## Journal of Differential Geometry

### Spectral rigidity and invariant distributions on Anosov surfaces

#### Abstract

This article considers inverse problems on closed Riemannian surfaces whose geodesic flow is Anosov. We prove spectral rigidity for any Anosov surface and injectivity of the geodesic ray transform on solenoidal 2-tensors. We also establish surjectivity results for the adjoint of the geodesic ray transform on solenoidal tensors. The surjectivity results are of independent interest and imply the existence of many geometric invariant distributions on the unit sphere bundle. In particular, we show that on any Anosov surface $(M, g)$, given a smooth function $f$ on $M$, there is a distribution in the Sobolev space $H^{-1}(SM)$ that is invariant under the geodesic flow and whose projection to $M$ is the given function $f$.

#### Article information

Source
J. Differential Geom., Volume 98, Number 1 (2014), 147-181.

Dates
First available in Project Euclid: 23 July 2014

https://projecteuclid.org/euclid.jdg/1406137697

Digital Object Identifier
doi:10.4310/jdg/1406137697

Mathematical Reviews number (MathSciNet)
MR3263517

Zentralblatt MATH identifier
1304.37021

#### Citation

Paternain, Gabriel P.; Salo, Mikko; Uhlmann, Gunther. Spectral rigidity and invariant distributions on Anosov surfaces. J. Differential Geom. 98 (2014), no. 1, 147--181. doi:10.4310/jdg/1406137697. https://projecteuclid.org/euclid.jdg/1406137697