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.
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:
⟨ α , δ β ⟩ :=
⟨ d α , β ⟩
where α is any p-form and β is any (p + 1)-form, and
is the metric induced on the bundle of (p + 1)-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.
The Bochner Laplacian is given by
This is also known as the connection or rough Laplacian.
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 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.
If M is a compact Kähler manifold, there is a Weitzenböck formula relating the
-Laplacian (see Dolbeault complex) and the Euclidean Laplacian on (p,q)-forms.
in a unitary frame at each point.
According to the Weitzenböck formula, if
is an operator of order zero involving the curvature.
in a unitary frame, then
with k in the s-th place.