## Journal of Differential Geometry

### Invariant distributions and X-ray transform for Anosov flows

Colin Guillarmou

#### Abstract

For Anosov flows preserving a smooth measure on a closed manifold $\mathcal{M}$, we define a natural self-adjoint operator $\Pi$ which maps into the space of flow invariant distributions in $\cap_{r \lt 0} H^r (\mathcal{M})$ and whose kernel is made of coboundaries in $\cup_{s \gt 0} H^s (\mathcal{M})$. We describe relations to the Livsic theorem and recover regularity properties of cohomological equations using this operator. For Anosov geodesic flows on the unit tangent bundle $\mathcal{M}= SM$ of a compact manifold $\mathcal{M}$, we apply this theory to study X-ray transform on symmetric tensors on $\mathcal{M}$. In particular, we prove existence of flow invariant distributions on $SM$ with prescribed push-forward on $\mathcal{M}$ and a similar version for tensors. This allows us to show injectivity of the X-ray transform on an Anosov surface: any divergence-free symmetric tensor on $\mathcal{M}$ which integrates to $0$ along all closed geodesics is zero.

#### Article information

Source
J. Differential Geom., Volume 105, Number 2 (2017), 177-208.

Dates
Received: 1 December 2014
First available in Project Euclid: 8 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1486522813

Digital Object Identifier
doi:10.4310/jdg/1486522813

Mathematical Reviews number (MathSciNet)
MR3606728

Zentralblatt MATH identifier
1372.37059

#### Citation

Guillarmou, Colin. Invariant distributions and X-ray transform for Anosov flows. J. Differential Geom. 105 (2017), no. 2, 177--208. doi:10.4310/jdg/1486522813. https://projecteuclid.org/euclid.jdg/1486522813