Knowlpedia

Definition

Laplace–Beltrami operator

The negative metric trace of the covariant Hessian, with a sign chosen to be nonnegative in Riemannian signature.

Definition

Let (M,g)(M,g) be a with ∇\nabla. In the convention used here, the Laplace–Beltrami operator on a smooth function ff is

Δgf=−tr⁡g(∇df)=−1∣det⁡g∣ ∂i ⁣(∣det⁡g∣ gij∂jf).\Delta_g f=-\operatorname{tr}_g(\nabla df) =-\frac{1}{\sqrt{|\det g|}}\, \partial_i\!\left(\sqrt{|\det g|}\,g^{ij}\partial_j f\right).

Its , with no factors of ii, is

σ2(Δg)(x,ξ)=−gx−1(ξ,ξ).\sigma_2(\Delta_g)(x,\xi)=-g_x^{-1}(\xi,\xi).
Riemannian signature

If gg is positive definite, Δg\Delta_g is . On compactly supported functions it satisfies

∫Mf‾ Δgf dμg=∫M∣df∣g2 dμg,\int_M \overline f\,\Delta_g f\,d\mu_g =\int_M |df|_g^2\,d\mu_g,

so the convention adopted here gives a nonnegative operator. On Euclidean space it is −∑i∂i2-\sum_i\partial_i^2. Authors who define Δ=div⁡grad⁡\Delta=\operatorname{div}\operatorname{grad} use the negative of this operator.

Indefinite signature

For an indefinite metric, the same coordinate expression is still defined but is not elliptic because its symbol vanishes on nonzero null covectors. In Lorentzian signature it is the normally hyperbolic □g\Box_g. The name “Laplace–Beltrami operator” is most often reserved for the Riemannian case, while “wave operator” or “d’Alembertian” emphasizes the Lorentzian case.

Related operators

This scalar operator should not be confused with the on differential forms or with a on sections of a vector bundle. They agree on functions when their sign conventions are aligned, but their domains and lower-order geometry differ.

References
  1. Peter Petersen, Riemannian Geometry, 3rd ed., Springer, 2016. Publisher record. Relevant: Chapters 2 and 4.
  2. Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, European Mathematical Society, 2007. Publisher record. Relevant: §1.5 and Chapter 3.