Knowlpedia

Definition

Codifferential

The degree-lowering formal adjoint of the exterior derivative on an oriented Riemannian manifold.

Let (M,g)(M,g) be an oriented nn-dimensional without boundary. The codifferential is the of the ,

δ:Ωk(M)⟶Ωk−1(M),\delta:\Omega^k(M)\longrightarrow\Omega^{k-1}(M),

characterized by ⟨dα,β⟩L2=⟨α,δβ⟩L2\langle d\alpha,\beta\rangle_{L^2}=\langle\alpha,\delta\beta\rangle_{L^2} for compactly supported forms. With the real convention used here,

δβ=(−1)n(k+1)+1∗d∗β\delta\beta=(-1)^{n(k+1)+1}*d*\beta

for every kk-form β\beta. This definition is formal: it does not by itself specify a domain for an unbounded Hilbert-space operator.

Local formula and elementary properties

For a local orthonormal frame e1,…,ene_1,\ldots,e_n and the Levi–Civita connection ∇\nabla,

δβ=−∑j=1nιej∇ejβ.\delta\beta=-\sum_{j=1}^n\iota_{e_j}\nabla_{e_j}\beta.

Consequently δ2=0\delta^2=0. A form is called coclosed when δβ=0\delta\beta=0. On a function, δ\delta vanishes because there are no forms of degree −1-1.

Relationship to the Hodge Laplacian

The is

Δ=dδ+δd.\Delta=d\delta+\delta d.

It is formally self-adjoint and degree-preserving. On a compact manifold without boundary, a form is harmonic precisely when it is both closed and coclosed. This equivalence follows from

⟨Δα,α⟩L2=∥dα∥L22+∥δα∥L22.\langle\Delta\alpha,\alpha\rangle_{L^2} =\lVert d\alpha\rVert_{L^2}^2+\lVert\delta\alpha\rVert_{L^2}^2.
Conventions and scope
References
  1. Jürgen Jost, Riemannian Geometry and Geometric Analysis, 7th ed., Springer, 2017. Publisher record. Relevant: Chapter 3, “The Laplace Operator and Harmonic Differential Forms.”
  2. Raymond O. Wells Jr., Differential Analysis on Complex Manifolds, 3rd ed., Springer, 2008. Publisher record. Relevant: Chapter III, Hermitian exterior algebra and the Hodge-star operator.