Weitzenböck identity

In mathematics, in particular in differential geometry, mathematical physics, and representation theory a Weitzenbock identity expresses a relationship between two second-order elliptic operators on a manifold with the same leading symbol. Usually Weitzenbock 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. Instead of attempting to be completely general, then, this article presents three examples of Weitzenbock identities: from Riemannian geometry, spin geometry, and complex analysis.

Riemannian geometry

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 $langle\; -,-\; angle$ is the metric induced on the bundle of ("p"+1)-forms. The usual form Laplacian is then given by::Δ = dδ + δd.

On the other hand, the Levi-Civita connection supplies a differential operator::$abla:Omega^pM\; ightarrow\; T^*MotimesOmega^pM$where Ω^p"M" is the bundle of "p"-forms and "T"^*M is the cotangent bundle of "M". The Bochner Laplacian is given by ::$Delta\text{'}=\; abla^*\; abla$where $abla^*$ is the adjoint of $abla$.

The Weitzenbock formula then asserts that::Δ' - Δ = "A"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::$A=frac\{1\}\{2\}langle\; R(\; heta,\; heta,\#),\#\; angle\; +\; Ric(\; heta,\#)$where :*"R" is the Riemann curvature tensor, :*"Ric" is the Ricci tensor,:*$heta:T^*MotimesOmega^pM\; ightarrowOmega^\{p+1\}M$ is the alternation map,:*$\#:Omega^\{p+1\}M\; ightarrow\; T^*MotimesOmega^pM$ is the universal derivation inverse to θ on 1-forms.

pin geometry

If "M" is an oriented spin manifold with Dirac operator ð, then one may form the spin Laplacian Δ = ð^*ð + ðð^* on the spin bundle. On the other hand, the Levi-Civita connection extends to the spin bundle to yield a differential operator::$abla:SM\; ightarrow\; T^*Motimes\; SM$As in the case of Riemannian manifolds, let $Delta\text{'}=\; abla^*\; abla$. This is another self-adjoint operator and, moreover, has the same leading symbol as the spin Laplacian. The Weitzenbock formula yields:::$Delta\text{'}-Delta=-frac\{1\}\{4\}Sc$where "Sc" is the scalar curvature.

Complex differential geometry

If "M" is a compact Kähler manifold, there is a Weitzenbock 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 Weitzenbock formula, if α ε Ω^(p,q)"M", then::Δ'α-Δα = "A"(α)where "A" is an operator of order zero involving the curvature. Specifically, if:: in a unitary frame, then:: with "k" in the "s"-th place.

Other Weitzenbock identities

*In conformal geometry there is a Weitzenbock 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.

References

*citation|title=Principles of algebraic geometry|first1=Philip|last1=Griffiths|first2=Joe|last2=Harris|authorlink1=Philip A. Griffiths|authorlink2=Joe Harris (mathematician)|publisher=Wiley-Interscience|publication-date=1994|isbn=978-0471050599|year=1978