Operators on Riemannian Manifolds: Hodge Laplacian, Laplace-Beltrami Laplacian, the Bochner Laplacian, and Weitzenböck Formulae

Abstract

The Laplacian is a very important operator because it shows up in many of the equations used in physics to describe natural phenomena such as heat diffusion or wave propagation. Therefore, it is highly desirable to generalize the Laplacian to functions defined on a manifold. Furthermore, in the late 1930s, Georges de Rham (inspired by Elie Cartan) realized that it was fruitful to define a version of the Laplacian operating on differential forms, because of a fundamental and almost miraculous relationship between harmonics forms (those in the kernel of the Laplacian) and the de Rham cohomology groups on a (compact, orientable) smooth manifold. Indeed, as we will see in Section 8.6, for every cohomology group \(H^k_{\mathrm {DR}}(M)\), every cohomology class \([\omega ]\in H^k_{\mathrm {DR}}(M)\) is represented by a unique harmonic k-form ω. The connection between analysis and topology lies deep and has many important consequences. For example, Poincare duality follows as an “easy” consequence of the Hodge theorem.