On the characterization of the kernel of the geodesic X-ray transform

Author:
Eduardo Chappa

Journal:
Trans. Amer. Math. Soc. **358** (2006), 4793-4807

MSC (2000):
Primary 58Jxx; Secondary 44A12, 53Cxx

Published electronically:
June 20, 2006

MathSciNet review:
2231872

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Abstract: Let be a compact manifold with boundary. We consider covariant symmetric tensor fields of order two that belong to a Sobolev space . We prove, under the assumption that the metric is simple, that solenoidal tensor fields that belong to the kernel of the geodesic X-ray transform are smooth up to the boundary. As a corollary we obtain that they form a finite-dimensional set in .

**1.****Yu. E. Anikonov***Some Methods for the Study of Multidimensional Inverse Problems*Nauka, Sibirsk Otdel., Novosibirsk (1978).**2.**Yu. E. Anikonov and V. G. Romanov,*On uniqueness of determination of a form of first degree by its integrals along geodesics*, J. Inverse Ill-Posed Probl.**5**(1997), no. 6, 487–490 (1998). MR**1623603**, 10.1515/jiip.1997.5.6.487**3.****I.N. Bernstein, M.L. Gerver***Conditions of distinguishability of metrics by hodographs*Methods and algorithms of interpretation of seismological information. Computerized Seismology. Vol. 13. Nauka, Moscow 50-73 (1980).**4.**Jacques Chazarain and Alain Piriou,*Introduction to the theory of linear partial differential equations*, Studies in Mathematics and its Applications, vol. 14, North-Holland Publishing Co., Amsterdam-New York, 1982. Translated from the French. MR**678605****5.**Louis Boutet de Monvel,*Boundary problems for pseudo-differential operators*, Acta Math.**126**(1971), no. 1-2, 11–51. MR**0407904****6.**Peter B. Gilkey,*The index theorem and the heat equation*, Publish or Perish, Inc., Boston, Mass., 1974. Notes by Jon Sacks; Mathematics Lecture Series, No. 4. MR**0458504****7.**Lars Hörmander,*The analysis of linear partial differential operators. I*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR**717035****8.****R.G. Mukhometov***On the Problem of Integral Geometry*Math. Problems in Geophysics. Akad. Nauk. SSSR, Sibirsk. Otdel., Vychisl. Tsentr, Novosibirsk, 6, No. 2, 212-242 (1975).**9.**R. G. Muhometov,*On a problem of reconstructing Riemannian metrics*, Sibirsk. Mat. Zh.**22**(1981), no. 3, 119–135, 237 (Russian). MR**621466****10.**L. N. Pestov and V. A. Sharafutdinov,*Integral geometry of tensor fields on a manifold of negative curvature*, Sibirsk. Mat. Zh.**29**(1988), no. 3, 114–130, 221 (Russian); English transl., Siberian Math. J.**29**(1988), no. 3, 427–441 (1989). MR**953028**, 10.1007/BF00969652**11.**V. A. Sharafutdinov,*Integral geometry of a tensor field on a manifold with upper-bounded curvature*, Sibirsk. Mat. Zh.**33**(1992), no. 3, 192–204, 221 (Russian, with Russian summary); English transl., Siberian Math. J.**33**(1992), no. 3, 524–533 (1993). MR**1178471**, 10.1007/BF00970902**12.**V. A. Sharafutdinov,*Integral geometry of tensor fields*, Inverse and Ill-posed Problems Series, VSP, Utrecht, 1994. MR**1374572****13.**Vladimir A. Sharafutdinov,*Finiteness theorem for the ray transform on a Riemannian manifold*, Inverse Problems**11**(1995), no. 5, 1039–1050. MR**1353801****14.****V.A. Sharafutdinov***Ray Transform on Riemannian Manifolds*University of Washington (1999).**15.**Vladimir Sharafutdinov and Gunther Uhlmann,*On deformation boundary rigidity and spectral rigidity of Riemannian surfaces with no focal points*, J. Differential Geom.**56**(2000), no. 1, 93–110. MR**1863022****16.****P. Stefanov and G. Uhlmann***Unpublished Notes*, 1998.

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Additional Information

**Eduardo Chappa**

Affiliation:
Department of Mathematics, University of Washington, Seattle, Washington 98195

Address at time of publication:
Department of Mathematical and Physical Sciences, Texas A&M International University, Laredo, Texas 78041-1900

DOI:
https://doi.org/10.1090/S0002-9947-06-04059-1

Received by editor(s):
December 20, 2002

Received by editor(s) in revised form:
August 3, 2004

Published electronically:
June 20, 2006

Additional Notes:
This work was partially supported by NSF grant #DMS-00-70488 and NSF grant #DMS-9705792

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.