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