Laplace–Beltrami operator
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In
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on
submanifold In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
s in
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
and, even more generally, on Riemannian and
pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
s. It is named after
Pierre-Simon Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
and Eugenio Beltrami. For any twice-
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
real-valued function ''f'' defined on Euclidean space R''n'', the Laplace operator (also known as the ''Laplacian'') takes ''f'' to the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of t ...
of its
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
vector field, which is the sum of the ''n'' pure second derivatives of ''f'' with respect to each vector of an orthonormal basis for R''n''. 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 form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s using the divergence and exterior derivative. The resulting operator is called the Laplace–de Rham operator (named after
Georges de Rham Georges de Rham (; 10 September 1903 – 9 October 1990) was a Swiss mathematician, known for his contributions to differential topology. Biography Georges de Rham was born on 10 September 1903 in Roche, a small village in the canton of Vaud in ...
).


Details

The Laplace–Beltrami operator, like the Laplacian, is the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of t ...
of the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
: :\Delta f = \nabla \cdot \nabla f. An explicit formula in
local coordinates Local coordinates are the ones used in a ''local coordinate system'' or a ''local coordinate space''. Simple examples: * Houses. In order to work in a house construction, the measurements are referred to a control arbitrary point that will allow ...
is possible. Suppose first that ''M'' is an oriented Riemannian manifold. The orientation allows one to specify a definite
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of th ...
on ''M'', given in an oriented coordinate system ''x''''i'' by :\operatorname_n := \sqrt \;dx^1\wedge \cdots \wedge dx^n where is the absolute value of the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the metric tensor, and the ''dxi'' are the 1-forms forming the dual frame to the frame :\partial_i := \frac of the tangent bundle TM and \wedge is the
wedge product A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by convert ...
. The divergence of a vector field ''X'' on the manifold is then defined as the scalar function with the property : (\nabla \cdot X) \operatorname_n := L_X \operatorname_n where ''LX'' is the
Lie derivative In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
along the vector field ''X''. In local coordinates, one obtains : \nabla \cdot X = \frac \partial_i \left(\sqrt X^i\right) where here and below 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 In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
\langle\cdot,\cdot\rangle on the manifold, as :\langle \operatorname f(x) , v_x \rangle = df(x)(v_x) for all vectors ''vx'' anchored at point ''x'' in the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
''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 : \left(\operatorname f\right)^i = \partial^i f = g^ \partial_j f where ''gij'' are the components of the inverse of the metric tensor, so that with δ''i''''k'' the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
. 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 :\Delta f = \frac \partial_i \left(\sqrt g^ \partial_j f \right). 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 In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form :dV ...
(a
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
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 −∇ . are formal adjoints, in the sense that for ''ƒ'' a compactly supported function :\int_M df(X) \operatorname_n = - \int_M f \nabla \cdot X \operatorname_n     (proof) where the last equality is an application of Stokes' theorem. Dualizing gives for all compactly supported functions ''ƒ'' and ''h''. Conversely, () 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'', :\int_M f\,\Delta h \operatorname_n = -\int_M \langle d f, d h \rangle \operatorname_n = \int_M h\,\Delta f \operatorname_n. 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, : -\Delta u=\lambda u, where u is the
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
associated with the eigenvalue \lambda. It can be shown using the self-adjointness proved above that the eigenvalues \lambda 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 \lambda, i.e. the
eigenspace In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s are all finite-dimensional. Notice by taking the constant function as an eigenfunction, we get \lambda=0 is an eigenvalue. Also since we have considered -\Delta an integration by parts shows that \lambda\geq 0. More precisely if we multiply the eigenvalue eqn. through by the eigenfunction u and integrate the resulting eqn. on M we get( using the notation dV=\operatorname_n) :-\int_M \Delta u\ u\ dV=\lambda\int_Mu^2\ dV Performing an integration by parts or what is the same thing as using the divergence theorem on the term on the left, and since M has no boundary we get :-\int_M\Delta u\ u\ dV=\int_M, \nabla u, ^2\ dV Putting the last two equations together we arrive at :\int_M, \nabla u, ^2\ dV=\lambda\int_Mu^2\ dV We conclude from the last equation that \lambda\geq 0. A fundamental result of
André Lichnerowicz André Lichnerowicz (January 21, 1915, Bourbon-l'Archambault – December 11, 1998, Paris) was a noted French differential geometer and mathematical physicist of Polish descent. He is considered the founder of modern Poisson geometry. Biograp ...
states that: Given a compact ''n''-dimensional Riemannian manifold with no boundary with n\geq 2. Assume the
Ricci curvature In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
satisfies the lower bound: : \operatorname(X,X)\geq \kappa g(X,X),\kappa>0, where g(\cdot,\cdot) is the metric tensor and X is any tangent vector on the manifold M. Then the first positive eigenvalue \lambda_1 of the eigenvalue equation satisfies the lower bound: : \lambda_1\geq \frac\kappa. This lower bound is sharp and achieved on the sphere \mathbb^n. In fact on \mathbb^2 the eigenspace for \lambda_1 is three dimensional and spanned by the restriction of the coordinate functions x_1,x_2,x_3 from \mathbb^3 to \mathbb^2. Using spherical coordinates (\theta,\phi), on \mathbb^2 the two dimensional sphere, set :x_3=\cos\phi=u_1, we see easily from the formula for the spherical Laplacian displayed below that : -\Delta _u_1=2u_1 Thus the lower bound in Lichnerowicz's theorem is achieved at least in two dimensions. Conversely it was proved by Morio Obata, that if the ''n''-dimensional compact Riemannian manifold without boundary were such that for the first positive eigenvalue \lambda_1 one has, :\lambda_1=\frac\kappa, then the manifold is isometric to the ''n''-dimensional sphere \mathbb^n(1/\sqrt), the sphere of radius 1/\sqrt. Proofs of all these statements may be found in the book by Isaac Chavel. Analogous sharp bounds also hold for other Geometries and for certain degenerate Laplacians associated with these geometries like the Kohn Laplacian (after Joseph J. Kohn) on a compact
CR manifold In mathematics, a CR manifold, or Cauchy–Riemann manifold, is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge. Form ...
. Applications there are to the global embedding of such CR manifolds in \mathbb^n.


Tensor Laplacian

The Laplace–Beltrami operator can be written using the trace (or contraction) of the iterated covariant derivative associated with the Levi-Civita connection. The Hessian (tensor) of a function f is the symmetric 2-tensor :\displaystyle \mbox f \in \mathbf \Gamma(\mathsf T^*M \otimes \mathsf T^*M), \mbox f := \nabla^2 f \equiv \nabla \nabla f \equiv \nabla \mathrm df, where ''df'' denotes the (exterior) derivative of a function ''f''. Let ''X''i be a basis of tangent vector fields (not necessarily induced by a coordinate system). Then the components of ''Hess f'' are given by :(\mbox f)_ = \mbox f(X_i, X_j) = \nabla_\nabla_ f - \nabla_ f 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 (or
contraction Contraction may refer to: Linguistics * Contraction (grammar), a shortened word * Poetic contraction, omission of letters for poetic reasons * Elision, omission of sounds ** Syncope (phonology), omission of sounds in a word * Synalepha, merged ...
) of the Hessian with respect to the metric: :\displaystyle \Delta f := \mathrm \nabla \mathrm df \in \mathsf C^\infty(M). More precisely, this means :\displaystyle \Delta f(x) = \sum_^n \nabla \mathrm df(X_i,X_i), or in terms of the metric :\Delta f = \sum_ g^ (\mbox f)_. In
abstract indices Abstract index notation (also referred to as slot-naming index notation) is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeho ...
, the operator is often written :\Delta f = \nabla^a \nabla_a f 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 In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
s, the Laplace–Beltrami operator defined on a tensor ''T'' by :\Delta T = g^\left( \nabla_\nabla_ T - \nabla_ T\right) is well-defined.


Laplace–de Rham operator

More generally, one can define a Laplacian differential operator on sections of the bundle of
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s on a
pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
. On a Riemannian manifold it is an elliptic operator, while on a
Lorentzian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
it is
hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
. The Laplace–de Rham operator is defined by :\Delta = \mathrm\delta + \delta\mathrm = (\mathrm+\delta)^2,\; where d is the exterior derivative or differential and ''δ'' is the codifferential, acting as on ''k''-forms, where ∗ is the
Hodge star In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the ...
. The first order operator \mathrm+\delta is the Hodge-Dirac operator. When computing the Laplace–de Rham operator on a scalar function ''f'', we have , so that :\Delta f = \delta \, \mathrm df. 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 the codifferential assures that the Laplace–de Rham operator is (formally)
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite f ...
, 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 In mathematics, in particular in differential geometry, mathematical physics, and representation theory a Weitzenböck identity, named after Roland Weitzenböck, expresses a relationship between two second-order elliptic operators on a manifo ...
that explicitly involves the
Ricci curvature tensor In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
.


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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, the metric is reduced to the Kronecker delta, and one therefore has , g, = 1. Consequently, in this case :\Delta f = \frac \partial_i \sqrt\partial^i f = \partial_i \partial^i f which is the ordinary Laplacian. In
curvilinear coordinates In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inve ...
, such as
spherical A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
or
cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference d ...
, one obtains alternative expressions. Similarly, the Laplace–Beltrami operator corresponding to the
Minkowski metric In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
with
signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
is the
d'Alembertian In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of Mi ...
.


Spherical Laplacian

The spherical Laplacian is the Laplace–Beltrami operator on the -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 ''f'' on ''S''''n''−1, the spherical Laplacian is defined by :\Delta _f(x) = \Delta f(x/, x, ) where ''f''(''x''/, ''x'', ) is the degree zero homogeneous extension of the function ''f'' to R''n'' − , and \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: :\Delta f = r^\frac\left(r^\frac\right) + r^\Delta _f. More generally, one can formulate a similar trick using the
normal bundle In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion). Definition Riemannian manifold Let (M,g) be a Riemannian m ...
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 coordinate system. Let 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''''n''−1. Then the spherical Laplacian has the form: :\Delta _ f(\xi,\phi) = (\sin\phi)^ \frac\left((\sin\phi)^\frac\right) + (\sin\phi)^\Delta _\xi f where \Delta _\xi is the Laplace–Beltrami operator on the ordinary unit -sphere. In particular, for the ordinary 2-sphere using standard notation for polar coordinates we get: :\Delta _ f(\theta,\phi) = (\sin\phi)^ \frac\left(\sin\phi\frac\right) + (\sin\phi)^ \fracf


Hyperbolic space

A similar technique works in
hyperbolic space In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...
. Here the hyperbolic space ''H''''n''−1 can be embedded into the ''n'' dimensional
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
, a real vector space equipped with the quadratic form :q(x) = x_1^2 - x_2^2-\cdots - x_n^2. Then ''H''''n'' is the subset of the future null cone in Minkowski space given by :H^n = \. \, Then :\Delta _ f = \left. \Box f\left(x/q(x)^\right) \ _ Here f(x/q(x)^) is the degree zero homogeneous extension of ''f'' to the interior of the future null cone and is the
wave operator In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of Mi ...
:\Box = \frac - \cdots - \frac. The operator can also be written in polar coordinates. Let be spherical coordinates on the sphere with respect to a particular point ''p'' of ''H''''n''−1 (say, the center of the
Poincaré disc Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science * Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré * Luc ...
). Here ''t'' represents the hyperbolic distance from ''p'' and ''ξ'' a parameter representing the choice of direction of the geodesic in ''S''''n''−2. Then the hyperbolic Laplacian has the form: :\Delta _ f(t,\xi) = \sinh(t)^ \frac\left(\sinh(t)^\frac\right) + \sinh(t)^\Delta _\xi f where \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: :\Delta _ f(r,\theta) = \sinh(r)^ \frac\left(\sinh(r)\frac\right) + \sinh(r)^ \fracf


See also

* Covariant derivative *
Laplacian operators in differential geometry In differential geometry there are a number of second-order, linear, elliptic differential operators bearing the name Laplacian. This article provides an overview of some of them. Connection Laplacian The connection Laplacian, also known as t ...
* Laplace operator


Notes


References

* * . * {{DEFAULTSORT:Laplace-Beltrami operator Differential operators Riemannian geometry de:Verallgemeinerter Laplace-Operator#Laplace-Beltrami-Operator