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:

${\displaystyle \int _{M}\langle \alpha ,\delta \beta \rangle :=\int _{M}\langle d\alpha ,\beta \rangle }$

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

${\displaystyle \Delta =d\delta +\delta d.}$

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

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

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

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

where ${\displaystyle \nabla ^{*}}$ is the adjoint of ${\displaystyle \nabla }$.

The Weitzenböck formula then asserts that

${\displaystyle \Delta '-\Delta =A}$

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

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

where

• R is the Riemann curvature tensor,
• Ric is the Ricci tensor,
• ${\displaystyle \theta :T^{*}M\otimes \Omega ^{p}M\rightarrow \Omega ^{p+1}M}$ is the map that takes the wedge product of a 1-form and p-form and gives a (p+1)-form,
• ${\displaystyle \#:\Omega ^{p+1}M\rightarrow T^{*}M\otimes \Omega ^{p}M}$ 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 = ð² on the spin bundle. On the other hand, the Levi-Civita connection extends to the spin bundle to yield a differential operator

${\displaystyle \nabla :SM\rightarrow T^{*}M\otimes SM.}$

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

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

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 ${\displaystyle {\bar {\partial }}}$-Laplacian (see Dolbeault complex) and the Euclidean Laplacian on (p,q)-forms. Specifically, let

${\displaystyle \Delta ={\bar {\partial }}^{*}{\bar {\partial }}+{\bar {\partial }}{\bar {\partial }}^{*}}$, and

${\displaystyle \Delta '=-\sum _{k}\nabla _{k}\nabla _{\bar {k}}}$ in a unitary frame at each point.

According to the Weitzenböck formula, if ${\displaystyle \alpha \in \Omega ^{(p,q)}M}$, then

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

where ${\displaystyle A}$ is an operator of order zero involving the curvature. Specifically,

if ${\displaystyle \alpha =\alpha _{i_{1}i_{2}\dots i_{p}{\bar {j}}_{1}{\bar {j}}_{2}\dots {\bar {j}}_{q}}}$ in a unitary frame, then

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

See also[]

• Bochner identity
• Bochner–Kodaira–Nakano identity
• Laplacian operators in differential geometry

References[]

• Griffiths, Philip; Harris, Joe (1978), Principles of algebraic geometry, Wiley-Interscience (published 1994), ISBN 978-0-471-05059-9