 Positive Laplace-Beltrami operator

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.