Indifferential geometry, theLaplace–Beltrami operator is a generalization of theLaplace operator to functions defined onsubmanifolds inEuclidean space and, even more generally, onRiemannian andpseudo-Riemannian manifolds. It is named afterPierre-Simon Laplace andEugenio Beltrami.

For any twice-differentiable real-valued functionf defined on Euclidean spaceRⁿ, the Laplace operator (also known as theLaplacian) takesf to thedivergence of itsgradient vector field, which is the sum of then pure second derivatives off with respect to each vector of anorthonormal basis forRⁿ. Like the Laplacian, the Laplace–Beltrami operator is defined as the divergence of the gradient, and is alinear operator taking functions into functions. The operator can be extended to operate on tensors as the divergence of the covariant derivative. Alternatively, the operator can be generalized to operate ondifferential forms using the divergence andexterior derivative. The resulting operator is called the Laplace–de Rham operator (named afterGeorges de Rham).

The Laplace–Beltrami operator, like the Laplacian, is the (Riemannian)divergence of the (Riemannian)gradient:

${\displaystyle \mathrm{\Delta }f=\mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{\nabla }f).}$

An explicit formula inlocal coordinates is possible.

Suppose first thatM is anorientedRiemannian manifold. The orientation allows one to specify a definitevolume form onM, given in an oriented coordinate systemxⁱ by

${\displaystyle {\mathrm{vol}}_n:=\sqrt{|g|}d{x}^1\wedge \cdots \wedge d{x}^n}$

where|g| := |det(g_ij)| is theabsolute value of thedeterminant of themetric tensor, and thedxⁱ are the1-forms forming thedual frame to the frame

${\displaystyle {\mathrm{\partial }}_i:=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^i}}$

of the tangent bundle${\displaystyle TM}$ and${\displaystyle \wedge }$ is thewedge product.

The divergence of a vector field${\displaystyle X}$ on the manifold is then defined as the scalar function${\displaystyle \mathrm{\nabla }\cdot X}$ with the property

${\displaystyle (\mathrm{\nabla }\cdot X){\mathrm{vol}}_n:=L_X{\mathrm{vol}}_n}$

whereL_X is theLie derivative along thevector fieldX. In local coordinates, one obtains

${\displaystyle \mathrm{\nabla }\cdot X=\frac{1}{\sqrt{|g|}}{\mathrm{\partial }}_i\left(\sqrt{|g|}{X}^i\right)}$

where here and below theEinstein notation is implied, so that the repeated indexi is summed over.

The gradient of a scalar function ƒ is the vector field gradf that may be defined through theinner product${\displaystyle ⟨\cdot ,\cdot ⟩}$ on the manifold, as

${\displaystyle ⟨\mathrm{grad}f(x),v_x⟩=df(x)(v_x)}$

for all vectorsv_x anchored at pointx in thetangent spaceT_xM of the manifold at pointx. Here,dƒ is theexterior derivative of the function ƒ; it is a 1-form taking argumentv_x. In local coordinates, one has

${\displaystyle {\left(\mathrm{grad}f\right)}^i={\mathrm{\partial }}^{i}f=g^{ij}{\mathrm{\partial }}_{j}f}$

whereg^ij are the components of the inverse of themetric tensor, so thatg^ijg_jk = δⁱ_k with δⁱ_k theKronecker delta.

Combining the definitions of the gradient and divergence, the formula for the Laplace–Beltrami operator applied to a scalar function ƒ is, in local coordinates

${\displaystyle \mathrm{\Delta }f=\frac{1}{\sqrt{|g|}}{\mathrm{\partial }}_i\left(\sqrt{|g|}{g}^{ij}{\mathrm{\partial }}_{j}f\right).}$

IfM is not oriented, then the above calculation carries through exactly as presented, except that the volume form must instead be replaced by avolume element (adensity rather than a form). Neither the gradient nor the divergence actually depends on the choice of orientation, and so the Laplace–Beltrami operator itself does not depend on this additional structure.

The exterior derivative${\displaystyle d}$ and${\displaystyle -\mathrm{\nabla }}$ are formal adjoints, in the sense that for a compactly supported function${\displaystyle f}$

${\displaystyle {\int }_{M}df(X){\mathrm{vol}}_n=-{\int }_{M}f\mathrm{\nabla }\cdot X{\mathrm{vol}}_n}$     (proof)

where the last equality is an application ofStokes' theorem. Dualizing gives

${\displaystyle {\int }_{M}f\mathrm{\Delta }h{\mathrm{vol}}_n=-{\int }_M⟨df,dh⟩{\mathrm{vol}}_n}$

2

for all compactly supported functions${\displaystyle f}$ and${\displaystyle h}$. Conversely, (2) characterizes the Laplace–Beltrami operator completely, in the sense that it is the only operator with this property.

As a consequence, the Laplace–Beltrami operator is negative and formally self-adjoint, meaning that for compactly supported functions${\displaystyle f}$ and${\displaystyle h}$,

${\displaystyle {\int }_{M}f\mathrm{\Delta }h{\mathrm{vol}}_n=-{\int }_M⟨df,dh⟩{\mathrm{vol}}_n={\int }_{M}h\mathrm{\Delta }f{\mathrm{vol}}_n.}$

Because the Laplace–Beltrami operator, as defined in this manner, is negative rather than positive, often it is defined with the opposite sign.

Let M denote a compact Riemannian manifold without boundary. We want to consider the eigenvalue equation,

${\displaystyle -\mathrm{\Delta }u=\lambda u,}$

where${\displaystyle u}$ is theeigenfunction associated with the eigenvalue${\displaystyle \lambda }$. It can be shown using the self-adjointness proved above that the eigenvalues${\displaystyle \lambda }$ are real. The compactness of the manifold${\displaystyle M}$ allows one to show that the eigenvalues are discrete and furthermore, the vector space of eigenfunctions associated with a given eigenvalue${\displaystyle \lambda }$, i.e. theeigenspaces are all finite-dimensional. Notice by taking the constant function as an eigenfunction, we get${\displaystyle \lambda =0}$ is an eigenvalue. Also since we have considered${\displaystyle -\mathrm{\Delta }}$ an integration by parts shows that${\displaystyle \lambda \ge 0}$. More precisely if we multiply the eigenvalue equation through by the eigenfunction${\displaystyle u}$ and integrate the resulting equation on${\displaystyle M}$ we get (using the notation${\displaystyle dV={\mathrm{vol}}_n}$):

${\displaystyle -{\int }_M\mathrm{\Delta }uudV=\lambda {\int }_M{u}^{2}dV}$

Performing an integration by parts or what is the same thing as using thedivergence theorem on the term on the left, and since${\displaystyle M}$ has no boundary we get

${\displaystyle -{\int }_M\mathrm{\Delta }uudV={\int }_M|\mathrm{\nabla }u{|}^{2}dV}$

Putting the last two equations together we arrive at

${\displaystyle {\int }_M|\mathrm{\nabla }u{|}^{2}dV=\lambda {\int }_M{u}^{2}dV}$

We conclude from the last equation that${\displaystyle \lambda \ge 0}$.

A fundamental result ofAndré Lichnerowicz^[1] states that: Given a compactn-dimensional Riemannian manifold with no boundary with${\displaystyle n\ge 2}$. Assume theRicci curvature satisfies the lower bound:

${\displaystyle \mathrm{Ric}(X,X)\ge \kappa g(X,X),\kappa >0,}$

where${\displaystyle g(\cdot ,\cdot )}$ is the metric tensor and${\displaystyle X}$ is any tangent vector on the manifold${\displaystyle M}$. Then the first positive eigenvalue${\displaystyle {\lambda }_1}$ of the eigenvalue equation satisfies the lower bound:

${\displaystyle {\lambda }_1\ge \frac{n}{n-1}\kappa .}$

This lower bound is sharp and achieved on the sphere${\displaystyle {\mathbb{S}}^n}$. In fact on${\displaystyle {\mathbb{S}}^2}$ the eigenspace for${\displaystyle {\lambda }_1}$ is three dimensional and spanned by the restriction of the coordinate functions${\displaystyle x_1,x_2,x_3}$ from${\displaystyle {\mathbb{R}}^3}$ to${\displaystyle {\mathbb{S}}^2}$. Using spherical coordinates${\displaystyle (\theta ,\varphi )}$, on${\displaystyle {\mathbb{S}}^2}$ the two dimensional sphere, set

${\displaystyle x_3=\cos \varphi =u_1,}$

we see easily from the formula for the spherical Laplacian displayed below that

${\displaystyle -{\mathrm{\Delta }}_{{\mathbb{S}}^2}{u}_1=2u_1}$

Thus the lower bound in Lichnerowicz's theorem is achieved at least in two dimensions.

Conversely it was proved byMorio Obata,^[2] that if then-dimensional compact Riemannian manifold without boundary were such that for the first positive eigenvalue${\displaystyle {\lambda }_1}$ one has,

${\displaystyle {\lambda }_1=\frac{n}{n-1}\kappa ,}$

then the manifold is isometric to then-dimensional sphere${\displaystyle {\mathbb{S}}^n(\sqrt{\frac{n-1}{\kappa }})}$, the sphere of radius${\displaystyle \sqrt{\frac{n-1}{\kappa }}}$. Proofs of all these statements may be found in the book by Isaac Chavel.^[3] Analogous sharp bounds also hold for other Geometries and for certain degenerate Laplacians associated with these geometries like theKohn Laplacian (afterJoseph J. Kohn) on a compactCR manifold. Applications there are to the global embedding of such CR manifolds in${\displaystyle {\mathbb{C}}^n.}$^[4]

The Laplace–Beltrami operator can be written using thetrace (or contraction) of the iteratedcovariant derivative associated with theLevi-Civita connection. TheHessian (tensor) of a function${\displaystyle f}$ is the symmetric 2-tensor

${\displaystyle {\displaystyle \text{Hess}f\in \mathbf{\Gamma }({\mathsf{T}}^{\ast }M\otimes {\mathsf{T}}^{\ast }M)}}$,${\displaystyle \text{Hess}f:={\mathrm{\nabla }}^{2}f\equiv \mathrm{\nabla }\mathrm{\nabla }f\equiv \mathrm{\nabla }\mathrm{d}f}$,

wheredf denotes the(exterior) derivative of a functionf.

LetX_i be a basis of tangent vector fields (not necessarily induced by a coordinate system). Then the components ofHess f are given by

${\displaystyle (\text{Hess}f{)}_{ij}=\text{Hess}f(X_i,X_j)={\mathrm{\nabla }}_{X_i}{\mathrm{\nabla }}_{X_j}f-{\mathrm{\nabla }}_{{\mathrm{\nabla }}_{X_i}{X}_j}f}$

This is easily seen to transform tensorially, since it is linear in each of the argumentsX_i,X_j. The Laplace–Beltrami operator is then the trace (orcontraction) of the Hessian with respect to the metric:

${\displaystyle {\displaystyle \mathrm{\Delta }f:=\mathrm{t}\mathrm{r}\mathrm{\nabla }\mathrm{d}f\in {\mathsf{C}}^{\mathrm{\infty }}(M)}}$.

More precisely, this means

${\displaystyle {\displaystyle \mathrm{\Delta }f(x)=\sum _{i=1}^n\mathrm{\nabla }\mathrm{d}f(X_i,X_i)}}$,

or in terms of the metric

${\displaystyle \mathrm{\Delta }f=\sum _{ij}{g}^{ij}(\text{Hess}f{)}_{ij}.}$

Inabstract indices, the operator is often written

${\displaystyle \mathrm{\Delta }f={\mathrm{\nabla }}^a{\mathrm{\nabla }}_{a}f}$

provided it is understood implicitly that this trace is in fact the trace of the Hessiantensor.

Because the covariant derivative extends canonically to arbitrarytensors, the Laplace–Beltrami operator defined on a tensorT by

${\displaystyle \mathrm{\Delta }T=g^{ij}\left({\mathrm{\nabla }}_{X_i}{\mathrm{\nabla }}_{X_j}T-{\mathrm{\nabla }}_{{\mathrm{\nabla }}_{X_i}{X}_j}T\right)}$

is well-defined.

More generally, one can define a Laplaciandifferential operator on sections of the bundle ofdifferential forms on apseudo-Riemannian manifold. On aRiemannian manifold it is anelliptic operator, while on aLorentzian manifold it ishyperbolic. TheLaplace–de Rham operator is defined by

${\displaystyle \mathrm{\Delta }=\mathrm{d}\delta +\delta \mathrm{d}=(\mathrm{d}+\delta {)}^2,}$

where d is theexterior derivative or differential andδ is thecodifferential, acting as(−1)^kn+n+1∗d∗ onk-forms, where ∗ is theHodge star. The first order operator${\displaystyle \mathrm{d}+\delta }$ is the Hodge–Dirac operator.^[5]

When computing the Laplace–de Rham operator on a scalar functionf, we haveδf = 0, so that

${\displaystyle \mathrm{\Delta }f=\delta \mathrm{d}f.}$

Up to an overall sign, the Laplace–de Rham operator is equivalent to the previous definition of the Laplace–Beltrami operator when acting on a scalar function; see the proof for details. On functions, the Laplace–de Rham operator is actually the negative of the Laplace–Beltrami operator, as the conventional normalization of thecodifferential assures that the Laplace–de Rham operator is (formally)positive definite, whereas the Laplace–Beltrami operator is typically negative. The sign is merely a convention, and both are common in the literature. The Laplace–de Rham operator differs more significantly from the tensor Laplacian restricted to act on skew-symmetric tensors. Apart from the incidental sign, the two operators differ by aWeitzenböck identity that explicitly involves theRicci curvature tensor.

Many examples of the Laplace–Beltrami operator can be worked out explicitly.

In the usual (orthonormal)Cartesian coordinatesxⁱ onEuclidean space, the metric is reduced to the Kronecker delta, and one therefore has${\displaystyle |g|=1}$. Consequently, in this case

${\displaystyle \mathrm{\Delta }f=\frac{1}{\sqrt{|g|}}{\mathrm{\partial }}_i\sqrt{|g|}{\mathrm{\partial }}^{i}f={\mathrm{\partial }}_i{\mathrm{\partial }}^{i}f}$

which is the ordinary Laplacian. Incurvilinear coordinates, such asspherical orcylindrical coordinates, one obtainsalternative expressions.

Similarly, the Laplace–Beltrami operator corresponding to theMinkowski metric withsignature(− + + +) is thed'Alembertian.

The spherical Laplacian is the Laplace–Beltrami operator on the(n − 1)-sphere with its canonical metric of constant sectional curvature 1. It is convenient to regard the sphere as isometrically embedded intoRⁿ as theunit sphere centred at the origin. Then for a functionf onSⁿ⁻¹, the spherical Laplacian is defined by

${\displaystyle {\mathrm{\Delta }}_{S^{n-1}}f(x)=\mathrm{\Delta }f(x/|x|)}$

wheref(x/|x|) is the degree zero homogeneous extension of the functionf toRⁿ − {0}, and${\displaystyle \mathrm{\Delta }}$ is the Laplacian of the ambient Euclidean space. Concretely, this is implied by the well-known formula for the Euclidean Laplacian in spherical polar coordinates:

${\displaystyle \mathrm{\Delta }f=r^{1-n}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r^{n-1}\frac{\mathrm{\partial }f}{\mathrm{\partial }r}\right)+r^{-2}{\mathrm{\Delta }}_{S^{n-1}}f.}$

More generally, one can formulate a similar trick using thenormal bundle to define the Laplace–Beltrami operator of any Riemannian manifold isometrically embedded as a hypersurface of Euclidean space.

One can also give an intrinsic description of the Laplace–Beltrami operator on the sphere in anormal coordinate system. Let(ϕ,ξ) be spherical coordinates on the sphere with respect to a particular pointp of the sphere (the "north pole"), that is geodesic polar coordinates with respect top. Hereϕ represents the latitude measurement along a unit speed geodesic fromp, andξ a parameter representing the choice of direction of the geodesic inSⁿ⁻¹. Then the spherical Laplacian has the form:

${\displaystyle {\mathrm{\Delta }}_{S^{n-1}}f(\xi ,\varphi )=(\sin \varphi {)}^{2-n}\frac{\mathrm{\partial }}{\mathrm{\partial }\varphi }\left((\sin \varphi {)}^{n-2}\frac{\mathrm{\partial }f}{\mathrm{\partial }\varphi }\right)+(\sin \varphi {)}^{-2}{\mathrm{\Delta }}_{\xi }f}$

where${\displaystyle {\mathrm{\Delta }}_{\xi }}$ is the Laplace–Beltrami operator on the ordinary unit(n − 2)-sphere. In particular, for the ordinary 2-sphere using standard notation for polar coordinates we get:

${\displaystyle {\mathrm{\Delta }}_{S^2}f(\theta ,\varphi )=(\sin \varphi {)}^{-1}\frac{\mathrm{\partial }}{\mathrm{\partial }\varphi }\left(\sin \varphi \frac{\mathrm{\partial }f}{\mathrm{\partial }\varphi }\right)+(\sin \varphi {)}^{-2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\theta }^2}f}$

A similar technique works inhyperbolic space. Here the hyperbolic spaceHⁿ⁻¹ can be embedded into then dimensionalMinkowski space, a real vector space equipped with the quadratic form

${\displaystyle q(x)=x_1^2-x_2^2-\cdots -x_n^2.}$

ThenHⁿ is the subset of the future null cone in Minkowski space given by

${\displaystyle H^n=\{x\mid q(x)=1,x_1>1\}.}$

Then

${\displaystyle {\mathrm{\Delta }}_{H^{n-1}}f={◻f\left(x/q(x{)}^{1/2}\right)|}_{H^{n-1}}}$

Here${\displaystyle f(x/q(x{)}^{1/2})}$ is the degree zero homogeneous extension off to the interior of the future null cone and□ is thewave operator

${\displaystyle ◻=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_1^2}-\cdots -\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_n^2}.}$

The operator can also be written in polar coordinates. Let(t,ξ) be spherical coordinates on the sphere with respect to a particular pointp ofHⁿ⁻¹ (say, the center of thePoincaré disc). Heret represents the hyperbolic distance fromp andξ a parameter representing the choice of direction of the geodesic inSⁿ⁻². Then the hyperbolic Laplacian has the form:

${\displaystyle {\mathrm{\Delta }}_{H^{n-1}}f(t,\xi )=\sinh (t{)}^{2-n}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\sinh (t{)}^{n-2}\frac{\mathrm{\partial }f}{\mathrm{\partial }t}\right)+\sinh (t{)}^{-2}{\mathrm{\Delta }}_{\xi }f}$

where${\displaystyle {\mathrm{\Delta }}_{\xi }}$ is the Laplace–Beltrami operator on the ordinary unit (n − 2)-sphere. In particular, for the hyperbolic plane using standard notation for polar coordinates we get:

${\displaystyle {\mathrm{\Delta }}_{H^2}f(r,\theta )=\sinh (r{)}^{-1}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(\sinh (r)\frac{\mathrm{\partial }f}{\mathrm{\partial }r}\right)+\sinh (r{)}^{-2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\theta }^2}f}$

• Covariant derivative
• Laplacian operators in differential geometry
• Laplace operator

• Flanders, Harley (1989),Differential forms with applications to the physical sciences, Dover,ISBN 978-0-486-66169-8
• Jost, Jürgen (2002),Riemannian Geometry and Geometric Analysis, Berlin: Springer-Verlag,ISBN 3-540-42627-2.
• Solomentsev, E.D.; Shikin, E.V. (2001) [1994],"Laplace–Beltrami equation",Encyclopedia of Mathematics,EMS Press

• Differential operators
• Riemannian geometry

• CS1 maint: multiple names: authors list
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