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 d\alpha ,\beta \rangle }
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
:\Omega ^{p}M\rightarrow \Omega ^{1}M\otimes \Omega ^{p}M,}
where ΩpM is the bundle of p-forms. The Bochner Laplacian is given by
where is the adjoint of . 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
- R is the Riemann curvature tensor,
- Ric is the Ricci tensor,
- is the map that takes the wedge product of a 1-form and p-form and gives a (p+1)-form,
- is the universal derivation inverse to θ on 1-forms.
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. Specifically, let
and
in a unitary frame at each point.
According to the Weitzenböck formula, if , then
where is an operator of order zero involving the curvature. Specifically, if
in a unitary frame, then
with k in the s-th place.
- 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.