Hodge Laplacian

Summary

The Hodge Laplacian is the operator $\Delta = d\delta + \delta d$ on differential forms of a Riemannian manifold, built from the exterior derivative $d$ and its adjoint $\delta$. On functions it is the Laplace–Beltrami operator, up to sign convention. It is elliptic, self-adjoint and nonnegative, and on a closed manifold its kernel, the harmonic forms, represents the de Rham cohomology. This is the content of Hodge theory, which links analysis on a manifold with its topology.