Spectral geometry studies which properties of a Riemannian manifold can be recovered from the spectrum of geometric differential operators, especially the Laplace-Beltrami operator. Explicit spherical harmonics, Fourier series on a flat torus, and isospectral manifolds illustrate three different aspects of the problem.
Spectral geometry is a field of mathematics that studies the relationship between the geometric properties of a manifold (a mathematical space that locally resembles Euclidean space) and the spectra of differential operators defined on that manifold, particularly the Laplace operator. Essentially, it connects the shape and structure of a geometric space to the eigenvalues and eigenfunctions of these operators.
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