Positive Laplace-Beltrami operator (${\Delta }_{+}=-{\mathrm{div}}_g{\mathrm{grad}}_g$)

The positive convention for the Laplace-Beltrami operator is ${\Delta }_{+}f=\delta df$, where $\delta$ is the codifferential. It is the restriction of the Hodge Laplacian to functions. In coordinates,On a compact boundaryless Riemannian manifold, or for compactly supported functions, $⟨f,{\Delta }_{+}f⟩=\Vert df{\Vert }^2\ge 0$. The opposite $\mathrm{div}\mathrm{grad}$ convention is also standard; formulas involving the sign must specify which is used.

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.