Harmonic Space

A harmonic space is a Riemannian manifold in which the Laplacian operator admits a complete system of eigenfunctions with desirable symmetry properties — a generalization of the familiar Fourier modes on Euclidean space to curved geometries. Arthur Walker contributed to their classification, showing how harmonic structure constrains the global geometry of a manifold. Harmonic spaces connect the local spectral behavior of the Laplacian to global topological constraints, and they appear in the study of symmetric spaces, quantum mechanical configuration spaces, and the spectral geometry underlying hearing the shape of a drum.