# nLab Laplace operator

Contents

### Context

#### Riemannian geometry

Riemannian geometry

## Applications

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

(..)

## 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

• $\partial_i \coloneqq \frac{\partial}{\partial x^i}$ 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.

## References

Textbook accounts: