  We obtain the asymptotic expansions of the traces of the thermoelastic
operators with the Dirichlet and Neumann boundary conditions on a Riemannian
manifold, and give an effective method to calculate all the coefficients of the
asymptotic expansions. These coefficients provide precise geometric
information. In particular, we explicitly calculate the first two coefficients
concerning the volumes of the manifold and its boundary. As an application, by
combining our results with the isoperimetric inequality we show that an
$n$-dimensional geodesic ball is uniquely determined up to isometry by its
thermoelastic spectrum among all bounded thermoelastic bodies with boundary.
