Relates 2 second-order elliptic operators on a manifold with the same principal symbol
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[edit]
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 (
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 Ω
pM 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.
Spin geometry[edit]
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[edit]
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.
Other Weitzenböck identities[edit]
- 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[edit]
References[edit]