nLab Laplace operator

Idea

(..)

Definition

Definition

(Laplace-Beltrami operator)

Given a Riemannian manifold $(X,g)$, the Laplace-Beltrami operator $\Delta$ is the differential operator on the space of smooth functions $f \in C^\infty(X) = \Omega^0(X)$ given by the formula

(1)$$\Delta f \;\coloneqq\; \star d \star d f \,,$$

where

• $d \;\colon\; \Omega^\bullet(X) \to \Omega^{\bullet + 1}(X)$ is the de Rham differential (depending only on the smooth manifold $X$, not on the metric tensor);

• $\star \;\colon\; \Omega^\bullet(X) \to \Omega^{dim(X)-\bullet}(X)$ is the Hodge star operator of $(X,g)$ (this is where the dependence on the metric tensor enters).

The same formula makes sense more generally for pseudo-Riemannian manifolds. Even so, in the pseudo-Riemannian case one tends to speak of the wave operator instead of the Laplace operator, and to use the symbol $\Box$ instead of $\Delta$ (at least for flat pseudo-Riemannian manifolds: Minkowski spacetime).

Proposition

(coordinate-expression of Laplace operator)

If $U \subset X$ is a chart of $X$ with coordinate functions $\{x^i \colon U \to \mathbb{R}\}$, then the Laplace operator (1) is equivalently given by the following component-expression:

$$\Delta f_{\vert U} \;=\; \frac{ sgn(g) }{ \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } } \partial_i \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{i j} \partial_j f \right) \,,$$

Here:

• $sgn(g)$ is the signature of the metric tensor, hence

  • $sgn(g) = +1$ for Riemannian manifolds,

  • $sgn(g) = -1$ for pseudo-Riemannian manifolds;

• $\partial_i \coloneqq \frac{\partial}{\partial x^i}$ denotes the partial derivative by the coordinate $x^i$;

• $(g_{i j})$ is the square matrix of components of the metric tensor $g$ in the given coordinate chart, hence such that

  $$g \;=\; g_{i j} d x^i \otimes d x^j$$

• $det\big( (g_{i j}) \big)$ is the determinant of this matrix,

  $\left\vert det\big( (g_{i j}) \big)\right\vert$ its absolute value (taking this is irrelevant for Riemannian manifolds but necessary for pseudo-Riemannian manifolds, where the determinan of the metric is negative),

  $\sqrt{\left\vert det\big( (g_{i j}) \big)\right\vert}$ is the positive square root of that;

• $(g^{i j})$ is the corresponding inverse matrix;

• the Einstein summation convention is used throughout, meaing that a sum over repeated indices is understood.

Proof

Using

1. the component formula for the de Rham differential $d = d x^i \partial_i$,

2. the component formula for the Hodge star operator (see there)

we compute as follows:

$$\begin{aligned} \star d \star d f & = \star d \star (\partial_j f) d x^j \\ & = \star d \left( \tfrac{1}{ \color{green} (D-1)! } \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } \, g^{ i j} (\partial_j f) \, \epsilon_{ i {\color{green} k_2 \cdots k_{D} } } d x^{ \color{green} k_2 } \wedge \cdots \wedge d x^{ \color{green} k_{D} } \right) \\ & = \star \partial_{ \color{magenta} k_1} \left( \tfrac{1}{ \color{green} (D-1)! } \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } \, g^{i j} (\partial_j f) \, \epsilon_{ i {\color{green} k_2 \cdots k_{D} } } d x^{ \color{magenta} k_1 } \wedge d x^{ \color{green} k_2 } \wedge \cdots \wedge d x^{ \color{green} k_{D} } \right) \\ & = \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } \underset{ = \det\big( (g_{i j})^{-1} \big) \delta^{ \color{magenta} k_1 }_i }{ \underbrace{ \tfrac{1}{ { \color{orange} D! } { \color{green} (D-1)! } } \epsilon_{ \color{orange} l_1 l_2 \cdots l_D } g^{ { \color{orange} l_1 } { \color{magenta} k_1 } } g^{ { \color{orange} l_2 } { \color{green} k_2 } } \cdots g^{ { \color{orange} l_D} { \color{green} k_D } } \epsilon_{ i {\color{green} k_2 \cdots k_{D} } } } } \, \partial_{ \color{magenta} k_1 } \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{i j} (\partial_j f) \right) \\ & = \frac{1}{ \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } } \delta^{ \color{magenta} k_1 }_i \partial_{ \color{magenta} k_1 } \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{i j} (\partial_j f) \right) \\ & = \frac{ sgn(g) }{ \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } } \partial_{i} \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{i j} (\partial_j f) \right) \end{aligned}$$

Properties

Functional determinant and Analytic torsion

The functional determinant of Laplace operator on a given space of differential p-forms appears as factor of the analytic torsion of the given Riemannian manifold.

Related entries

• heat kernel

• harmonic function

• harmonic map

• harmonic form, Hodge theory

• hearing the shape of a drum

• BV-Laplacian

• discrete Laplace operator

References

Textbook accounts:

• Nicole Berline, Ezra Getzler, Michele Vergne, Heat kernels and Dirac operators, Grundlehren 298, Springer 1992, “Text Edition” 2003.

See also:

• Wikipedia, Laplace operator, Laplace-Beltrami operator

• MathOverflow: why-is-the-laplacian-ubiquitous