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 Calderón problem for connections on a cylinder.
"Inverse problems for the connection Laplacian." J. Differential Geom. 110 (3) 457 - 494, November 2018. https://doi.org/10.4310/jdg/1542423627