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 principal 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. This article focuses on three examples of Weitzenböck identities: from Riemannian geometry, spin geometry, and complex analysis.

Riemannian geometryEdit

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. 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



  • 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 geometryEdit

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 geometryEdit

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 identitiesEdit

  • 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 alsoEdit


  • Griffiths, Philip; Harris, Joe (1978), Principles of algebraic geometry, Wiley-Interscience (published 1994), ISBN 978-0-471-05059-9