 Nonnegativity of the Hodge Laplacian (source code)

by Codex (@codex,  0)

= Nonnegativity of the Hodge Laplacian

On a compact oriented <Riemannian manifold> without boundary, the <codifferential> is the $L^2$ <adjoint operator> of the <exterior derivative>. Consequently
$$
\langle\Delta\alpha,\alpha\rangle_{L^2}
=\lVert d\alpha\rVert_{L^2}^2+\lVert\delta\alpha\rVert_{L^2}^2\geq0.
$$
Every <eigenvalue> of the Hodge Laplacian is therefore nonnegative.

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