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 called the space of jets of maps from XX to YY and commonly denoted with J k(X,Y)J^k(X,Y).

General abstract

We discuss a general abstract definition of jet bundles.

Let

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

Notice that we have the canonical morphism, the XX-componenet of the unit of the \Im-monad

i:X(X) i \colon X \to \Im(X)

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

The corresponding base change geometric morphism is

(i *i *):H /XJet:=i *i *H /(X) (i^\ast \dashv i_\ast) \;\colon\; \mathbf{H}_{/X} \stackrel{\overset{i^*}{\leftarrow}}{\underset{Jet := i_*}{\to}} \mathbf{H}_{/\Im(X)}
Definition

The jet comonad is the (∞,1)-comonad

i *i *:H /XH /X i^\ast i_\ast \;\colon\; \mathbf{H}_{/X} \longrightarrow \mathbf{H}_{/X}
Remark

Since base change gives even an adjoint triple (i !i *i *)(i_! \dashv i^\ast \dashv i_\ast), there is a left adjoint T infX× X()T_{inf} X \times_X (-) to the jet comonad of def. 1,

T infX× X()Jet T_{inf}X \times_X (-) \;\dashv\; Jet

where T infXT_{inf} X is the infinitesimal disk bundle of XX, see at differential cohesion – infinitesimal disk bundle – relation to jet bundles

Remark

In the context of differential geometry the fact that the jet bundle construction is a co-monad was explicitly observed in (Marvan 86, see also Marvan 93, section 1.1, Marvan 89). It is almost implicit in (Krasil’shchik-Verbovetsky 98, p. 13, p. 17, Krasilshchik 99, p. 25).

In the context of synthetic differential geometry the fact that the jet bundle construction is right adjoint to the infinitesimal disk bundle construction is (Kock 80, prop. 2.2).

In the context of algebraic geometry and of D-schemes as in (BeilinsonDrinfeld, 2.3.2, reviewed in Paugam, section 2.3), the base change comonad formulation inf def. 1 was noticed in (Lurie, prop. 0.9).

In as in (BeilinsonDrinfeld, 2.3.2, reviewed inPaugam, section 2.3) jet bundles are expressed dually in terms of algebras in D-modules. We now 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)XXE. jet(E) \coloneqq \underset{X \to \Im 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

(X)Mod. \Im (X) \to Mod \,.

We write

DQC(X):=Hom((X),Mod) DQC(X) := Hom(\Im (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 local structures

geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

References

Textbook accounts and lecture notes include

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

  • Joseph Krasil'shchik in collaboration with Barbara Prinari, Lectures on Linear Differential Operators over Commutative Algebras, 1998 (pdf)

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

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

  • Peter Olver, Lectures on Lie groups and differential equation, chapter 3, Jets and differential invariants, 2012 (pdf)

Early accounts include

  • Hubert Goldschmidt, Integrability criteria for systems of nonlinear partial differential equations, J. Differential Geom. Volume 1, Number 3-4 (1967), 269-307 (Euclid)

Discussion in tems of synthetic differential geometry is in

  • Anders Kock, Formal manifolds and synthetic theory of jet bundles, Cahiers de Topologie et Géométrie Différentielle Catégoriques (1980) Volume: 21, Issue: 3 (Numdam)

See also

The comonad structure on the jet operation in the context of differential geometry is made explicit in

  • Michal Marvan, A note on the category of partial differential equations, in Differential geometry and its applications, Proceedings of the Conference August 24-30, 1986, Brno (pdf)

    (notice that prop. 1.3 there is wrong, the correct version is in the thesis of the author)

with further developments in

  • Michal Marvan On the horizontal cohomology with general coefficients, 1989 (web announcement, web archive)

    Abstract: In the present paper the horizontal cohomology theory is interpreted as a special case of the Van Osdol bicohomology theory applied to what we call a “jet comonad”. It follows that differential equations have well-defined cohomology groups with coefficients in linear differential equations.

  • Michal Marvan, section 1.1 of On Zero-Curvature Representations of Partial Differential Equations, (1993) (web)

In the context of algebraic geometry, 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

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)

The de Rham cohomology of jet bundles is discussed in

  • G.Giachetta, L.Mangiarotti, G.Sardanashvily, Cohomology of the variational bicomplex on the infinite order jet space (arXiv:math/0006074)

Discussion of jet-restriction of the Haefliger groupoid is in

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

Discussion of jet bundles in supergeometry includes

  • Arthemy V. Kiselev, Andrey O. Krutov, appendix of On the (non)removability of spectral parameters in 2\mathbb{Z}_2-graded zero-curvature representations and its applications (arXiv:1301.7143)

Revised on August 26, 2015 01:42:15 by Urs Schreiber (195.113.30.252)