Knowlpedia

Definition

Codifferential

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

Definition

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.

The codifferential and this energy identity are developed in Jost, Chapter 3, “The Laplace Operator and Harmonic Differential Forms”.

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.