Classical Hodge Decomposition Theorem

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.

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