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 mult ...
, the Laplace–Beltrami operator is a generalization of the
Laplace operator to functions defined on
submanifolds in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
and, even more generally, on
Riemannian and
pseudo-Riemannian manifolds. 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 summarize ...
and
Eugenio Beltrami.
For any twice-
differentiable 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 ...
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 application ...
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
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 ...
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 ...
:
:
An explicit formula in
local coordinates is possible.
Suppose first that ''M'' is an
oriented
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
Riemannian manifold. The orientation allows one to specify a definite
volume form on ''M'', given in an oriented coordinate system ''x''
''i'' by
:
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 ''dx
i'' are the
1-form
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ...
s forming the
dual frame to the frame
:
of the tangent bundle
and
is the
wedge product.
The divergence of a vector field ''X'' on the manifold is then defined as the scalar function with the property
:
where ''L
X'' is the
Lie derivative along the
vector field ''X''. In local coordinates, one obtains
:
where here and below the
Einstein notation
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
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 ...
on the manifold, as
:
for all vectors ''v
x'' anchored at point ''x'' in the
tangent space ''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, so that with δ
''i''''k'' 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
:
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 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. Mathematicall ...
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
:
(proof)
where the last equality is an application of
Stokes' theorem
Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
. 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'',
:
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 associated with the eigenvalue
. It can be shown using the self-adjointness proved above that the eigenvalues
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
, 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
is an eigenvalue. Also since we have considered
an integration by parts shows that
. More precisely if we multiply the eigenvalue eqn. through by the eigenfunction
and integrate the resulting eqn. on
we get( using the notation
)
:
Performing an integration by parts or what is the same thing as using the
divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the ...
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 associated with the Levi-Civita connection. 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
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:
:
.
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 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
:
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 application ...
s on a
pseudo-Riemannian manifold. On a
Riemannian manifold it is an
elliptic operator, while on a
Lorentzian manifold it is
hyperbolic. 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
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 ...
, 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 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
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 ...
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 fu ...
, 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
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 measure ...
.
Examples
Many examples of the Laplace–Beltrami operator can be worked out explicitly.
Euclidean space
In the usual (orthonormal)
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
''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'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, the metric is reduced to the Kronecker delta, and one therefore has
. Consequently, in this case
:
which is the ordinary Laplacian. In
curvilinear coordinates, such as
spherical or
cylindrical coordinates, one obtains
alternative expressions.
Similarly, the Laplace–Beltrami operator corresponding to the
Minkowski metric with
signature
A signature (; from la, signare, "to sign") is a Handwriting, handwritten (and often Stylization, 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 ...
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 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 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, 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
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:
:
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
*
Laplacian operators in differential geometry
*
Laplace operator
Notes
References
*
* .
*
{{DEFAULTSORT:Laplace-Beltrami operator
Differential operators
Riemannian geometry
de:Verallgemeinerter Laplace-Operator#Laplace-Beltrami-Operator