Weitzenböck identity
# Weitzenböck identity

In mathematics, in particular in differential geometry, mathematical physics, and representation theory a Weitzenböck identity, named after Roland Weitzenböck, expresses a relationship between two second-order elliptic operators on a manifold with the same principal symbol. Usually Weitzenböck 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. This article focuses on three examples of Weitzenböck identities: from Riemannian geometry, spin geometry, and complex analysis.

## 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 $\langle \cdot ,\cdot \rangle$ is the metric induced on the bundle of (p + 1)-forms. The usual form Laplacian is then given by

$\Delta =d\delta +\delta d.$

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

$\nabla :\Omega ^{p}M\rightarrow \Omega ^{1}M\otimes \Omega ^{p}M,$

where $Ω^pM$ is the bundle of p-forms. The Bochner Laplacian is given by

$\Delta '=\nabla ^{*}\nabla$

where $\nabla ^{*}$ is the adjoint of $\nabla$. This is also known as the connection or rough Laplacian.

The Weitzenböck formula then asserts that

$\Delta '-\Delta =A$

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

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

where

* R is the Riemann curvature tensor,
* Ric is the Ricci tensor,
* $\theta :T^{*}M\otimes \Omega ^{p}M\rightarrow \Omega ^{p+1}M$ is the map that takes the wedge product of a 1-form and p-form and gives a (p+1)-form,
* $\#:\Omega ^{p+1}M\rightarrow T^{*}M\otimes \Omega ^{p}M$ is the universal derivation inverse to θ on 1-forms.

## Spin geometry

If M is an oriented spin manifold with Dirac operator ð, then one may form the spin Laplacian Δ = $ð^2$ 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 Weitzenböck formula yields:

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

where Sc is the scalar curvature. This result is also known as the Lichnerowicz formula.

## Complex differential geometry

If M is a compact Kähler manifold, there is a Weitzenböck 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 Weitzenböck formula, if $\alpha \in \Omega ^{(p,q)}M$, then

$\Delta ^{\prime }\alpha -\Delta \alpha =A(\alpha )$

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

$\alpha =\alpha _{i_{1}i_{2}\dots i_{p}{\bar {j}}_{1}{\bar {j}}_{2}\dots {\bar {j}}_{q}}$

in a unitary frame, then

$A(\alpha )=-\sum _{k,j_{s}}\operatorname {Ric} _{{\bar {j}}_{\alpha }}^{\bar {k}}\alpha _{i_{1}i_{2}\dots i_{p}{\bar {j}}_{1}{\bar {j}}_{2}\dots {\bar {k}}\dots {\bar {j}}_{q}}$

with k in the s-th place.

## Other Weitzenböck identities

* In conformal geometry there is a Weitzenböck 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.
