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?



Connection between the Hodge laplacian and the Laplace operator
A rather short proof can be found [here](http://en.wikipedia.org/wiki/Proofs_involving_the_Laplace%E2%80%93Beltrami_operator).


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](http://en.wikipedia.org/wiki/Weitzenb%C3%B6ck_identity).


