Weitzenböck identity

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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 leading symbol. (The origins of this terminology seem doubtful, however, as there does not seem to be any evidence that such identities ever appeared in Weitzenböck's work.) 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. Instead of attempting to be completely general, then, this article presents 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 (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 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


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 (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 α ε Ω(p,q)M, then

Δ'α − Δα = A(α)

where A 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

  • 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


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