## Analysis & PDE

• Anal. PDE
• Volume 9, Number 8 (2016), 1903-1930.

### Invariant distributions and the geodesic ray transform

#### Abstract

We establish an equivalence principle between the solenoidal injectivity of the geodesic ray transform acting on symmetric $m$-tensors and the existence of invariant distributions or smooth first integrals with prescribed projection over the set of solenoidal $m$-tensors. We work with compact simple manifolds, but several of our results apply to nontrapping manifolds with strictly convex boundary.

#### Article information

Source
Anal. PDE, Volume 9, Number 8 (2016), 1903-1930.

Dates
Received: 18 November 2015
Revised: 28 July 2016
Accepted: 12 September 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843378

Digital Object Identifier
doi:10.2140/apde.2016.9.1903

Mathematical Reviews number (MathSciNet)
MR3599521

Zentralblatt MATH identifier
1357.53088

#### Citation

Paternain, Gabriel P.; Zhou, Hanming. Invariant distributions and the geodesic ray transform. Anal. PDE 9 (2016), no. 8, 1903--1930. doi:10.2140/apde.2016.9.1903. https://projecteuclid.org/euclid.apde/1510843378

#### References

• Y. 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. 5:6 (1997), 487–490.
• N. S. Dairbekov and V. A. Sharafutdinov, “Conformal Killing symmetric tensor fields on Riemannian manifolds”, Mat. Tr. 13:1 (2010), 85–145. In Russian; translated in Siberian Advances in Mathematics 21:1 (2011), 1–41.
• N. Dairbekov and G. Uhlmann, “Reconstructing the metric and magnetic field from the scattering relation”, Inverse Probl. Imaging 4:3 (2010), 397–409.
• E. Delay, “Smooth compactly supported solutions of some underdetermined elliptic PDE, with gluing applications”, Comm. Partial Differential Equations 37:10 (2012), 1689–1716.
• C. Guillarmou, “Invariant distributions and X-ray transform for Anosov flows”, preprint, 2014. To appear in J. Differential Geom.
• V. Guillemin and D. Kazhdan, “Some inverse spectral results for negatively curved $2$-manifolds”, Topology 19:3 (1980), 301–312.
• V. Guillemin and D. Kazhdan, “Some inverse spectral results for negatively curved $n$-manifolds”, pp. 153–180 in Geometry of the Laplace operator (Honolulu, HI, 1979), Proc. Sympos. Pure Math. 36, Amer. Math. Soc., Providence, RI, 1980.
• Q. Han, A basic course in partial differential equations, Graduate Studies in Mathematics 120, American Mathematical Society, Providence, RI, 2011.
• S. Helgason, The Radon transform, 2nd ed., Progress in Mathematics 5, Birkhäuser, Boston, 1999.
• T. Kato, M. Mitrea, G. Ponce, and M. Taylor, “Extension and representation of divergence-free vector fields on bounded domains”, Math. Res. Lett. 7:5–6 (2000), 643–650.
• B. Kruglikov and V. S. Matveev, “The geodesic flow of a generic metric does not admit nontrivial integrals polynomial in momenta”, Nonlinearity 29:6 (2016), 1755–1768.
• W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, 2000.
• R. G. Muhometov, “The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry”, Dokl. Akad. Nauk SSSR 232:1 (1977), 32–35. In Russian.
• G. P. Paternain, M. Salo, and G. Uhlmann, “The attenuated ray transform for connections and Higgs fields”, Geom. Funct. Anal. 22:5 (2012), 1460–1489.
• G. P. Paternain, M. Salo, and G. Uhlmann, “Tensor tomography on surfaces”, Invent. Math. 193:1 (2013), 229–247.
• G. P. Paternain, M. Salo, and G. Uhlmann, “Spectral rigidity and invariant distributions on Anosov surfaces”, J. Differential Geom. 98:1 (2014), 147–181.
• G. P. Paternain, M. Salo, and G. Uhlmann, “Invariant distributions, Beurling transforms and tensor tomography in higher dimensions”, Math. Ann. 363:1–2 (2015), 305–362.
• G. P. Paternain, M. Salo, and G. Uhlmann, “On the range of the attenuated ray transform for unitary connections”, Int. Math. Res. Not. 2015:4 (2015), 873–897.
• L. N. Pestov and V. A. Sharafutdinov, “Integral geometry of tensor fields on a manifold of negative curvature”, Sibirsk. Mat. Zh. 29:3 (1988), 114–130. In Russian; translated in Siberian Mathematical Journal 29:3 (1988), 427–441.
• L. Pestov and G. Uhlmann, “Two dimensional compact simple Riemannian manifolds are boundary distance rigid”, Ann. of Math. $(2)$ 161:2 (2005), 1093–1110.
• M. Salo and G. Uhlmann, “The attenuated ray transform on simple surfaces”, J. Differential Geom. 88:1 (2011), 161–187.
• V. A. Sharafutdinov, Integral geometry of tensor fields, VSP, Utrecht, 1994.
• V. Sharafutdinov, “Ray transform on Riemannian manifolds”, lecture notes, University of Oulu, 1999, hook http://math.nsc.ru/~sharafutdinov/files/Lectures.pdf \posturlhook.
• V. A. Sharafutdinov, “An integral geometry problem in a nonconvex domain”, Sibirsk. Mat. Zh. 43:6 (2002), 1430–1442. In Russian; translated in Siberian Math. J. 43:6 (2002), 1159–1168.
• V. Sharafutdinov, M. Skokan, and G. Uhlmann, “Regularity of ghosts in tensor tomography”, J. Geom. Anal. 15:3 (2005), 499–542.
• P. Stefanov and G. Uhlmann, “Stability estimates for the X-ray transform of tensor fields and boundary rigidity”, Duke Math. J. 123:3 (2004), 445–467.
• P. Stefanov and G. Uhlmann, “Boundary rigidity and stability for generic simple metrics”, J. Amer. Math. Soc. 18:4 (2005), 975–1003.
• P. Stefanov and G. Uhlmann, “Integral geometry on tensor fields on a class of non-simple Riemannian manifolds”, Amer. J. Math. 130:1 (2008), 239–268.
• P. Stefanov, G. Uhlmann, and A. Vasy, “Inverting the local geodesic X-ray transform on tensors”, preprint, 2014. To appear in J. Anal. Math.
• G. Uhlmann and A. Vasy, “The inverse problem for the local geodesic ray transform”, Invent. Math. 205:1 (2016), 83–120. With an appendix by H. Zhou.