Friday, January 16, 2015 - 10:00 to 10:50

Thack 703

### Abstract or Additional Information

I follow the proof in Chapter 6 of Frank Warner's *Foundations of **Differential Manifolds and Lie Groups* of the Hodge Decomposition Theorem. To state the theorem, we take an elegant generalization of the Laplacian operator "Delta" to the Laplace-Beltrami operator on *p*-forms on Riemannian manifolds. The decomposition theorem states that this space of p-forms orthogonally decomposes into the kernel and image of the Laplace-Beltrami operator. The short but pretty proof uses the ellipticity of the Laplace Beltrami operator together with a Rellich-Kondrachov-type theorem and the Hahn Banach theorem.