In
differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, the Laplace–Beltrami operator is a generalization of the
Laplace operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a Scalar field, scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \ ...
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 S \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly ...
s in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
and, even more generally, on
Riemannian and
pseudo-Riemannian manifold
In mathematical physics, 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 polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
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 ...
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 rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to ...
of its
gradient
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
vector field, which is the sum of the ''n'' pure second derivatives of ''f'' with respect to each vector of an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit vec ...
for R
''n''. Like the Laplacian, the Laplace–Beltrami operator is defined as the divergence of the gradient, and is a
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
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
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
. 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 (Riemannian)
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to ...
of the (Riemannian)
gradient
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
:
:
An explicit formula in
local coordinates is possible.
Suppose first that ''M'' is an
oriented Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
. 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 t ...
on ''M'', given in an oriented coordinate system ''x''
''i'' by
:
where is the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
of the
determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
of the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
, and the ''dx
i'' are the
1-form
In differential geometry, a one-form (or covector field) on a differentiable manifold is a differential form of degree one, that is, a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the t ...
s forming the
dual frame to the frame
:
of the tangent bundle
and
is the
wedge product.
The divergence of a vector field
on the manifold is then defined as the scalar function
with the property
:
where ''L
X'' is the
Lie derivative along the
vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
''X''. In local coordinates, one obtains
:
where here and below the
Einstein notation
In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies ...
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, ofte ...
on the manifold, as
:
for all vectors ''v
x'' anchored at point ''x'' in the
tangent space
In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...
''T
xM'' of the manifold at point ''x''. Here, ''d''ƒ is the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
of the function ƒ; it is a 1-form taking argument ''v
x''. In local coordinates, one has
:
where ''g
ij'' are the components of the inverse of the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
, 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
:
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
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
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
and
are formal adjoints, in the sense that for a compactly supported function
:
(proof)
where the last equality is an application of
Stokes' theorem
Stokes' theorem, also known as the Kelvin–Stokes theorem after Lord Kelvin and George Stokes, the fundamental theorem for curls, or simply the curl theorem, is a theorem in vector calculus on \R^3. Given a vector field, the theorem relates th ...
. Dualizing gives
for all compactly supported functions
and
. 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
and
,
:
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,
:
where
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
. It can be shown using the self-adjointness proved above that the eigenvalues
are real. The compactness of the manifold
allows one to show that the eigenvalues are discrete and furthermore, the vector space of eigenfunctions associated with a given eigenvalue
, i.e. the
eigenspace
In linear algebra, an eigenvector ( ) or characteristic vector is a Vector (mathematics and physics), vector that has its direction (geometry), direction unchanged (or reversed) by a given linear map, linear transformation. More precisely, an e ...
s are all finite-dimensional. Notice by taking the constant function as an eigenfunction, we get
is an eigenvalue. Also since we have considered
an integration by parts shows that
. More precisely if we multiply the eigenvalue equation through by the eigenfunction
and integrate the resulting equation on
we get (using the notation
):
:
Performing an integration by parts or what is the same thing as using the
divergence theorem on the term on the left, and since
has no boundary we get
:
Putting the last two equations together we arrive at
:
We conclude from the last equation that
.
A fundamental result of
André Lichnerowicz states that: Given a compact ''n''-dimensional Riemannian manifold with no boundary with
. Assume the
Ricci curvature satisfies the lower bound:
:
where
is the metric tensor and
is any tangent vector on the manifold
. Then the first positive eigenvalue
of the eigenvalue equation satisfies the lower bound:
:
This lower bound is sharp and achieved on the sphere
. In fact on
the eigenspace for
is three dimensional and spanned by the restriction of the coordinate functions
from
to
. Using spherical coordinates
, on
the two dimensional sphere, set
:
we see easily from the formula for the spherical Laplacian displayed below that
:
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
one has,
:
then the manifold is isometric to the ''n''-dimensional sphere
, the sphere of radius
. 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. Applications there are to the global embedding of such CR manifolds in
Tensor Laplacian
The Laplace–Beltrami operator can be written using the
trace (or contraction) of the iterated
covariant derivative
In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to:
Statistics
* Covariance matrix, a matrix of covariances between a number of variables
* Covariance or cross-covariance between ...
associated with the
Levi-Civita connection
In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the ( pseudo-) Riemannian ...
. The
Hessian (tensor) of a function
is the symmetric 2-tensor
:
,
,
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
:
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) of the Hessian with respect to the metric:
:
.
More precisely, this means
:
,
or in terms of the metric
:
In
abstract indices, the operator is often written
:
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 associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
s, the Laplace–Beltrami operator defined on a tensor ''T'' by
:
is well-defined.
Laplace–de Rham operator
More generally, one can define a Laplacian
differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
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 mathematical physics, 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
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
it is an
elliptic operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which im ...
, while on a
Lorentzian manifold
In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere non-degenerate bilinear form, nondegenerate. This is a generalization of a Riema ...
it is
hyperbolic
Hyperbolic may refer to:
* of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics
** Hyperbolic geometry, a non-Euclidean geometry
** Hyperbolic functions, analogues of ordinary trigonometric functions, defined u ...
. The Laplace–de Rham operator is defined by
:
where d is the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
or differential and ''δ'' is the
codifferential, acting as on ''k''-forms, where ∗ is the
Hodge star. The first order operator
is the Hodge–Dirac operator.
When computing the Laplace–de Rham operator on a scalar function ''f'', we have , so that
:
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, 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
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
''x''
''i'' on
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
, the metric is reduced to the Kronecker delta, and one therefore has
. Consequently, in this case
:
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 invertible, l ...
, such as
spherical or
cylindrical coordinates
A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infinite ...
, one obtains
alternative expressions.
Similarly, the Laplace–Beltrami operator corresponding to the
Minkowski metric
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of general_relativity, gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model ...
with
signature
A signature (; from , "to sign") is a 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. Signatures are often, but not always, Handwriting, handwritt ...
is the
d'Alembertian.
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
In mathematics, a unit sphere is a sphere of unit radius: the locus (mathematics), set of points at Euclidean distance 1 from some center (geometry), center point in three-dimensional space. More generally, the ''unit -sphere'' is an n-sphere, -s ...
centred at the origin. Then for a function ''f'' on ''S''
''n''−1, the spherical Laplacian is defined by
:
where ''f''(''x''/, ''x'', ) is the degree zero homogeneous extension of the function ''f'' to R
''n'' − , and
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:
:
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 ...
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:
:
where
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:
:
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
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model helps show how a ...
, a real vector space equipped with the quadratic form
:
Then ''H''
''n'' is the subset of the future null cone in Minkowski space given by
:
Then
:
Here
is the degree zero homogeneous extension of ''f'' to the interior of the future null cone and is the
wave operator
:
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). 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:
:
where
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:
:
See also
*
Covariant derivative
In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to:
Statistics
* Covariance matrix, a matrix of covariances between a number of variables
* Covariance or cross-covariance between ...
*
Laplacian operators in differential geometry
*
Laplace operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a Scalar field, scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \ ...
Notes
References
*
* .
*
{{DEFAULTSORT:Laplace-Beltrami operator
Differential operators
Riemannian geometry
de:Verallgemeinerter Laplace-Operator#Laplace-Beltrami-Operator