nLab Laplace operator



Riemannian geometry

Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

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



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular 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}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)






(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 \,,


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).


(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) \,,




  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}


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.


Textbook accounts:

See also:

Last revised on May 6, 2020 at 11:33:35. See the history of this page for a list of all contributions to it.