Salo and G. Uhlmann, Spectral rigidity and invariant distributions on Anosov surfaces, J. Diff. Geom., 98 (2014), 147-181.

[22]

G. P. Paternain, M. Salo, G. Uhlmann, On the range of the attenuated ray transform for unitary connections, Int. Math. Res. Not., (2015), 873-897. doi: 10.1093/imrn/rnt228.

[23]

G. P. Paternain, M. Salo and G. Uhlmann, Tensor tomography on surfaces, Invent. Math., 193 (2013), 229-247. doi: 10.1007/s00222-012-0432-1.

[24]

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.

[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), 4331-4347. doi: 10.1155/S1073792804142116.

[26]

L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math., 161 (2005), 1093-1110. doi: 10.4007/annals.2005.161.1093.

[27]

E. Powell, Boundary Rigidity, unpublished draft, 2014.

[28]

M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces, J. Diff. Geom., 88 (2011), 161-187.

[29]

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

[30]

M. E. Taylor, Partial Differential Equations I. Basic Theory, Second edition, Applied Mathematical Sciences, 115, Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8.

[31]

F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, 1967.

show all references

References:
[1]

G. Ainsworth, The attenuated magnetic ray transform on surfaces, Inverse Probl. Imaging, 7 (2013), 27-46. doi: 10.3934/ipi.2013.7.27.

[2]

D. V. Anosov and Y. G. Sinai, Certain smooth ergodic systems [Russian], Uspekhi Mat. Nauk., 22 (1967), 107-172.

[3]

V. I. Arnold, Some remarks on flows of line elements and frames, Dokl. Akad. Nauk SSSR, 138 (1961), 255-257.

[4]

V. I. Arnold and A. B. Givental, Symplectic geometry, in Dynamical Systems IV, Encyclopaedia of Mathematical Sciences, 4, Springer Verlag, Berlin, 1990, 1-136. doi: 10.1007/978-3-662-06793-2.

[5]

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

[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), 535-609. doi: 10.1016/j.aim.2007.05.014.

[7]

N. Dairbekov and G. Uhlmann, Reconstructing the metric and magnetic field from the scattering relation, Inverse Probl. Imaging, 4 (2010), 397-409. doi: 10.3934/ipi.2010.4.397.

[8]

M. Dunajski, Solitons, Instantons, and Twistors, Oxford Graduate Texts in Mathematics, 19, Oxford University Press, Oxford, 2010.

[9]

N. J. Hitchin, G. B. Segal and R. S. Ward, Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces, Oxford Graduate Texts in Mathematics, 4, The Clarendon Press, Oxford University Press, New York, 1997.

[10]

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.

[11]

S. Kobayashi, Differential Geometry of Complex Vector Bundles, Publications of the Mathematical Society of Japan 15, Kanô Memorial Lectures 5, Princeton University Press, Princeton, NJ; Iwanami Shoten, Tokyo, 1987. doi: 10.1515/9781400858682.

[12]

V. V. Kozlov, Calculus of variations in the large and classical mechanics, Uspekhi Mat. Nauk, 40 (1985), 33-60, 237.

[13]

N. Manton and P. Sutcliffe, Topological Solitons, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511617034.

[14]

L. J. Mason and N. M. J. Woodhouse, Integrability, Self-duality, and Twistor Theory, London Mathematical Society Monographs, New Series, 15, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996.

[15]

R. Michel, Sur la rigidité imposée par la longueur des géodésiques, Invent. Math., 65 (1981), 71-83. doi: 10.1007/BF01389295.

[16]

S. P. Novikov, Variational methods and periodic solutions of equations of Kirchhoff type. II, (Russian) Funktsional. Anal. i Prilozhen, 15 (1981), 37-52, 96.

[17]

S. P. Novikov, Hamiltonian formalism and a multivalued analogue of Morse theory, (Russian) Uspekhi Mat. Nauk, 37 (1982), 3-49, 248.

[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), 54-66.

[19]

G. P. Paternain, Transparent connections over negatively curved surfaces, J. Mod. Dyn., 3 (2009), 311-333. doi: 10.3934/jmd.2009.3.311.

[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, J. Lewowicz, S. Newhouse), Pitman Research Notes in Math., 362, Longman, Harlow, 1996, 132-145.

[21]

G. P. Paternain, M. Salo and G. Uhlmann, Spectral rigidity and invariant distributions on Anosov surfaces, J. Diff. Geom., 98 (2014), 147-181.

[22]

G. P. Paternain, M. Salo, G. Uhlmann, On the range of the attenuated ray transform for unitary connections, Int. Math. Res. Not., (2015), 873-897. doi: 10.1093/imrn/rnt228.

[23]

G. P. Paternain, M. Salo and G. Uhlmann, Tensor tomography on surfaces, Invent. Math., 193 (2013), 229-247. doi: 10.1007/s00222-012-0432-1.

[24]

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.

[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), 4331-4347. doi: 10.1155/S1073792804142116.

[26]

L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math., 161 (2005), 1093-1110. doi: 10.4007/annals.2005.161.1093.

[27]

E. Powell, Boundary Rigidity, unpublished draft, 2014.

[28]

M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces, J. Diff. Geom., 88 (2011), 161-187.

[29]

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

[30]

M. E. Taylor, Partial Differential Equations I. Basic Theory, Second edition, Applied Mathematical Sciences, 115, Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8.

[31]

F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, 1967.

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