structural:expansion, linguistic:voice_change
Laplace–Beltrami operator

The Laplacian also can be generalized to an elliptic operator called the Laplace–Beltrami operator defined on a Riemannian manifold. The Laplace–Beltrami operator, when applied to a function, is the trace (tr) of the function's Hessian: Δ f = tr (H (f)) {\displaystyle \Delta f=\operatorname {tr} {\big (}H(f){\big)}} where the trace is taken with respect to the inverse of the metric tensor. The Laplace–Beltrami operator also can be generalized to an operator (also called the Laplace–Beltrami operator) which operates on tensor fields, by a similar formula.

Another generalization of the Laplace operator that is available on pseudo-Riemannian manifolds uses the exterior derivative, in terms of which the "geometer's Laplacian" is expressed as Δ f = δ d f. {\displaystyle \Delta f=\delta df.}

Here δ is the codifferential, which can also be expressed in terms of the Hodge star and the exterior derivative. This operator differs in sign from the "analyst's Laplacian" defined above. More generally, the "Hodge" Laplacian is defined on differential forms α by Δ α = δ d α + d δ α. {\displaystyle \Delta \alpha =\delta d\alpha +d\delta \alpha.}

This is known as the Laplace–de Rham operator, which is related to the Laplace–Beltrami operator by the Weitzenböck identity.