View source for Laplace–Beltrami operator

{{Short description|operator generalizing the Laplacian in differential geometry}}
{{distinguish|Beltrami operator}}
For any twice-differentiable real-valued function f defined on Euclidean space <b>R</b><sup>n</sup>, the Laplace operator takes f to the divergence of its gradient vector field, which is the sum of the n second derivatives of f with respect to each vector of an orthonormal basis for <b>R</b><sup>n</sup>. In the field of [[differential geometry]], this operator is generalized to operate on functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. 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).

==Details==
The Laplace–Beltrami operator, like the Laplacian, is the [[divergence]] of the [[gradient]]:
:<math>\Delta f = \nabla \cdot \nabla 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''<sup>''i''</sup> by

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

where {{nowrap|1={{!}}''g''{{!}} := {{!}}det(''g<sub>ij</sub>''){{!}}}} is the [[absolute value]] of the [[determinant]] of the [[metric tensor]], and the ''dx<sup>i</sup>'' are the [[1-form]]s forming the [[dual basis]] to the basis vectors
:<math>\partial_i := \frac {\partial}{\partial x^i}</math>
of the tangent space <math>T_p M</math> and <math>\wedge</math> is the [[wedge product]]. 

The divergence of a vector field ''X'' on the manifold is then defined as the scalar function with the property

:<math>
(\nabla \cdot X) \operatorname{vol}_n := L_X \operatorname{vol}_n
</math>

where ''L<sub>X</sub>'' is the [[Lie derivative]] along the [[vector field]] ''X''. In local coordinates, one obtains

:<math>
\nabla \cdot 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 \operatorname{grad} f(x) , v_x \rangle = df(x)(v_x)</math>

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

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

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>'' = &delta;<sup>''i''</sup><sub>''k''</sub>}} with δ<sup>''i''</sup><sub>''k''</sub> the [[Kronecker 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

:<math>\Delta  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 &minus;&nabla; .  are formal adjoints, in the sense that for ''ƒ'' a compactly supported function

:<math>\int_M df(X) \operatorname{vol}_n = - \int_M f  \nabla \cdot X \operatorname{vol}_n </math> &nbsp;&nbsp;&nbsp;&nbsp;<small>[[Laplace–Beltrami operator/Proofs|(proof)]]</small>

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

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

for all compactly supported functions ''ƒ'' and ''h''.  Conversely, ({{EquationRef|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 ƒ and ''h'',

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

==Eigenvalues of the Laplace–Beltrami operator (Lichnerowicz–Obata theorem)==
Let M denote a compact Riemannian manifold without boundary. We want to consider the eigenvalue equation,
:<math> -\Delta  u=\lambda u,</math>
where <math>u</math> is the [[eigenfunction]] associated with the eigenvalue <math>\lambda</math>. It can be shown using the self-adjointness proved above that the eigenvalues <math>\lambda</math> are real. The compactness of the manifold M allows one to show that the eigenvalues are discrete and furthermore, the vector space of eigenfunctions associated with a given eigenvalue <math>\lambda</math>, i.e. the [[eigenspace]]s are all finite-dimensional. Notice by taking the constant function as an eigenfunction, we get <math>\lambda=0</math> is an eigenvalue. Also since we have considered <math>-\Delta </math> an integration by parts shows that <math>\lambda\geq 0</math>. More precisely if we multiply the eigenvalue eqn. through by the eigenfunction <math>u</math> and integrate the resulting eqn. on <math>M</math> we get( using the notation <math>dV=\operatorname{vol}_n</math>)
:<math>-\int_M \Delta  u\ u\ dV=\lambda\int_Mu^2\ dV</math>
Performing an integration by parts or what is the same thing as using the [[divergence theorem]] on the term on the left, and since <math>M</math> has no boundary we get
:<math>-\int_M\Delta  u\ u\ dV=\int_M|\nabla u|^2\ dV</math>
Putting the last two equations together we arrive at
:<math>\int_M|\nabla u|^2\ dV=\lambda\int_Mu^2\ dV</math>
We conclude from the last equation that <math>\lambda\geq 0</math>.

A fundamental result of [[Andre Lichnerowicz]] <ref>{{cite book|author=Lichnerowicz, Andre|title=Geometrie des groupes de transformations|year=1958|publisher=Dunod|location=Paris}}</ref> states that: Given a compact ''n''-dimensional Riemannian manifold with no boundary with <math>n\geq 2</math>. Assume the [[Ricci curvature]] satisfies the lower bound:
:<math> \operatorname{Ric}(X,X)\geq \kappa g(X,X),\kappa>0,</math>
where <math>g(\cdot,\cdot)</math> is the metric tensor and <math>X</math> is any tangent vector on the manifold <math>M</math>. Then the first positive eigenvalue <math>\lambda_1</math> of the eigenvalue equation satisfies the lower bound:
:<math> \lambda_1\geq \frac{n}{n-1}\kappa.</math>
This lower bound is sharp and achieved on the sphere <math>\mathbb{S}^n</math>. In fact on <math>\mathbb{S}^2</math> the eigenspace for <math>\lambda_1</math> is three dimensional and spanned by the restriction of the coordinate functions <math>x_1,x_2,x_3</math> from <math>\mathbb{R}^3</math> to <math>\mathbb{S}^2</math>. Using spherical coordinates <math>(\theta,\phi)</math>, on <math>\mathbb{S}^2</math> the two dimensional sphere, set
:<math>x_3=\cos\phi=u_1,</math>
we see easily from the formula for the spherical Laplacian displayed below that
:<math> -\Delta _{\mathbb{S}^2}u_1=2u_1</math>
Thus the lower bound in Lichnerowicz's theorem is achieved at least in two dimensions.

Conversely it was proved by Morio Obata,<ref>{{cite journal|author=Obata, Morio|title=Certain conditions for a Riemannian manifold to be isometric with a sphere|journal=J. Math. Soc. Jpn.|volume=14|issue=3|year=1962|pages=333–340|doi=10.2969/jmsj/01430333}}</ref> that if the ''n''-dimensional compact Riemannian manifold without boundary were such that for the first positive eigenvalue <math>\lambda_1</math> one has,
:<math>\lambda_1=\frac{n}{n-1}\kappa,</math>

then the manifold is isometric to the ''n''-dimensional sphere <math> \mathbb{S}^n(1/\sqrt{\kappa})</math>, the sphere of radius <math>1/\sqrt{\kappa}</math>. Proofs of all these statements may be found in the book by Isaac Chavel.<ref>{{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}}</ref> Analogous sharp bounds also hold for other Geometries and for certain degenerate Laplacians associated with these geometries like the [[CR manifold#The tangential Cauchy.E2.80.93Riemann complex.28Kohn Laplacian.2C Kohn-Rossi Complex.29|Kohn Laplacian]] ( after [[Joseph J. Kohn]]) on a compact [[CR manifold]]. Applications there are to the global embedding of such CR manifolds in <math>\mathbb{C}^n.</math> <ref>{{cite journal|author=Chanillo, Sagun, Chiu, Hung-Lin and Yang, Paul C.|title=Embeddability for 3-dimensional CR manifolds and CR Yamabe Invariants|journal=Duke Mathematical Journal|volume=161|issue=15|year=2012|pages=2909–2921|doi=10.1215/00127094-1902154|arxiv=1007.5020}}</ref>

==Tensor Laplacian==

The Laplace–Beltrami operator can be written using the [[Tensor_contraction|trace (or contraction)]] of the iterated [[covariant derivative]] associated with the Levi-Civita connection. The [[Hessian_matrix#Generalizations_to_Riemannian_manifolds|Hessian (tensor)]] of a function <math>f</math> is the symmetric 2-tensor

:<math>\displaystyle \mbox{Hess} f \in \mathbf \Gamma(\mathsf T^*M \otimes \mathsf T^*M)</math>, <math>\mbox{Hess} f := \nabla^2 f \equiv \nabla \nabla f \equiv \nabla \mathrm df</math>,

where ''df'' denotes the [[Exterior_derivative#Definition|(exterior) derivative]] of a function ''f''.

Let ''X''<sub>i</sub> be a basis of tangent vector fields (not necessarily induced by a coordinate system). Then the components of ''Hess f'' are given by

:<math>(\mbox{Hess} f)_{ij} = \mbox{Hess} 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''<sub>i</sub>, ''X''<sub>j</sub>.  The Laplace–Beltrami operator is then the trace (or [[Tensor_contraction|contraction]]) of the Hessian with respect to the metric:

:<math>\displaystyle \Delta f := \mathrm{tr} \nabla \mathrm df \in \mathsf C^\infty(M)</math>.

More precisely, this means

:<math>\displaystyle \Delta f(x) = \sum_{i=1}^n \nabla \mathrm df(X_i,X_i)</math>,

or in terms of the metric

:<math>\Delta  f = \sum_{ij} g^{ij} (\mbox{Hess} 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=(&minus;1)<sup>''kn''+''n''+1</sup>&lowast;d&lowast;}} on ''k''-forms, where ∗ is the [[Hodge star]].

When computing the Laplace–Beltrami operator on a scalar function ''f'', we have {{nowrap|1=''δf'' = 0}}, so that

:<math>\Delta  f = \delta \, \mathrm 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 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 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''<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

:<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]] {{nowrap|(&minus; + + +)}} is the [[d'Alembertian]].

;Spherical Laplacian
The spherical Laplacian is the Laplace–Beltrami operator on the {{nowrap|(''n'' &minus; 1)}}-sphere with its canonical metric of constant sectional curvature&nbsp;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 ''f'' on ''S''<sup>''n''&minus;1</sup>, the spherical Laplacian is defined by
:<math>\Delta _{S^{n-1}}f(x) = \Delta  f(x/|x|)</math>
where ''f''(''x''/|''x''|) is the degree zero homogeneous extension of the function ''f'' to '''R'''<sup>''n''</sup>&nbsp;&minus;&nbsp;{0}, and <math>\Delta</math> 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 {{nowrap|(''ϕ'', ''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 ''ϕ'' 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''&minus;1</sup>. Then the spherical Laplacian has the form:
:<math>\Delta _{S^{n-1}} f(\xi,\phi) = (\sin\phi)^{2-n} \frac{\partial}{\partial \phi}\left((\sin\phi)^{n-2}\frac{\partial f}{\partial \phi}\right) + (\sin\phi)^{-2}\Delta _\xi f</math>
where <math>\Delta _\xi</math> is the Laplace&ndash;Beltrami operator on the ordinary unit {{nowrap|(''n'' &minus; 2)}}-sphere.  In particular, for the ordinary 2-sphere using standard notation for polar coordinates we get:
:<math>\Delta _{S^{2}} f(\theta,\phi) = (\sin\phi)^{-1} \frac{\partial}{\partial \phi}\left(\sin\phi\frac{\partial f}{\partial \phi}\right) + (\sin\phi)^{-2} \frac{\partial^2}{\partial \theta^2}f</math>

;Hyperbolic space
A similar technique works in [[hyperbolic space]].  Here the hyperbolic space ''H''<sup>''n''&minus;1</sup> 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''<sup>''n''</sup> is the subset of the future null cone in Minkowski space given by
:<math>H^n = \{ x \mid q(x) = 1, x_1>1\}. \, </math>
Then
:<math>\Delta _{H^{n-1}} f = \left. \Box f\left(x/q(x)^{1/2}\right) \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 {{math|□}} 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 {{nowrap|(''t'', ''ξ'')}} be spherical coordinates on the sphere with respect to a particular point ''p'' of ''H''<sup>''n''&minus;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''&minus;2</sup>. Then the hyperbolic Laplacian has the form:
:<math>\Delta _{H^{n-1}} f(t,\xi) = \sinh(t)^{2-n} \frac{\partial}{\partial t}\left(\sinh(t)^{n-2}\frac{\partial f}{\partial t}\right) + \sinh(t)^{-2}\Delta _\xi f</math>
where <math>\Delta _\xi</math> is the Laplace&ndash;Beltrami operator on the ordinary unit (''n''&nbsp;&minus;&nbsp;2)-sphere.  In particular, for the hyperbolic plane using standard notation for polar coordinates we get:
:<math>\Delta _{H^{2}} f(r,\theta) = \sinh(r)^{-1} \frac{\partial}{\partial r}\left(\sinh(r)\frac{\partial f}{\partial r}\right) + \sinh(r)^{-2} \frac{\partial^2}{\partial \theta^2}f</math>

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

== Notes ==
{{Reflist}}

== References ==
* {{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|authorlink=Jürgen 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]]

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