# Geodesic ray transform with matrix weights for piecewise constant functions

## TL;DR

This paper proves the injectivity of the geodesic ray transform with matrix weights for piecewise constant functions on certain manifolds, extending previous unweighted results without requiring conjugate point assumptions.

## Contribution

It establishes injectivity of the weighted geodesic ray transform for piecewise constant functions on compact, nontrapping manifolds with minimal assumptions, including higher dimensions.

## Key findings

- Injectivity holds for continuous matrix weights.
- Results extend unweighted transform cases.
- Applicable in dimensions three and higher with foliation condition.

## Abstract

We show injectivity of the geodesic X-ray transform on piecewise constant functions when the transform is weighted by a continuous matrix weight. The manifold is assumed to be compact and nontrapping of any dimension, and in dimension three and higher we assume a foliation condition. We make no assumption regarding conjugate points or differentiability of the weight. This extends recent results for unweighted transforms.