nLab
jet bundle

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Idea

A jet can be thought of as the infinitesimal germ of a section of some bundle or of a map between spaces. Jets are a coordinate free version of Taylor-polynomials and Taylor series.

Definition

Concrete

For p:PXp : P \to X a surjective submersion of smooth manifolds and kk \in \mathbb{N}, the bundle J kPXJ^k P \to X of kk- order jets of sections of pp is the bundle whose fiber over a point xXx \in X is the space of equivalence classes of germs of sections of pp, where two germs are considered equivalent if their first kk partial derivatives at xx coincide.

In the case when pp is a trivial bundle p:X×YXp:X\times Y \to X its sections are canonically in bijection with maps from XX to YY and two sections have the same partial derivatives iff the partial derivatives of the corresponding maps from XX to YY agree. So in this case the jet space J kPJ^k P is the space of jets of maps from XX to YY and commonly denoted with J k(X,Y)J^k(X,Y).

The infinite order jet bundle as the cofree J 1J^1-coalgebra

Let J 1J^1 be the functor on bundles over XX sending a bundle p:PXp:P\to X to the bundle J 1p:J 1PXJ^1 p: J^1 P \to X of first order jets of sections of pp. This functor is co-pointed: the natural projection J 1PPJ^1 P \to P forgets first order information. A J 1J^1-coalgebra is then by definition a bundle p:PXp:P\to X with a bundle map PJ 1PP\to J^1 P, which is the same as a connection on PP. With this, the infinite order jet bundle of sections of pp may be defined as the cofree J 1 J^1-coalgebra on PP. By definition it comes with a bundle map p :J PPp_\infty: J^\infty P \to P (“forgetting all higher order information”) and a universal connection C:J PJ 1(J P)C: J^\infty P \to J^1( J^\infty P) sometimes called the Cartan connection or infinite order contact structure on J PJ^\infty P. Its universal property states that for any bundle q:QXq:Q\to X with connection K:QJ 1QK: Q \to J^1 Q and bundle map φ:QP\varphi: Q\to P, there exists a unique prolongation of φ\varphi denoted with j φ:QJ Pj^\infty \varphi: Q \to J^\infty P commuting with all given maps: p j φ=φp_\infty \circ j^\infty \varphi =\varphi and J 1(j φ)K=Cj φJ^1(j^\infty \varphi)\circ K= C \circ j^\infty \varphi.

In the particular case when q:QXq:Q\to X is the identity XXX\to X with its canonical connection, a morphisms of bundles φ\varphi from qq to pp is the same as a section of pp and j φj^\infty \varphi is traditionally called the infinite jet prolongation of the section φ\varphi.

When the the functor J 1J^1 preserves sequential limits and the category of bundles over XX has equalizers, the cofree J 1J^1-coalgebra can be constructed as a sequential limit:

J 3PJ 2PJ 1PJ 0P=P \cdots \to J^3 P \to J^2 P \to J^1 P \to J^0 P = P

where J n+1PJ^{n+1} P is recursively defined as the equalizer between J 1(J nP)J nPJ 1(J n1P)J^1 (J^n P ) \to J^n P \to J^1 ( J^{n-1} P) and J 1(J nP)J 1(J n1P)J^1 (J^n P) \to J^1 (J^{n-1} P), where the first arrow in J 1(J nP)J nPJ 1(J n1P)J^1 (J^n P ) \to J^n P \to J^1 ( J^{n-1} P) is the standard projection and the second one is the previously obtained equalizer (starting the induction with the identity J 1PJ 1PJ^1 P \to J^1 P), while the arrow J 1(J nP)J 1(J n1P)J^1(J^n P ) \to J^1( J^{n-1} P) is just the map obtained applying the functor J 1J^1 to J nPJ n1PJ^n P \to J^{n-1} P.

This sequantial limit construction should be a special case (and dual) to the one described in transfinite construction of free algebras.

other definitions

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General abstract

We discuss a general abstract definition of jet bundles.

Let

For XHX \in \mathbf{H}, write Π inf(X)\mathbf{\Pi}_{inf}(X) for the corresponding de Rham space object.

Notice that we have the canonical morphism

i:XΠ inf(X) i : X \to \mathbf{\Pi}_{inf}(X)

(“inclusion of constant paths into all infinitesimal paths”).

Definition

Write

Jet:H /XJet:=i *i *H /Π inf(X) Jet \;\colon\; \mathbf{H}_{/X} \stackrel{\overset{i^*}{\leftarrow}}{\underset{Jet := i_*}{\to}} \mathbf{H}_{/\mathbf{\Pi}_{inf}(X)}

for the corresponding base change geometric morphism.

Its direct image may be called the jet bundle (∞,1)-functor . Or rather, if one regards the jet bundle again as a bundle over XX, then the jet construction is the (∞,1)-comonad i *i *i^\ast i_\ast.

In the context of D-schemes this is (BeilinsonDrinfeld, 2.3.2). The abstract formulation as used here appears in (Lurie, prop. 0.9). See also (Paugam, section 2.3) for a review. There this is expressed dually in terms of algebras in D-modules. We indicate how the translation works

Remark

In terms of differential homotopy type theory this means that forming “jet types” of dependent types over XX is the dependent product operation along the unit of the infinitesimal shape modality

jet(E)XΠ infXE. jet(E) \coloneqq \underset{X \to \Pi_{inf}X}{\prod} E \,.
Definition

A quasicoherent (∞,1)-sheaf on XX is a morphism of (∞,2)-sheaves

XMod. X \to Mod \,.

We write

QC(X):=Hom(X,Mod) QC(X) := Hom(X, Mod)

for the stable (∞,1)-category of quasicoherent (∞,1)-sheaves.

A D-module on XX is a morphism of (∞,2)-sheaves

Π inf(X)Mod. \mathbf{\Pi}_{inf}(X) \to Mod \,.

We write

DQC(X):=Hom(Π inf(X),Mod) DQC(X) := Hom(\mathbf{\Pi}_{inf}(X), Mod)

for the stable (∞,1)-category of D-modules.

The Jet algebra functor is the left adjoint to the forgetful functor from commutative algebras over 𝒟(X)\mathcal{D}(X) to those over the structure sheaf 𝒪(X)\mathcal{O}(X)

(JetF):Alg 𝒟(X)FJetAlg 𝒪(X). (Jet \dashv F) : Alg_{\mathcal{D}(X)} \stackrel{\overset{Jet}{\leftarrow}}{\underset{F}{\to}} Alg_{\mathcal{O}(X)} \,.

Application

Typical Lagrangians in quantum field theory are defined on jet bundles. Their variational calculus is governed by Euler-Lagrange equations.

Examples of sequences of infinitesimal and local structures

first order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
derivativeTaylor seriesgermsmooth function
tangent vectorjetgerm of curvecurve
square-0 ring extensionnilpotent ring extensionring extension
Lie algebraformal grouplocal Lie groupLie group
Poisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

References

The abstract characterization of jet bundles as the direct images of base change along the de Rham space projection is noticed on p. 6 of

The explicit description in terms of formal duals of commutative monoids in D-modules is in

An exposition of this is in section 2.3 of

Standard textbook references include

  • G. Sardanashvily, Fibre bundles, jet manifolds and Lagrangian theory, Lectures for theoreticians, arXiv:0908.1886

  • Shihoko Ishii, Jet schemes, arc spaces and the Nash problem, arXiv:math.AG/0704.3327

  • D. J. Saunders, The geometry of jet bundles, London Mathematical Society Lecture Note Series 142, Cambridge Univ. Press 1989.

A discussion of jet bundles with an eye towards discussion of the variational bicomplex on them is in chapter 1, section A of

  • Ian Anderson, The variational bicomplex (pdf)

Discussion of jet-restriction of the Haefliger groupoid is in

  • Arne Lorenz, Jet Groupoids, Natural Bundles and the Vessiot Equivalence Method, Thesis (pdf)

Revised on April 23, 2014 07:10:26 by Michael Bächtold (147.88.200.8)