the Hodge Laplacian
/ HOJ /
The ordinary Laplacian acts on functions. But on a manifold the natural objects to differentiate and integrate are differential forms — the things you wedge and integrate over surfaces and volumes. The Hodge Laplacian is the right second-order operator for forms: it extends the Laplacian from functions (0-forms) to forms of every degree, and its 'kernel' — the forms it annihilates — turns out to be the perfect, canonical representatives of cohomology classes. It is the analytic heart of Hodge theory.
On a compact oriented Riemannian manifold the exterior derivative d: Omega^k -> Omega^{k+1} has a formal adjoint d^* (the codifferential, built from d and the Hodge star *), going Omega^k -> Omega^{k-1}. The Hodge Laplacian on k-forms is Delta = d d^* + d^* d. A form omega is harmonic when Delta omega = 0; on a compact manifold this is equivalent, by integration by parts, to omega being both closed (d omega = 0) and coclosed (d^* omega = 0) simultaneously. On functions d^* of a 0-form is zero, so Delta reduces to d^* d, matching the Laplace-Beltrami operator (up to sign convention). The crucial theorem is that every de Rham cohomology class has exactly one harmonic representative — the unique form of least norm in the class — so harmonic forms are a finite-dimensional model of cohomology realized by an elliptic PDE. This is why dim of the space of harmonic k-forms equals the k-th Betti number.
The Hodge Laplacian is the operator behind the Hodge decomposition and the whole bridge from analysis to topology on manifolds; it underlies Hodge theory on Kahler manifolds, the Bochner technique, and index theory. Two honest cautions. First, the Hodge Laplacian is not the same as the rough (connection) Laplacian nabla^* nabla even though both are 'Laplacians on forms' — they differ by a curvature term, and that difference is precisely the content of the Bochner-Weitzenbock formula; never substitute one for the other. Second, the clean statement 'each cohomology class has a unique harmonic representative' needs compactness (and a chosen metric): on a noncompact manifold harmonic forms can fail to be L^2 or to represent cohomology, and the harmonic representative depends on the metric even though the cohomology class does not.
On a flat 2-torus the harmonic 1-forms are exactly the constant-coefficient forms a dx + b dy; there is a two-dimensional space of them, matching the first Betti number b_1 = 2 of the torus and the two independent loops you can wind around.
Harmonic 1-forms on the torus realize its first cohomology as a 2-dimensional space.
On functions the Hodge Laplacian and Laplace-Beltrami operator agree, but on higher forms they are tied to curvature through Bochner-Weitzenbock and differ from the rough Laplacian; treating 'the Laplacian on forms' as unambiguous is a mistake.