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:
On the other hand, the Levi-Civita connection supplies a differential operator
The Weitzenböck formula then asserts that
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 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
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
According to the Weitzenböck formula, if , then
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]
- Griffiths, Philip; Harris, Joe (1978), Principles of algebraic geometry, Wiley-Interscience (published 1994), ISBN 978-0-471-05059-9