# Functions of constant geodesic X-ray transform

## TL;DR

This paper investigates the geometric conditions under which functions with constant geodesic X-ray transform exist, revealing restrictions on the manifold's shape and symmetry, especially in Euclidean domains.

## Contribution

It establishes geometric constraints for manifolds admitting functions with constant geodesic X-ray transform, including boundary conditions and symmetry requirements.

## Key findings

- Boundary must be umbilical for such functions to exist
- In Euclidean domains, the boundary must be a sphere
- Existence of such functions on rotationally symmetric manifolds

## Abstract

We show that the existence of a function in $L^{1}$ with constant geodesic X-ray transform imposes geometrical restrictions on the manifold. The boundary of the manifold has to be umbilical and in the case of a strictly convex Euclidean domain, it must be a ball. Functions with constant geodesic X-ray transform always exist on manifolds with rotational symmetry.