Laplace–Beltrami operator

Summary

The Laplace–Beltrami operator is the Laplacian of a Riemannian manifold: the divergence of the gradient of a function, both taken with respect to the metric. It is a second-order elliptic operator that depends only on the metric, not on coordinates. Its eigenvalues and eigenfunctions form the subject of spectral geometry, and it governs heat flow, waves and harmonic functions on manifolds.