Product rule for the positive Laplace-Beltrami operator (${\Delta }_{+}(fh)=f{\Delta }_{+}h+h{\Delta }_{+}f-2⟨df,dh{⟩}_g$)

For a function $f$ and one-form $\alpha$, the codifferential obeys $\delta (f\alpha )=f\delta \alpha -⟨df,\alpha {⟩}_g$. This follows from the graded Leibniz rule and the definition of the Hodge star operator. Applying it to $d(fh)=fdh+hdf$ gives the product formula. In particular ${\Delta }_{+}(f^2)=2f{\Delta }_{+}f-2|df{|}_g^2$. The cross-term sign reverses for the opposite Laplace-Beltrami operator convention.