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.
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:
![{\displaystyle \int _{M}\langle \alpha ,\delta \beta \rangle :=\int _{M}\langle d\alpha ,\beta \rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/67460d523d9ed8948484871c57bdec8ac24da716)
where
α is any
p-form and
β is any (
p + 1)-form, and
![{\displaystyle \langle \cdot ,\cdot \rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a50080b735975d8001c9552ac2134b49ad534c0)
is the metric induced on the bundle of (
p + 1)-forms. The usual
form Laplacian is then given by
![{\displaystyle \Delta =d\delta +\delta d.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31d3bccb8f6be42b1bf194fe940e5e07667332f8)
On the other hand, the Levi-Civita connection supplies a differential operator
![{\displaystyle \nabla :\Omega ^{p}M\rightarrow \Omega ^{1}M\otimes \Omega ^{p}M,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b48cdbd15a2065d86f7ee71cf5ea3bf879e911e)
where Ω
pM is the bundle of
p-forms. The
Bochner Laplacian is given by
![{\displaystyle \Delta '=\nabla ^{*}\nabla }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea2d43b03f4d4d7aee024ff0fbdb492c001fd7a)
where
![{\displaystyle \nabla ^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be5195d4fd1329c52f53368df15aeb65e4e32e72)
is the adjoint of
![{\displaystyle \nabla }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2)
. This is also known as the connection or rough Laplacian.
The Weitzenböck formula then asserts that
![{\displaystyle \Delta '-\Delta =A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7aff019ee8e6a136a8ba1a0a3e6d41ac60cb6d2)
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
![{\displaystyle A={\frac {1}{2}}\langle R(\theta ,\theta )\#,\#\rangle +\operatorname {Ric} (\theta ,\#),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87928302acde07e856a7561d73bc5f777179d46d)
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[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
![{\displaystyle \nabla :SM\rightarrow T^{*}M\otimes SM.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ffdb607b30c770f0573b9df92b84519c5a138d2)
As in the case of Riemannian manifolds, let
![{\displaystyle \Delta '=\nabla ^{*}\nabla }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea2d43b03f4d4d7aee024ff0fbdb492c001fd7a)
. This is another self-adjoint operator and, moreover, has the same leading symbol as the spin Laplacian. The Weitzenböck formula yields:
![{\displaystyle \Delta '-\Delta =-{\frac {1}{4}}Sc}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bef5f5cf6bd7fd883ab542393d5b64ff57dd104e)
where
Sc is the scalar curvature. This result is also known as the
Lichnerowicz formula.
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
![{\displaystyle \Delta ={\bar {\partial }}^{*}{\bar {\partial }}+{\bar {\partial }}{\bar {\partial }}^{*},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77edc1c1b9f70dc95ae769eb799f36a09bc117dc)
and
![{\displaystyle \Delta '=-\sum _{k}\nabla _{k}\nabla _{\bar {k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24fbccc821c97e3ac394a0f46b736d15b0bac1fb)
in a unitary frame at each point.
According to the Weitzenböck formula, if
, then
![{\displaystyle \Delta ^{\prime }\alpha -\Delta \alpha =A(\alpha )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c98f215189560188256d4e0332d5ec59a5c4a753)
where
![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
is an operator of order zero involving the curvature. Specifically, if
![{\displaystyle \alpha =\alpha _{i_{1}i_{2}\dots i_{p}{\bar {j}}_{1}{\bar {j}}_{2}\dots {\bar {j}}_{q}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c27302fcc857530c650dc6f590c6bd92c44448d)
in a unitary frame, then
![{\displaystyle A(\alpha )=-\sum _{k,j_{s}}\operatorname {Ric} _{{\bar {j}}_{\alpha }}^{\bar {k}}\alpha _{i_{1}i_{2}\dots i_{p}{\bar {j}}_{1}{\bar {j}}_{2}\dots {\bar {k}}\dots {\bar {j}}_{q}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85b831b5642e421135117ef9b0c376bab7efee01)
with
k in the
s-th place.
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]