Two dimensional compact simple Riemannian manifolds are boundary distance rigid by Leonid Pestov and Gunther Uhlmann (2005)

By Leonid Pestov, Gunther Uhlmann.

Primary paper: annals.math.princeton.edu/wp-content/uploads/annals-v161-n2-p12.pdf

Annals of Mathematics 161, 1093–1110. Introduction and Theorem 1.1: boundary distances determine a simple compact surface metric up to a boundary-fixing diffeomorphism.