Weitzenböck identity

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.

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.

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.