nLab
jet bundle

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

          </semantics></math></div>

          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          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

          pEX p \coloneqq E \to X

          a surjective submersion of smooth manifolds and kk \in \mathbb{N}, the bundle

          J kPX J^k P \to X

          of order-kk 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).

          In order to pass to kk \to \infty to form the infinite jet bundle J PJ^\infty P one forms the projective limit over the finite-order jet bundles,

          J Elim kJ kE=lim(J 3EJ 2EJ 1EE) J^\infty E \coloneqq \underset{\longleftarrow}{\lim}_k J^k E = \underset{\longleftarrow}{\lim} \left( \cdots J^3 E \to J^2 E \to J^1 E \to E \right)

          but one has to decide in which category of infinite-dimensional manifolds to take this limit:

          1. one may form the limit formally, i.e. in pro-manifolds. This is what is implicit for instance in Anderson, p.3-5;

          2. one may form the limit in Fréchet manifolds, this is farily explicit in (Saunders 89, chapter 7). See at Fréchet manifold – Projective limits of finite-dimensional manifolds. Beware that this is not equivalent to the pro-manifold structure (see the remark here). It makes sense to speak of locally pro-manifolds.

          General abstract

          We discuss a general abstract definition of jet bundles.

          Let H\mathbf{H} be an (∞,1)-topos equipped with differential cohesion with infinitesimal shape modality \Im (or rather a tower k\Im_k of such, for each infinitesimal order k{}k \in \mathbb{N} \cup \{\infty\} ).

          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-component 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^*}{\longleftarrow}}{\underset{Jet := i_*}{\longrightarrow}} \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. ,

          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 comonad 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. 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

          Exposition of variational calculus in terms of jet bundles and Lepage forms and aimed at examples from physics is in

          Textbook accounts and lecture notes include

          • Peter Michor, Manifolds of differentiable mappings, Shiva Publishing (1980) pdf

          • 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)

          The algebra of smooth functions of just locally finite order on the jet bundle (“locally pro-manifold”) was maybe first considered in

          • Floris Takens, A global version of the inverse problem of the calculus of variations, J. Differential Geom. Volume 14, Number 4 (1979), 543-562. (Euclid)

          Discussion of the Fréchet manifold structure on infinite jet bundles includes

          • David Saunders, chapter 7 Infinite jet bundles of The geometry of jet bundles, London Mathematical Society Lecture Note Series 142, Cambridge Univ. Press 1989.

          • M. Bauderon, Differential geometry and Lagrangian formalism in the calculus of variations, in Differential Geometry, Calculus of Variations, and their Applications, Lecture Notes in Pure and Applied Mathematics, 100, Marcel Dekker, Inc., N.Y., 1985, pp. 67-82.

          • C. T. J. Dodson, George Galanis, Efstathios Vassiliou,, p. 109 and section 6.3 of Geometry in a Fréchet Context: A Projective Limit Approach, Cambridge University Press (2015)

          • Andrew Lewis, The bundle of infinite jets (2006) (pdf)

          Discussion of finite-order jet bundles 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)

          • Anders Kock, section 2.7 of Synthetic geometry of manifolds, Cambridge Tracts in Mathematics 180 (2010). (pdf)

          The jet 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

          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

          The de Rham complex and variational bicomplex of jet bundles is discussed in

          • G. Giachetta, L. Mangiarotti, Gennadi Sardanashvily, Cohomology of the variational bicomplex on the infinite order jet space, Journal of Mathematical Physics 42, 4272-4282 (2001) (arXiv:math/0006074)

          where both versions (smooth functions being globally or locally of finite order) are discussed and compared.

          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)

          See also

          In the context of supermanifolds, discussion is in

          Last revised on November 13, 2017 at 09:40:01. See the history of this page for a list of all contributions to it.