Laplace-Beltrami Operator

The Laplace-Beltrami operator is a generalization of the ordinary Laplacian differential operator to functions defined on a Riemannian manifold, taking into account the manifold's own metric structure. It is built from the divergence and gradient operators adapted to the manifold and reduces to the familiar Laplacian on flat Euclidean space. The operator is central to the study of harmonic functions, heat diffusion and eigenvalue problems on curved spaces, and its eigenvalues and eigenfunctions describe the natural vibration modes of a manifold, connecting geometry to spectral theory.