Laplace–Beltrami operator

Remarks

In local coordinates $x^1, \dots, x^n$, with $g_{ij}$ the metric, $g^{ij}$ its inverse and $\abs{g} = \det(g_{ij})$, $$\Delta f = \frac{1}{\sqrt{\abs{g}}} \sum_{i,j} \partial_i \bigl(\sqrt{\abs{g}} \, g^{ij} \partial_j f\bigr).$$ Equivalently $\Delta f = \sum_{i,j} g^{ij} \bigl(\partial_i \partial_j f - \sum_k \Gamma^k_{ij} \partial_k f\bigr)$, with $\Gamma^k_{ij}$ the Christoffel symbols.

Example: on the unit sphere $S^2$ with coordinates $(\theta, \varphi)$ and metric $d\theta^2 + \sin^2\theta \, d\varphi^2$, one has $\sqrt{\abs{g}} = \sin\theta$ and $$\Delta f = \frac{1}{\sin\theta} \partial_\theta(\sin\theta \, \partial_\theta f) + \frac{1}{\sin^2\theta} \partial_\varphi^2 f.$$ For $f = \cos\theta$ this gives $\frac{1}{\sin\theta} \partial_\theta(-\sin^2\theta) = -2\cos\theta$.

Non-example: on $\R^2 \setminus \set{0}$ in polar coordinates $(r, \theta)$, the operator $\partial_r^2 + \partial_\theta^2$ is not the Laplace–Beltrami operator of the Euclidean metric $dr^2 + r^2 d\theta^2$. The formula above gives $\Delta f = \partial_r^2 f + \frac{1}{r} \partial_r f + \frac{1}{r^2} \partial_\theta^2 f$.

With this sign convention $\Delta$ is the usual Laplacian $\sum_i \partial_i^2$ on $\R^n$ and $-\Delta$ has nonnegative spectrum on a compact manifold. Many geometers use the opposite sign, $\Delta = -\operatorname{div} \operatorname{grad}$. On functions, $-\Delta$ equals the Hodge Laplacian $\delta d$.