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. It is named after Pierre-Simon Laplace and Eugenio Beltrami. 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. 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).
Details The Laplace–Beltrami operator, like the Laplacian, is the (Riemannian) divergence of the (Riemannian) gradient:
Δ f = d i v ( ∇ f ) . {\displaystyle \Delta f={\rm {div}}(\nabla f).}
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 xi by
vol n := | g | d x 1 ∧ ⋯ ∧ d x n {\displaystyle \operatorname {vol} _{n}:={\sqrt {|g|}}\;dx^{1}\wedge \cdots \wedge dx^{n}}
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
∂ i := ∂ ∂ x i {\displaystyle \partial _{i}:={\frac {\partial }{\partial x^{i}}}}
of the tangent bundle T M {\displaystyle TM} and ∧ {\displaystyle \wedge } is the wedge product. The divergence of a vector field X {\displaystyle X} on the manifold is then defined as the scalar function ∇ ⋅ X {\displaystyle \nabla \cdot X} with the property
( ∇ ⋅ X ) vol n := L X vol n {\displaystyle (\nabla \cdot X)\operatorname {vol} _{n}:=L_{X}\operatorname {vol} _{n}}
where LX is the Lie derivative along the vector field X. In local coordinates, one obtains
∇ ⋅ X = 1 | g | ∂ i ( | g | X i ) {\displaystyle \nabla \cdot X={\frac {1}{\sqrt {|g|}}}\partial _{i}\left({\sqrt {|g|}}X^{i}\right)}
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 ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } on the manifold, as
⟨ grad f ( x ) , v x ⟩ = d f ( x ) ( v x ) {\displaystyle \langle \operatorname {grad} f(x),v_{x}\rangle =df(x)(v_{x})}
for all vectors vx anchored at point x in the tangent space TxM of the manifold at point x. Here, dƒ is the exterior derivative of the function ƒ; it is a 1-form taking argument vx. In local coordinates, one has
( grad f ) i = ∂ i f = g i j ∂ j f {\displaystyle \left(\operatorname {grad} f\right)^{i}=\partial ^{i}f=g^{ij}\partial _{j}f}
where gij are the components of the inverse of the metric tensor, so that gijgjk = δik with δik 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
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