Geometrics Analysis Seminar Archives

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