Spectral geometry is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators.
One of the earliest results of this kind was due to Hermann Weyl who used David Hilbert's theory of integral equation in 1911 to show that the volume of a bounded domain in Euclidean space can be determined from the asymptotic behavior of the eigenvalues for the Dirichlet boundary value problem of the Laplace operator.
However as the example given by John Milnor tells us, the information of eigenvalues is not enough to determine the isometry class of a manifold (see isospectral).
A general and systematic method due to Toshikazu Sunada gave rise to a plethora of such examples which clarifies the phenomenon of isospectral manifolds.
Direct problems attempt to infer the behavior of the eigenvalues of a Riemannian manifold from knowledge of the geometry.