The elastic Dirichlet-to-Neumann map determines the real-analytic metric up to isometry, and its heat trace expansion gives explicit spectral invariants such as boundary volume and total mean curvature.
Artin, Galois theory , Notre Dame Univ., Indiana, 1942
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Determination of isometric real-analytic metric and spectral invariants for elastic Dirichlet-to-Neumann map on Riemannian manifolds
The elastic Dirichlet-to-Neumann map determines the real-analytic metric up to isometry, and its heat trace expansion gives explicit spectral invariants such as boundary volume and total mean curvature.