# Local rigidity of manifolds with hyperbolic cusps I. Linear theory and microlocal tools

## TL;DR

This paper establishes the linear spectral rigidity of manifolds with hyperbolic cusps by proving the injectivity of the X-ray transform on symmetric solenoidal tensors, extending rigidity results to non-compact settings.

## Contribution

It develops microlocal tools to invert pseudodifferential operators on Sobolev and H"older-Zygmund spaces, enabling analysis of spectral rigidity in manifolds with hyperbolic cusps.

## Key findings

- Proves injectivity of the X-ray transform on symmetric solenoidal 2-tensors.
- Extends microlocal calculus to invert pseudodifferential operators on advanced function spaces.
- Establishes spectral rigidity for compactly supported deformations of hyperbolic cusp manifolds.

## Abstract

This paper is the first in a series of two articles whose aim is to extend a recent result of Guillarmou-Lefeuvre on the local rigidity of the marked length spectrum from the case of compact negatively-curved Riemannian manifolds to the case of manifolds with hyperbolic cusps. In this first paper, we deal with the linear (or infinitesimal) version of the problem and prove that such manifolds are spectrally rigid for compactly supported deformations. More precisely, we prove that the X-ray transform on symmetric solenoidal 2-tensors is injective. In order to do so, we expand the microlocal calculus developed by Bonthonneau and Bonthonneau-Weich to be able to invert pseudodifferential operators on Sobolev and H\"older-Zygmund spaces modulo compact remainders. This theory has an interest on its own and is extensively used in the second paper [arXiv:1910.02154] in order to deal with the nonlinear problem.