Let M a riemannian manifold. How can I show that the hodgelaplaceoperator of a function $f$ is the negative of the laplaceoperator?
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A rather short proof can be found here. I assume you are interested in the case when $f$ is a scalar function. Otherwise the Hodge Laplacian differs from the Laplace–Beltrami operator not only by a sign due to the Ricci curvature. See the Weitzenböck identity. 

