nLab
Laplace operator

Contents

Context

Riemannian geometry

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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}{}& ʃ &\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

Idea

(..)

Definition

Definition

(Laplace-Beltrami operator)

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

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

where

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 Δ\Delta (at least for flat pseudo-Riemannian manifolds: Minkowski spacetime).

Proposition

(coordinate-expression of Laplace operator)

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

Δf |U=sgn(g)|det((g ij))| i(|det((g ij))|g ij jf), \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:

Proof

Using

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

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

we compute as follows:

ddf =d( jf)dx j =d(1(D1)!|det((g ij))|g ij( jf)ϵ ik 2k Ddx k 2dx k D) = k 1(1(D1)!|det((g ij))|g ij( jf)ϵ ik 2k Ddx k 1dx k 2dx k D) =|det((g ij))|1D!(D1)!ϵ l 1l 2l Dg l 1k 1g l 2k 2g l Dk Dϵ ik 2k D=det((g ij) 1)δ i k 1 k 1(|det((g ij))|g ij( jf)) =1|det((g ij))|δ i k 1 k 1(|det((g ij))|g ij( jf)) =sgn(g)|det((g ij))| i(|det((g ij))|g ij( jf)) \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:

See also:

Last revised on May 6, 2020 at 07:33:35. See the history of this page for a list of all contributions to it.