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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?

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up vote 3 down vote accepted

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.

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Your link to the Weitzenböck formula is broken, btw. It should be en.wikipedia.org/wiki/Weitzenböck_identity –  Willie Wong Jun 23 '10 at 12:44
    
Thank you, Willie. I stand corrected. –  Andrey Rekalo Jun 23 '10 at 12:48
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