Recovery of a time-dependent potential in hyperbolic equations on conformally transversally anisotropic manifolds

TL;DR

This paper addresses the inverse problem of uniquely recovering a time-dependent potential in wave equations on specific Riemannian manifolds, using partial boundary data under certain geometric conditions.

Contribution

It establishes the unique determination of time-dependent potentials in conformally transversally anisotropic manifolds from partial Cauchy data, assuming injectivity of the attenuated geodesic ray transform.

Findings

Proves unique recovery of potentials under specified geometric conditions.

Extends inverse problem results to time-dependent potentials in higher-dimensional manifolds.

Utilizes properties of the attenuated geodesic ray transform for the proof.

Abstract

We study an inverse problem of determining a time-dependent potential appearing in the wave equation in conformally transversally anisotropic manifolds of dimension three or higher. These are compact Riemannian manifolds with boundary that are conformally embedded in a product of the real line and a transversal manifold. Under the assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove the unique determination of time-dependent potentials from the knowledge of a certain partial Cauchy data set.