Inverse problems for the connection Laplacian

TL;DR

This paper addresses inverse problems for the connection Laplacian, reconstructing geometric and bundle data from boundary wave measurements, and extends results to elliptic Calderon problems for connections.

Contribution

It introduces a method to reconstruct a Riemannian manifold and Hermitian bundle with connection from hyperbolic boundary data, up to gauge transformations.

Findings

Reconstruction of manifold and bundle from boundary data

Extension to elliptic Calderon problem for connections

Reconstruction is unique up to gauge transformations

Abstract

We reconstruct a Riemannian manifold and a Hermitian vector bundle with compatible connection from the hyperbolic Dirichlet-to-Neumann operator associated with the wave equation of the connection Laplacian. The boundary data is local and the reconstruction is up to the natural gauge transformations of the problem. As a corollary we derive an elliptic analogue of the main result which solves a Calderon problem for connections on a cylinder.