Weitzenbock identity

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In mathematics, in particular in differential geometry, mathematical physics, and representation theory a Weitzenbock identity expresses a relationship between two second-order elliptic operators on a manifold with the same leading symbol. Usually Weitzenbock formulae are implemented for G-invariant self-adjoint operators between vector bundles associated to some principal G-bundle, although the precise conditions under which such a formula exists are difficult to formulate. Instead of attempting to be completely general, then, this article presents three examples of Weitzenbock identities: from Riemannian geometry, spin geometry, and complex analysis.

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[edit] Riemannian geometry

In Riemannian geometry there are two notions of the Laplacian on differential forms over an oriented compact Riemannian manifold M. The first definition uses the divergence operator δ defined as the formal adjoint of the de Rham operator d:

\int_M \langle \alpha,\delta\beta\rangle := \int_M\langle d\alpha,\beta\rangle

where α is any p-form and β is any (p+1)-form, and \langle -,-\rangle is the metric induced on the bundle of (p+1)-forms. The usual form Laplacian is then given by

Δ = dδ + δd.

On the other hand, the Levi-Civita connection supplies a differential operator

\nabla:\Omega^pM\rightarrow T^*M\otimes\Omega^pM

where ΩpM is the bundle of p-forms and T*M is the cotangent bundle of M. The Bochner Laplacian is given by

\Delta'=\nabla^*\nabla

where \nabla^* is the adjoint of \nabla.

The Weitzenbock formula then asserts that

Δ' - Δ = A

where A is a linear operator of order zero involving only the curvature.

The precise form of A is given, up to an overall sign depending on curvature conventions, by

A=\frac{1}{2}\langle R(\theta,\theta,\#),\#\rangle + Ric(\theta,\#)

where

  • R is the Riemann curvature tensor,
  • Ric is the Ricci tensor,
  • \theta:T^*M\otimes\Omega^pM\rightarrow\Omega^{p+1}M is the alternation map,
  • \#:\Omega^{p+1}M\rightarrow T^*M\otimes\Omega^pM is the universal derivation inverse to θ on 1-forms.

[edit] Spin geometry

If M is an oriented spin manifold with Dirac operator ð, then one may form the spin Laplacian Δ = ð*ð + ðð* on the spin bundle. On the other hand, the Levi-Civita connection extends to the spin bundle to yield a differential operator

\nabla:SM\rightarrow T^*M\otimes SM

As in the case of Riemannian manifolds, let \Delta'=\nabla^*\nabla. This is another self-adjoint operator and, moreover, has the same leading symbol as the spin Laplacian. The Weitzenbock formula yields:

\Delta'-\Delta=-\frac{1}{4}Sc

where Sc is the scalar curvature.

[edit] Complex differential geometry

If M is a compact Kähler manifold, there is a Weitzenbock formula relating the \bar{\partial}-Laplacian (see Dolbeault complex) and the Euclidean Laplacian on (p,q)-forms. Specifically, let

\Delta=\bar{\partial}^*\bar{\partial}+\bar{\partial}\bar{\partial}^*, and
\Delta'=-\sum_k\nabla_k\nabla_{\bar{k}} in a unitary frame at each point.

According to the Weitzenbock formula, if α ε Ω(p,q)M, then

Δ'α-Δα = A(α)

where A is an operator of order zero involving the curvature. Specifically, if

\alpha=\alpha_{i_1i_2\dots i_p\bar{j}_1\bar{j}_2\dots\bar{j}_q} in a unitary frame, then
A(\alpha)=-\sum_{k,j_s} Ric_{\bar{j}_\alpha}^{\bar{k}}\alpha_{i_1i_2\dots i_p\bar{j}_1\bar{j}_2\dots\bar{k}\dots\bar{j}_q} with k in the s-th place.

[edit] Other Weitzenbock identities

  • In conformal geometry there is a Weitzenbock formula relating a particular pair of differential operators defined on the tractor bundle. See Branson, T. and Gover, A.R., "Conformally Invariant Operators, Differential Forms, Cohomology and a Generalisation of Q-Curvature", Communications in Partial Differential Equations, 30 (2005) 1611-1669.
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