In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, in particular in
differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
,
mathematical physics
Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...
, and
representation theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
, a Weitzenböck identity, named after
Roland Weitzenböck, expresses a relationship between two second-order
elliptic operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which im ...
s on a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
with the same principal symbol. Usually Weitzenböck formulae are implemented for ''G''-invariant
self-adjoint operator
In mathematics, a self-adjoint operator on a complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle is a linear map ''A'' (from ''V'' to itself) that is its own adjoint. That is, \langle Ax,y \rangle = \langle x,Ay \rangle for al ...
s between
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...
s 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
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
,
spin geometry, and
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
.
Riemannian geometry
In
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
there are two notions of the
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...
on
differential forms
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
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 ()-form, and
is the metric induced on the bundle of ()-forms. The usual form Laplacian is then given by
On the other hand, the
Levi-Civita connection
In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the ( pseudo-) Riemannian ...
supplies a differential operator
where Ω
''p''''M'' 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
If ''M'' is an oriented
spin manifold with
Dirac operator
In mathematics and in quantum mechanics, a Dirac operator is a first-order differential operator that is a formal square root, or half-iterate, of a second-order differential operator such as a Laplacian. It was introduced in 1847 by William Ham ...
ð, then one may form the spin Laplacian Δ = ð
2 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
If ''M'' is a compact
Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnol ...
, 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
*In
conformal geometry
In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space.
In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two di ...
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
*
Bochner identity
*
Bochner–Kodaira–Nakano identity
*
Laplacian operators in differential geometry
References
*
{{DEFAULTSORT:Weitzenbock identity
Mathematical identities
Differential operators
Differential geometry