In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds.
For any twice-differentiable real-valued function f defined on Euclidean space Rn, the Laplace operator (also known as the Laplacian) takes f to the divergence of its gradient vector field, which is the sum of the n pure second derivatives of f with respect to each vector of an orthonormal basis for Rn.
The orientation allows one to specify a definite volume form on M, given in an oriented coordinate system xi by where |g| := |det(gij)| is the absolute value of the determinant of the metric tensor, and the dxi are the 1-forms forming the dual frame to the frame of the tangent bundle
with the property where LX is the Lie derivative along the vector field X.
In local coordinates, one obtains 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
Here, dƒ is the exterior derivative of the function ƒ; it is a 1-form taking argument vx.
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 rather than a form).
are formal adjoints, in the sense that for a compactly supported function
As a consequence, the Laplace–Beltrami operator is negative and formally self-adjoint, meaning that for compactly supported functions
): Performing an integration by parts or what is the same thing as using the divergence theorem on the term on the left, and since
A fundamental result of André Lichnerowicz[1] states that: Given a compact n-dimensional Riemannian manifold with no boundary with
Assume the Ricci curvature satisfies the lower bound: where
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,[2] that if the n-dimensional compact Riemannian manifold without boundary were such that for the first positive eigenvalue
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 with these geometries like the Kohn Laplacian (after Joseph J. Kohn) on a compact CR manifold.
[4] The Laplace–Beltrami operator can be written using the trace (or contraction) of the iterated covariant derivative associated with the Levi-Civita connection.
is the symmetric 2-tensor where df denotes the (exterior) derivative of a function f. Let Xi 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 Xi, Xj.
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.
Apart from the incidental sign, the two operators differ by a Weitzenböck identity that explicitly involves the Ricci curvature tensor.
In the usual (orthonormal) Cartesian coordinates xi on Euclidean space, the metric is reduced to the Kronecker delta, and one therefore has
Similarly, the Laplace–Beltrami operator corresponding to the Minkowski metric with signature (− + + +) is the d'Alembertian.
The spherical Laplacian is the Laplace–Beltrami operator on the (n − 1)-sphere with its canonical metric of constant sectional curvature 1.
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 (ϕ, 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 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 Sn−1.
In particular, for the ordinary 2-sphere using standard notation for polar coordinates we get: A similar technique works in hyperbolic space.
Let (t, ξ) be spherical coordinates on the sphere with respect to a particular point p of Hn−1 (say, the center of the Poincaré disc).