- Laplace-Beltrami operator
In
differential geometry , theLaplace operator can be generalized to operate on functions defined onsurface s, or more generally on Riemannian andpseudo-Riemannian manifold s. This more general operator goes by the name Laplace-Beltrami operator. As the Laplacian, the Laplace-Beltrami operator is defined as the divergence of thegradient . The operator can be extended to operate on tensors as the divergence of thecovariant derivative . Alternatively, the operator can be generalized to operate ondifferential forms using the divergence andexterior derivative . The resulting operator is called the Laplace-de Rham operator.Laplace-Beltrami operator
One defines the Laplace-Beltrami operator, just as the Laplacian, as the divergence of the gradient. To be able to find a formula for this operator, one will need to first write the divergence and the gradient on a manifold.
If g denotes the (pseudo)-
metric tensor on the manifold, one finds that thevolume form inlocal coordinates is given by:mathrm{vol}_n := sqrt X^i ight).
Here (and below) we use the
Einstein notation , so the above is actually a sum in "i". The gradient of a scalar function "f" may be defined through theinner product langlecdot,cdot angle on the manifold, as:langle mbox{grad} f(x) , v_x angle = df(x)(v_x)
for all vectors v_x anchored at point "x" in the
tangent bundle T_xM of the manifold at point "x". Here, "df" is theexterior derivative of the function "f"; it is a 1-form taking argument v_x. In local coordinates, one has:left(mbox{grad} f ight)^i = partial^i f = g^{ij} partial_j f.
Combining these, the formula for the Laplace-Beltrami operator applied to a scalar function "f" is, in local coordinates
:Delta f = mbox{div grad} ; f = frac{1}{sqrt .
When g| = 1, such as in the case of
Euclidean space with Cartesian coordinates, one then easily obtains:Delta f = partial_i partial^i f
which is the ordinary Laplacian. Using the
Minkowski metric with signature (+++-), one regains theD'Alembertian given previously. Under local parametrization u^1, u^2, the Laplace-Beltrami operator can be expanded in terms of the metric tensor andChristoffel symbols as follows::Delta f = g^{ij}left(frac{partial^2 f}{partial u^i, partial u^j} - Gamma_{ij}^k frac{partial f}{partial u^k} ight).
Note that by using the metric tensor for spherical and cylindrical coordinates, one can similarly regain the expressions for the Laplacian in spherical and cylindrical coordinates. The Laplace-Beltrami operator is handy not just in curved space, but also in ordinary flat space endowed with a non-linear coordinate system.
Note also that the exterior derivative "d" and -div are adjoint:
:int_M df(X) ;mathrm{vol}_n = - int_M f mbox{div} X ;mathrm{vol}_n (proof)
where the last equality is an application of
Stokes' theorem . Note also, the Laplace-Beltrami operator is negative and symmetric::int_M fDelta h ;mathrm{vol}_n = -int_M langle mbox{grad} f, mbox{grad} h angle ;mathrm{vol}_n = int_M hDelta f ;mathrm{vol}_n
for functions "f" and "h" . For this reason, several authors prefer to define the Laplace-Beltrami operator as the present one with a minus sign in front, so that it is positive .
Using the covariant derivative
The Laplace-Beltrami operator can be written using the trace of the iterated
covariant derivative associated to 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:H(f)_{ij}=H_f(X_i, X_j) = abla_{X_i} abla_{X_j} f - abla_{ abla_{X_i}X_j} 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 of the Hessian with respect to the metric:
:Delta f = sum_{ij} g^{ij} H(f)_{ij}.
In
abstract indices , the operator is often written:Delta f = abla^a abla_a f
provided it is understood implicitly that this trace is in fact the trace of the Hessian "tensor".
Laplace-de Rham operator
More generally, one can define a Laplacian
differential operator on theexterior algebra of adifferentiable manifold . On aRiemannian manifold it is anelliptic operator , while on aLorentzian manifold it is hyperbolic. The Laplace-de Rham operator is defined by:Delta= mathrm{d}delta+deltamathrm{d} = (mathrm{d}+delta)^2,;
where d is the
exterior derivative or differential and δ is thecodifferential . When acting on scalar functions, the codifferential may be defined as δ = −d, where ; is theHodge star ; more generally, the codifferential may include a sign that depends on the order of the "k"-form being acted on.One may prove that the Laplace-de Rham operator is equivalent to the previous definition of the Laplace-Beltrami operator when acting on a scalar function "f"; see the proof for details. Notice that the Laplace-de Rham operator is actually minus the Laplace-Beltrami operator; this minus sign follows from the conventional definition of the properties of the
codifferential . Unfortunately, Δ is used to denote both; which can sometimes be a source of confusion.Properties
Given scalar functions "f" and "h", and a real number "a", the Laplace-de Rham operator has the following properties:
#Delta(af + h) = a,Delta f + Delta h!
#Delta(fh) = f ,Delta h + 2 (partial_i f) (partial^i h) + h, Delta f (proof)Laplace operators on tensors
The Laplace-Beltrami operator can be extended to an operator on arbitrary
tensor s on a pseudo-Riemannian manifold using thecovariant derivative associated to theLevi-Civita connection . This extended operator may then act on skew-symmetric tensors. However, the resulting operator is not the same one as that given by the Laplace-de Rham operator: the two are related by theWeitzenböck identity .See also
*
Laplacian operators in differential geometry References
*
* Jürgen Jost, "Riemannian Geometry and Geometric Analysis", (2002) Springer-Verlag, Berlin ISBN 3-540-42627-2 . "(Provides a general introduction to curved surfaces)."
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