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:
where α is any p-form and β is any -form, and is the metric induced on the bundle of -forms. The usual form Laplacian is then given by
On the other hand, the Levi-Civita connection supplies a differential operator
where ΩpM is the bundle of p-forms and T∗M is the cotangent bundle of M. The Bochner Laplacian is given by
where is the adjoint of.
The Weitzenböck formula then asserts that
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
where

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
As in the case of Riemannian manifolds, let. This is another self-adjoint operator and, moreover, has the same leading symbol as the spin Laplacian. The Weitzenböck formula yields:
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 -Laplacian and the Euclidean Laplacian on -forms. Specifically, let
According to the Weitzenböck formula, if, then
where is an operator of order zero involving the curvature. Specifically,

Other Weitzenböck identities