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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. 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

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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 . This is also known as the connection or rough Laplacian.

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 where

  • 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

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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

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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 identities

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  • 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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References

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