# Laplace–Beltrami operator

Template:Distinguish
In differential geometry, the Laplace operator, named after Pierre-Simon Laplace, can be generalized to operate on functions defined on surfaces in Euclidean space and, 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).

The Laplace–Beltrami operator, like the Laplacian, is the divergence of the gradient:

An explicit formula in local coordinates is possible.

Suppose first that *M* is an oriented Riemannian manifold. The orientation allows one to specify a definite volume form on *M*, given in an oriented coordinate system *x*^{i} by

where the *dx ^{i}* are the 1-forms forming the dual basis to the basis vectors

and is the wedge product. Here |*g*| := |det(*g _{ij}*)| is the absolute value of the determinant of the metric tensor

*g*

_{ij}. The divergence div

*X*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 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 on the manifold, as

for all vectors *v _{x}* anchored at point

*x*in the tangent space

*T*of the manifold at point

_{x}M*x*. Here,

*d*ƒ is the exterior derivative of the function ƒ; it is a 1-form taking argument

*v*. In local coordinates, one has

_{x}where *g ^{ij}* are the components of the inverse of the metric tensor, so that

*g*= δ

^{ij}g_{jk}^{i}

_{k}with δ

^{i}

_{k}the Kronecker delta.

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

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 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 −div are formal adjoints, in the sense that for *ƒ* a compactly supported function

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

for all compactly supported functions *ƒ* and *h*. Conversely, (2) characterizes Δ 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 now M denote a compact Riemannian manifold with no boundary. We want to consider the eigenvalue equation,

is the eigenfunction associated to 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 to a given eigenvalue i.e the eigenspace 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 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 Andre Lichnerowicz ^{[1]} 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 the fact that on and using spherical coordinates , 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,^{[2]} 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.^{[3]} Analogous sharp bounds also hold for other Geometries and for certain degenerate Laplacians associated to 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 ^{[4]}

## Tensor Laplacian

The Laplace–Beltrami operator can be written using the trace of the iterated covariant derivative associated with the Levi-Civita connection. From this perspective, let *X*_{i} be a basis of tangent vector fields (not necessarily induced by a coordinate system). Then the **Hessian** of a function *f* is the symmetric 2-tensor whose components 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 of the Hessian with respect to 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 tensors, 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 forms 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 or differential and δ is the codifferential, acting as (−1)^{kn+n+1}∗*d*∗ on *k*-forms where ∗ is the Hodge star.

When computing Δƒ for a scalar function ƒ, we have δƒ = 0, 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 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.

## 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, 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 (−+++) is the D'Alembertian.

- Spherical Laplacian

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**^{n} as the unit sphere centred at the origin. Then for a function *ƒ* on *S*^{n−1}, the spherical Laplacian is defined by

where *ƒ*(*x*/|*x*|) is the degree zero homogeneous extension of the function *ƒ* to **R**^{n} − {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:

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 (*, **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 * 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 (*n* − 2)-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 (*t*, *ξ*) 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−1}. 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

## References

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}.

de:Verallgemeinerter Laplace-Operator#Laplace-Beltrami-Operator