Mathematische Annalen

, Volume 363, Issue 1–2, pp 305–362 | Cite as

Invariant distributions, Beurling transforms and tensor tomography in higher dimensions

  • Gabriel P. Paternain
  • Mikko Salo
  • Gunther Uhlmann


In the recent articles Paternain et al. (J. Differ Geom, 98:147–181, 2014, Invent Math 193:229–247, 2013), a number of tensor tomography results were proved on two-dimensional manifolds. The purpose of this paper is to extend some of these methods to manifolds of any dimension. A central concept is the surjectivity of the adjoint of the geodesic ray transform, or equivalently the existence of certain distributions that are invariant under the geodesic flow. We prove that on any Anosov manifold, one can find invariant distributions with controlled first Fourier coefficients. The proof is based on subelliptic type estimates and a Pestov identity. We present an alternative construction valid on manifolds with nonpositive curvature, based on the fact that a natural Beurling transform on such manifolds turns out to be essentially a contraction. Finally, we obtain uniqueness results in tensor tomography both on simple and Anosov manifolds that improve earlier results by assuming a condition on the terminator value for a modified Jacobi equation.



M.S. was supported in part by the Academy of Finland and an ERC Starting Grant (grant agreement no 307023), and G.U. was partly supported by NSF and a Simons Fellowship. The authors would like to express their gratitude to the Banff International Research Station (BIRS) for providing an excellent research environment via the Research in Pairs program and the workshop Geometry and Inverse Problems, where part of this work was carried out. We are also grateful to Hanming Zhou for several corrections to earlier drafts, and to Joonas Ilmavirta for helping with a numerical calculation. Finally we thank the referee for numerous suggestions that improved the presentation.


  1. 1.
    Anantharaman, N., Zelditch, S., Patterson-Sullivan distributions and quantum ergodicity, Ann. Henri Poincaré 8: 361–426, MR2314452. Zbl 1187, 81175 (2007)Google Scholar
  2. 2.
    Astala, K., Iwaniec, T., Martin, G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton University Press, Princeton (2009)MATHGoogle Scholar
  3. 3.
    Ballmann, W., Brin, M., Burns, K.: On surfaces with no conjugate points. J. Differ. Geom. 25, 249–273 (1987)MATHMathSciNetGoogle Scholar
  4. 4.
    Contreras, G., Gambaudo, J.-M., Iturriaga, R., Paternain, G.P.: The asymptotic Maslov index and its applications. Ergod. Theory Dynam. Syst. 23, 1415–1443 (2003)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Contreras, G., Iturriaga, R.: Convex Hamiltonians without conjugate points. Ergod. Theory Dynam. Syst. 19, 901–952 (1999)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Croke, C., Sharafutdinov, V.A.: Spectral rigidity of a compact negatively curved manifold. Topology 37, 1265–1273 (1998)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Dairbekov, N.S.: Integral geometry problem for nontrapping manifolds. Inverse Problems 22, 431–445 (2006)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Dairbekov, N.S., Paternain, G.P.: Rigidity properties of Anosov optical hypersurfaces. Ergod. Theory Dynam. Syst. 28, 707–737 (2008)MATHMathSciNetGoogle Scholar
  9. 9.
    Dairbekov, N.S., Paternain, G.P.: On the cohomological equation of magnetic flows. Mat. Contemp. 34, 155–193 (2008)MATHMathSciNetGoogle Scholar
  10. 10.
    Dairbekov, N.S., Sharafutdinov, V.A.: Some problems of integral geometry on Anosov manifolds. Ergod. Theory Dynam. Syst. 23, 59–74 (2003)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Dairbekov, N.S., Sharafutdinov, V.A.: On conformal Killing symmetric tensor fields on Riemannian manifolds. Sib. Adv. Math. 21, 1–41 (2011)CrossRefGoogle Scholar
  12. 12.
    Dairbekov, N.S., Uhlmann, G.: Reconstructing the metric and magnetic field from the scattering relation. Inverse Probl. Imag. 4, 397–409 (2010)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    de la Llave, R., Marco, J.M., Moriyón, R.: Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation. Ann. Math. 123, 537–611 (1986)MATHCrossRefGoogle Scholar
  14. 14.
    Dos Santos Ferreira, D., Kenig, C.E., Salo, M., Uhlmann, G.: Uhlmann, limiting carleman weights and anisotropic inverse problems. Invent. Math. 178, 119–171 (2009)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Duistermaat, J.J., Hörmander, L.: Fourier integral operators II. Acta Math. 128, 183–269 (1972)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Eberlein, P.: Geodesic flows on negatively curved manifolds. I. Ann. Math. 95, 492–510 (1972)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Eberlein, P.: When is a geodesic flow of Anosov type? I. J. Differ. Geom. 8, 437–463 (1973)MATHMathSciNetGoogle Scholar
  18. 18.
    Green, L.W.: A theorem of E. Hopf. Mich. Math. J. 5, 31–34 (1958)MATHCrossRefGoogle Scholar
  19. 19.
    Guillarmou, C.: Invariant distributions and X-ray transforms for Anosov flows. arXiv:1408.4732
  20. 20.
    Guillarmou, C.: Lens rigidity for manifolds with hyperbolic trapped set. arXiv:1412.1760
  21. 21.
    Guillemin, V., Kazhdan, D.: Some inverse spectral results for negatively curved 2-manifolds. Topology 19, 301–312 (1980)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Guillemin, V., Kazhdan, D.: On the cohomology of certain dynamical systems. Topology 19, 291–299 (1980)MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Guillemin, V., Kazhdan, D.: Some inverse spectral results for negatively curved n-manifolds, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), pp. 153–180,In: Proceedings of symposium Pure Mathematics, XXXVI, Amer. Math. Soc., Providence, R.I., (1980)Google Scholar
  24. 24.
    Hopf, E.: Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung. Ber. Verh. Sächs. Akad. Wiss. Leipzig 91, 261–304 (1939)MathSciNetGoogle Scholar
  25. 25.
    Hedenmalm, H.: The Beurling operator for the hyperbolic plane. Ann. Acad. Scient. Fenn. Math. 37, 3–18 (2012)MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Hörmander, L.: The analysis of linear partial differential operators, vol. I–IV, Springer-Verlag, Berlin Heidelberg (1983–1985)Google Scholar
  27. 27.
    Iwaniec, T., Martin, G.: Geometric function theory and non-linear analysis. Oxford University Press, Oxford (2001)Google Scholar
  28. 28.
    Iturriaga, R.: A geometric proof of the existence of the Green bundles. Proc. Am. Math. Soc. 130, 2311–2312 (2002)MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Jost, J.: Riemannian geometry and geometric analysis, 4th edn. Springer, Berlin Heidelberg (2005)MATHGoogle Scholar
  30. 30.
    Katok, A.: Cocycles, cohomology and combinatorial constructions in ergodic theory. In collaboration with E. A. Robinson, Jr. In: Proceedings of symposium Pure Mathematics, 69, Smooth ergodic theory and its applications (Seattle, WA, : 107–173 1999). Amer. Math. Soc, Providence, RI (2001)Google Scholar
  31. 31.
    Katok, A., Hasselblatt, B.: Introduction to the modern theory of dynamical systems. With a supplementary chapter by Katok and Leonardo Mendoza. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge (1995)Google Scholar
  32. 32.
    Klingenberg, W.: Riemannian manifolds with geodesic flow of Anosov type. Ann. Math. 99, 1–13 (1974)MATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Knieper, G.: Hyperbolic dynamics and Riemannian geometry. Handbook of dynamical systems, Vol. 1A, 453–545, North-Holland, Amsterdam (2002)Google Scholar
  34. 34.
    Li, X.: On the weak \(L^p\)-Hodge decomposition and Beurling-Ahlfors transforms on complete Riemannian manifolds. Probab. Theory Relat. Fields 150, 111–144 (2011)MATHCrossRefGoogle Scholar
  35. 35.
    Michel, R.: Sur la rigidité imposée par la longueur des géodésiques. Invent. Math. 65, 71–83 (1981)MATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Paternain, G.P.: Geodesic flows, Progress in Mathematics 180, Birkhäuser (1999)Google Scholar
  37. 37.
    Paternain, G.P., Salo, M., Uhlmann, G.: The attenuated ray transform for connections and Higgs fields. Geom. Funct. Anal. 22, 1460–1489 (2012)MATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Paternain, G.P., Salo, M., Uhlmann, G.: Tensor tomography on simple surfaces. Invent. Math. 193, 229–247 (2013)MATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Paternain, G.P., Salo, M., Uhlmann, G.: On the range of the attenuated ray transform for unitary connections, Int. Math. Res. Not. (to appear)Google Scholar
  40. 40.
    Paternain, G.P., Salo, M., Uhlmann, G.: Tensor tomography: progress and challenges. Chin. Ann. Math. Ser. B 35, 399–428 (2014)MATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Paternain, G.P., Salo, M., Uhlmann, G.: Spectral rigidity and invariant distributions on Anosov surfaces. J. Differ. Geom. 98, 147–181 (2014)MATHMathSciNetGoogle Scholar
  42. 42.
    Pestov, L.: Well-Posedness Questions of the Ray Tomography Problems, (Russian). Siberian Science Press, Novosibirsk (2003)Google Scholar
  43. 43.
    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
  44. 44.
    Pestov, L., Uhlmann, G.: Two dimensional compact simple Riemannian manifolds are boundary distance rigid. Ann. Math. 161, 1089–1106 (2005)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Reid, W.T.: Ordinary Differential Equations. Wiley, New York (1971)MATHGoogle Scholar
  46. 46.
    Ruggiero, R.O.: On the creation of conjugate points. Math. Z. 208, 41–55 (1991)MATHMathSciNetCrossRefGoogle Scholar
  47. 47.
    Sharafutdinov, V.A.: Integral Geometry of Tensor Fields. Inverse and Ill-posed Problems Series. VSP, Utrecht (1994)CrossRefGoogle Scholar
  48. 48.
    Sharafutdinov, V.A.: Ray transform on Riemannian manifolds, Eight lectures on integral geometry.
  49. 49.
    Sharafutdinov, V.A., Skokan, M., Uhlmann, G.: Regularity of ghosts in tensor tomography. J. Geom. Anal. 15, 517–560 (2005)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Sharafutdinov, V.A., Uhlmann, G.: On deformation boundary rigidity and spectral rigidity of Riemannian surfaces with no focal points. J. Differ. Geom. 56, 93–110 (2000)MATHMathSciNetGoogle Scholar
  51. 51.
    Salo, M., Uhlmann, G.: The attenuated ray transform on simple surfaces. J. Differ. Geom. 88, 161–187 (2011)MATHMathSciNetGoogle Scholar
  52. 52.
    Stefanov, P., Uhlmann, G.: Boundary rigidity and stability for generic simple metrics. J. Am. Math. Soc. 18, 975–1003 (2005)MATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    Stefanov, P., Uhlmann, G., Vasy, A.: Inverting the local geodesic X-ray transform on tensors. arXiv:1410.5145

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Gabriel P. Paternain
    • 1
  • Mikko Salo
    • 2
  • Gunther Uhlmann
    • 3
    • 4
  1. 1.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK
  2. 2.Department of Mathematics and StatisticsUniversity of JyväskyläJyväskyläFinland
  3. 3.Department of MathematicsUniversity of WashingtonSeattleUSA
  4. 4.University of HelsinkiHelsinkiFinland

Personalised recommendations