Let M a riemannian manifold. How can I show that the hodge-laplace-operator of a function $f$ is the negative of the laplace-operator?
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.