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{{distinguish|Beltrami operator}}
In [[differential geometry]], the [[Laplace operator]], named after [[Pierre-Simon Laplace]], can be generalized to operate on functions defined on [[surface]]s in [[Euclidean space]] and, more generally, on [[Riemannian manifold|Riemannian]] and [[pseudo-Riemannian manifold]]s. This more general operator goes by the name '''Laplace–Beltrami operator''', after Laplace and [[Eugenio Beltrami]]. Like the Laplacian, the Laplace–Beltrami operator is defined as the [[divergence]] of the [[gradient]], and is a [[linear 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 on [[differential forms]] using the divergence and [[exterior derivative]]. The resulting operator is called the '''Laplace–de Rham operator''' (named after [[Georges de Rham]]).

The Laplace–Beltrami operator, like the Laplacian, is the [[divergence]] of the [[gradient]]:
:<math>\Delta f = \operatorname{div}\operatorname{grad} f.</math>
An explicit formula in [[local coordinates]] is possible.

Suppose first that ''M'' is an [[oriented manifold|oriented]] [[Riemannian manifold]]. The orientation allows one to specify a definite [[volume form]] on ''M'', given in an oriented coordinate system ''x''''i'' by

:<math>\mathrm{vol}_n := \sqrt{|g|} \;dx^1\wedge \ldots \wedge dx^n</math>

where the ''dxi'' are the [[1-form]]s forming the [[dual basis]] to the basis vectors
:<math>\partial_i := \frac {\partial}{\partial x^i}</math>
and <math>\wedge</math> is the [[wedge product]]. Here {{nowrap|1={{!}}''g''{{!}} := {{!}}det(''gij''){{!}}}} is the [[absolute value]] of the [[determinant]] of the [[metric tensor]] ''g''''ij''. The divergence div ''X'' of a vector field ''X'' on the manifold is then defined as the scalar function with the property

:<math>
(\mbox{div} X) \; \mathrm{vol}_n := L_X \mathrm{vol}_n
</math>

where ''LX'' is the [[Lie derivative]] along the [[vector field]] ''X''. In local coordinates, one obtains

:<math>
\mbox{div} X = \frac{1}{\sqrt{|g|}} \partial_i \left(\sqrt {|g|} X^i\right)
</math>

where the [[Einstein notation]] is implied, so that the repeated index ''i'' is summed over. The gradient of a scalar function ƒ is the vector field grad ''f'' that may be defined through the [[inner product]] <math>\langle\cdot,\cdot\rangle</math> on the manifold, as

:<math>\langle \mbox{grad} f(x) , v_x \rangle = df(x)(v_x)</math>

for all vectors ''vx'' anchored at point ''x'' in the [[tangent space]] ''TxM'' of the manifold at point ''x''. Here, ''d''ƒ is the [[exterior derivative]] of the function ƒ; it is a 1-form taking argument ''vx''. In local coordinates, one has

:<math> \left(\mbox{grad} f\right)^i =
\partial^i f = g^{ij} \partial_j f</math>

where ''gij'' are the components of the inverse of the metric tensor, so that {{nowrap|1=''gijgjk'' = δ''i''''k''}} with δ''i''''k'' the [[Kronecker delta]].

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

:<math>\Delta f = \operatorname{div}\;\operatorname{grad} f =
\frac{1}{\sqrt {|g|}} \partial_i \left(\sqrt{|g|} g^{ij} \partial_j f \right).</math>

If ''M'' is not oriented, then the above calculation carries through exactly as presented, except that the volume form must instead be replaced by a [[volume element]] (a [[density on a manifold|density]] 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.

==Formal self-adjointness==

The exterior derivative ''d'' and −div are formal adjoints, in the sense that for ''ƒ'' a compactly supported function

:<math>\int_M df(X) \;\mathrm{vol}_n = - \int_M f \mathrm{div} X \;\mathrm{vol}_n </math>     [[Laplace–Beltrami operator/Proofs|(proof)]]

where the last equality is an application of [[Stokes' theorem]]. Dualizing gives

{{NumBlk|:|<math>\int_M f \,\Delta h\,\mathrm{vol}_n = -\int_M \langle df, dh\rangle\, \mathrm{vol}_n</math>|{{EquationRef|2}}}}

for all compactly supported functions ''ƒ'' and ''h''. Conversely, ({{EquationRef|2}}) characterizes Δ 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 ƒ and ''h'',

:<math>\int_M f\,\Delta h \;\mathrm{vol}_n = -\int_M \langle d f, d h \rangle \;\mathrm{vol}_n = \int_M h\,\Delta f \;\mathrm{vol}_n.</math>

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

==Tensor Laplacian==

The Laplace–Beltrami operator can be written using the [[trace of a matrix|trace]] of the iterated [[covariant derivative]] associated with the Levi-Civita connection. From this perspective, let ''X''i be a basis of tangent vector fields (not necessarily induced by a coordinate system). Then the '''[[Hessian matrix|Hessian]]''' of a function ''f'' is the symmetric 2-tensor whose components are given by

:<math>H(f)_{ij}=H_f(X_i, X_j) =\nabla_{X_i}\nabla_{X_j} f - \nabla_{\nabla_{X_i}X_j} f</math>

This is easily seen to transform tensorially, since it is linear in each of the arguments ''X''i, ''X''j. The Laplace–Beltrami operator is then the trace of the Hessian with respect to the metric:

:<math>\Delta f = \sum_{ij} g^{ij} H(f)_{ij}.</math>

In [[abstract indices]], the operator is often written

:<math>\Delta f = \nabla^a \nabla_a f</math>

provided it is understood implicitly that this trace is in fact the trace of the Hessian ''tensor''.

Because the covariant derivative extends canonically to arbitrary [[tensor]]s, the Laplace–Beltrami operator defined on a tensor ''T'' by
:<math>\Delta T = g^{ij}\left( \nabla_{X_i}\nabla_{X_j} T - \nabla_{\nabla_{X_i}X_j} T\right)</math>
is well-defined.

==Laplace–de Rham operator==
More generally, one can define a Laplacian [[differential operator]] on sections of the bundle of [[differential form]]s on a [[pseudo-Riemannian manifold]]. On a [[Riemannian manifold]] it is an [[elliptic operator]], while on a [[Lorentzian manifold]] it is [[hyperbolic operator|hyperbolic]]. The Laplace–de Rham operator is defined by

:<math>\Delta= \mathrm{d}\delta+\delta\mathrm{d} = (\mathrm{d}+\delta)^2,\;</math>

where d is the [[exterior derivative]] or differential and δ is the [[codifferential]], acting as {{nowrap|1=(−1)''kn''+''n''+1&lowast;''d''&lowast;}} on ''k''-forms where ∗ is the [[Hodge star]].

When computing Δƒ for a scalar function ƒ, we have δƒ = 0, so that

:<math>\Delta f = \delta \, df. \,</math>

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 [[Laplace–Beltrami operator/Proofs|proof]] for details. On functions, the Laplace–de Rham operator is actually the negative of the Laplace–Beltrami operator, as the conventional normalization of the [[codifferential]] assures that the Laplace–de Rham operator is (formally) [[positive definite]], whereas the Laplace–Beltrami operator is typically negative. The sign is a pure convention, however, 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 a [[Weitzenböck identity]] that explicitly involves the [[Ricci curvature tensor]].

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

;Euclidean space
In the usual (orthonormal) [[Cartesian coordinates]] ''x''''i'' on [[Euclidean space]], the metric is reduced to the Kronecker delta, and one therefore has <math>|g| = 1</math>. Consequently, in this case

:<math>\Delta f = \frac{1}{\sqrt{|g|}} \partial_i \sqrt{|g|}\partial^i f = \partial_i \partial^i f</math>

which is the ordinary Laplacian. In [[curvilinear coordinates]], such as [[spherical coordinates|spherical]] or [[cylindrical coordinates]], one obtains [[Laplacian#Coordinate expressions|alternative expressions]].

Similarly, the Laplace–Beltrami operator corresponding to the [[Minkowski metric]] with [[metric signature|signature]] (−+++) is the [[D'Alembertian]].

;Spherical Laplacian
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 into '''R'''''n'' as the unit sphere centred at the origin. Then for a function ''ƒ'' on ''S''''n''−1, the spherical Laplacian is defined by
:<math>\Delta_{S^{n-1}}f(x) = \Delta f(x/|x|)</math>
where ''ƒ''(''x''/|''x''|) is the degree zero homogeneous extension of the function ''ƒ'' to '''R'''''n'' − {0}, and Δ 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:
:<math>\Delta f = r^{1-n}\frac{\partial}{\partial r}\left(r^{n-1}\frac{\partial f}{\partial r}\right) + r^{-2}\Delta_{S^{n-1}}f.</math>

More generally, one can formulate a similar trick using the [[normal 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 a [[normal coordinates|normal coordinate system]]. Let (''t'', ''ξ'') be spherical coordinates on the sphere with respect to a particular point ''p'' of the sphere (the "north pole"), that is geodesic polar coordinates with respect to ''p''. Here ''t'' represents the latitude measurement along a unit speed geodesic from ''p'', and ''ξ'' a parameter representing the choice of direction of the geodesic in ''S''''n''−1. Then the spherical Laplacian has the form:
:<math>\Delta_{S^{n-1}} f(t,\xi) = \sin^{2-n}t \frac{\partial}{\partial t}\left(\sin^{n-2}t\frac{\partial f}{\partial t}\right) + \sin^{-2}t\Delta_\xi f</math>
where <math>\Delta_\xi</math> is the Laplace–Beltrami operator on the ordinary unit (''n'' − 2)-sphere.

;Hyperbolic space
A similar technique works in [[hyperbolic space]]. Here the hyperbolic space ''H''''n''−1 can be embedded into the ''n'' dimensional [[Minkowski space]], a real vector space equipped with the quadratic form
:<math>q(x) = x_1^2 - x_2^2-\cdots - x_n^2.</math>
Then ''H''''n'' is the subset of the future null cone in Minkowski space given by
:<math>H^n = \{ x | q(x) = 1, x_1>1\}. \, </math>
Then
:<math>\Delta_{H^{n-1}} f = \Box f\left(x/q(x)^{1/2}\right)|_{H^{n-1}}</math>
Here <math>f(x/q(x)^{1/2})</math> is the degree zero homogeneous extension of ''f'' to the interior of the future null cone and □ is the [[wave operator]]
:<math>\Box = \frac{\partial^2}{\partial x_1^2} - \cdots - \frac{\partial^2}{\partial x_n^2}.</math>

The operator can also be written in polar coordinates. Let (''t'', ''ξ'') be spherical coordinates on the sphere with respect to a particular point ''p'' of ''H''''n''−1 (say, the center of the [[Poincaré disc]]). Here ''t'' represents the hyperbolic distance from ''p'' and ''ξ'' a parameter representing the choice of direction of the geodesic in ''S''''n''−1. Then the spherical Laplacian has the form:
:<math>\Delta_{H^{n-1}} f(t,\xi) = \sinh^{2-n}t \frac{\partial}{\partial t}\left(\sinh^{n-2}t\frac{\partial f}{\partial t}\right) + \sinh^{-2}t\Delta_\xi f</math>
where <math>\Delta_\xi</math> is the Laplace–Beltrami operator on the ordinary unit (''n'' − 2)-sphere.

== See also ==
* [[Laplacian operators in differential geometry]]

== References ==
* {{citation|first=Isaac|last=Chavel|title=Eigenvalues in Riemannian Geometry|publisher=Academic Press|year=1984|volume=115|edition=2nd|series=Pure and Applied Mathematics|isbn=978-0-12-170640-1}}.
* {{citation|last=Flanders|first=Harley|title=Differential forms with applications to the physical sciences|publisher=Dover|year=1989|isbn=978-0-486-66169-8}}
* {{citation|first=Jürgen|last=Jost|title=Riemannian Geometry and Geometric Analysis|year=2002|publisher=Springer-Verlag|publication-place=Berlin|isbn=3-540-42627-2}}.
*{{eom|id=l/l057450|title=Laplace–Beltrami equation|first=E.D.|last= Solomentsev|first2=E.V.|last2= Shikin}}

{{DEFAULTSORT:Laplace-Beltrami operator}}
[[Category:Differential operators]]
[[Category:Riemannian geometry]]

[[de:Verallgemeinerter_Laplace-Operator#Laplace-Beltrami-Operator]]

Complex convexity - Revision history

en>David Eppstein: source

en>David Eppstein: unreferenced