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| {{distinguish|Beltrami operator}}
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| 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]]).
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| The Laplace–Beltrami operator, like the Laplacian, is the [[divergence]] of the [[gradient]]:
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| :<math>\Delta f = \operatorname{div}\operatorname{grad} f.</math>
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| An explicit formula in [[local coordinates]] is possible.
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| 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''<sup>''i''</sup> by
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| :<math>\mathrm{vol}_n := \sqrt{|g|} \;dx^1\wedge \ldots \wedge dx^n</math>
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| where the ''dx<sup>i</sup>'' are the [[1-form]]s forming the [[dual basis]] to the basis vectors
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| :<math>\partial_i := \frac {\partial}{\partial x^i}</math>
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| and <math>\wedge</math> is the [[wedge product]]. Here {{nowrap|1={{!}}''g''{{!}} := {{!}}det(''g<sub>ij</sub>''){{!}}}} is the [[absolute value]] of the [[determinant]] of the [[metric tensor]] ''g''<sub>''ij''</sub>. The divergence div ''X'' of a vector field ''X'' on the manifold is then defined as the scalar function with the property
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| :<math>
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| (\mbox{div} X) \; \mathrm{vol}_n := L_X \mathrm{vol}_n
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| </math>
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| where ''L<sub>X</sub>'' is the [[Lie derivative]] along the [[vector field]] ''X''. In local coordinates, one obtains
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| :<math>
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| \mbox{div} X = \frac{1}{\sqrt{|g|}} \partial_i \left(\sqrt {|g|} X^i\right)
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| </math>
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| 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
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| :<math>\langle \mbox{grad} f(x) , v_x \rangle = df(x)(v_x)</math>
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| for all vectors ''v<sub>x</sub>'' anchored at point ''x'' in the [[tangent space]] ''T<sub>x</sub>M'' of the manifold at point ''x''. Here, ''d''ƒ is the [[exterior derivative]] of the function ƒ; it is a 1-form taking argument ''v<sub>x</sub>''. In local coordinates, one has
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| :<math> \left(\mbox{grad} f\right)^i =
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| \partial^i f = g^{ij} \partial_j f</math>
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| where ''g<sup>ij</sup>'' are the components of the inverse of the metric tensor, so that {{nowrap|1=''g<sup>ij</sup>g<sub>jk</sub>'' = δ<sup>''i''</sup><sub>''k''</sub>}} with δ<sup>''i''</sup><sub>''k''</sub> the [[Kronecker delta]].
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| 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
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| :<math>\Delta f = \operatorname{div}\;\operatorname{grad} f =
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| \frac{1}{\sqrt {|g|}} \partial_i \left(\sqrt{|g|} g^{ij} \partial_j f \right).</math>
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| 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.
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| ==Formal self-adjointness==
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| The exterior derivative ''d'' and −div are formal adjoints, in the sense that for ''ƒ'' a compactly supported function
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| :<math>\int_M df(X) \;\mathrm{vol}_n = - \int_M f \mathrm{div} X \;\mathrm{vol}_n </math> <small>[[Laplace–Beltrami operator/Proofs|(proof)]]</small>
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| where the last equality is an application of [[Stokes' theorem]]. Dualizing gives
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| {{NumBlk|:|<math>\int_M f \,\Delta h\,\mathrm{vol}_n = -\int_M \langle df, dh\rangle\, \mathrm{vol}_n</math>|{{EquationRef|2}}}}
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| 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.
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| As a consequence, the Laplace–Beltrami operator is negative and formally self-adjoint, meaning that for compactly supported functions ƒ and ''h'',
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| :<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>
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| Because the Laplace–Beltrami operator, as defined in this manner, is negative rather than positive, often it is defined with the opposite sign.
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| ==Tensor Laplacian==
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| 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''<sub>i</sub> 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
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| :<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>
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| This is easily seen to transform tensorially, since it is linear in each of the arguments ''X''<sub>i</sub>, ''X''<sub>j</sub>. The Laplace–Beltrami operator is then the trace of the Hessian with respect to the metric:
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| :<math>\Delta f = \sum_{ij} g^{ij} H(f)_{ij}.</math>
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| In [[abstract indices]], the operator is often written
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| :<math>\Delta f = \nabla^a \nabla_a f</math>
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| provided it is understood implicitly that this trace is in fact the trace of the Hessian ''tensor''.
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| Because the covariant derivative extends canonically to arbitrary [[tensor]]s, the Laplace–Beltrami operator defined on a tensor ''T'' by
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| :<math>\Delta T = g^{ij}\left( \nabla_{X_i}\nabla_{X_j} T - \nabla_{\nabla_{X_i}X_j} T\right)</math>
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| is well-defined.
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| ==Laplace–de Rham operator==
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| 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
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| :<math>\Delta= \mathrm{d}\delta+\delta\mathrm{d} = (\mathrm{d}+\delta)^2,\;</math>
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| where d is the [[exterior derivative]] or differential and δ is the [[codifferential]], acting as {{nowrap|1=(−1)<sup>''kn''+''n''+1</sup>∗''d''∗}} on ''k''-forms where ∗ is the [[Hodge star]].
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| When computing Δƒ for a scalar function ƒ, we have δƒ = 0, so that
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| :<math>\Delta f = \delta \, df. \,</math>
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| 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]].
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| == Examples ==
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| Many examples of the Laplace–Beltrami operator can be worked out explicitly.
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| ;Euclidean space
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| In the usual (orthonormal) [[Cartesian coordinates]] ''x''<sup>''i''</sup> on [[Euclidean space]], the metric is reduced to the Kronecker delta, and one therefore has <math>|g| = 1</math>. Consequently, in this case
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| :<math>\Delta f = \frac{1}{\sqrt{|g|}} \partial_i \sqrt{|g|}\partial^i f = \partial_i \partial^i f</math>
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| which is the ordinary Laplacian. In [[curvilinear coordinates]], such as [[spherical coordinates|spherical]] or [[cylindrical coordinates]], one obtains [[Laplacian#Coordinate expressions|alternative expressions]].
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| Similarly, the Laplace–Beltrami operator corresponding to the [[Minkowski metric]] with [[metric signature|signature]] (−+++) is the [[D'Alembertian]].
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| ;Spherical Laplacian
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| 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'''<sup>''n''</sup> as the unit sphere centred at the origin. Then for a function ''ƒ'' on ''S''<sup>''n''−1</sup>, the spherical Laplacian is defined by
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| :<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'''<sup>''n''</sup> − {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:
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| :<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>
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| 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.
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| 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''<sup>''n''−1</sup>. Then the spherical Laplacian has the form:
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| :<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>
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| where <math>\Delta_\xi</math> is the Laplace–Beltrami operator on the ordinary unit (''n'' − 2)-sphere.
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| ;Hyperbolic space
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| A similar technique works in [[hyperbolic space]]. Here the hyperbolic space ''H''<sup>''n''−1</sup> can be embedded into the ''n'' dimensional [[Minkowski space]], a real vector space equipped with the quadratic form
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| :<math>q(x) = x_1^2 - x_2^2-\cdots - x_n^2.</math>
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| Then ''H''<sup>''n''</sup> is the subset of the future null cone in Minkowski space given by
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| :<math>H^n = \{ x | q(x) = 1, x_1>1\}. \, </math>
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| Then
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| :<math>\Delta_{H^{n-1}} f = \Box f\left(x/q(x)^{1/2}\right)|_{H^{n-1}}</math>
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| 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]]
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| :<math>\Box = \frac{\partial^2}{\partial x_1^2} - \cdots - \frac{\partial^2}{\partial x_n^2}.</math>
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| 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''<sup>''n''−1</sup> (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''<sup>''n''−1</sup>. Then the spherical Laplacian has the form:
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| :<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>
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| where <math>\Delta_\xi</math> is the Laplace–Beltrami operator on the ordinary unit (''n'' − 2)-sphere.
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| == See also ==
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| * [[Laplacian operators in differential geometry]]
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| == References ==
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| * {{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}}.
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| * {{citation|last=Flanders|first=Harley|title=Differential forms with applications to the physical sciences|publisher=Dover|year=1989|isbn=978-0-486-66169-8}}
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| * {{citation|first=Jürgen|last=Jost|title=Riemannian Geometry and Geometric Analysis|year=2002|publisher=Springer-Verlag|publication-place=Berlin|isbn=3-540-42627-2}}.
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| *{{eom|id=l/l057450|title=Laplace–Beltrami equation|first=E.D.|last= Solomentsev|first2=E.V.|last2= Shikin}}
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| {{DEFAULTSORT:Laplace-Beltrami operator}}
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| [[Category:Differential operators]]
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| [[Category:Riemannian geometry]]
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| [[de:Verallgemeinerter_Laplace-Operator#Laplace-Beltrami-Operator]]
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