I know of two ways of defining the (negative - depending on your convention) Laplace-Beltrami operator on the differential forms of a compact, orientable Riemannian manifold $M$.
- The Levi-Civita connection extends to a connection tensor bundles by Leibniz rule $\nabla(a\otimes b)=\nabla a\otimes b + (-1)^aa\otimes\nabla b$ (and similarly for wedges) and by defining it on $1$-forms by $$(\nabla\alpha)(X,Y) = \nabla_X\alpha(Y)-\nabla_Y\alpha(X)-\alpha([X,Y])$$ (is this correct?). In particular, we have a connection $\nabla:\Omega^k(M)\to\Gamma(M,T^*M\otimes\Lambda^kM)$ and another $\nabla:\Gamma(M,T^*M\otimes\Lambda^kM)\to\Gamma(M,T^*M^{\otimes 2}\otimes\Lambda^kM)$. We can now concatenate them and take the negative of the trace with respect to the metric $$\Delta=-tr_g(\nabla\nabla).$$
- Using Hodge theory, we can define $\Delta=-(dd^\star+d^\star d)$.
Is there an easy way to see whether these two definitions are equal (possibly without computing in coordinates)? A reference where it is done would be awesome!
The two Laplacians are not the same. One is the other plus a curvature term. The formula is known as the Weitzenböck formula. It is surprisingly difficult to calculate. It and its proof can be found in this note of Petersen.