nLab jet bundle

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

$$p \coloneqq E \to X$$

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

$$J^k P \to X$$

of order-$k$ jets of sections of $p$ is the bundle whose fiber over a point $x \in X$ is the space of equivalence classes of germs of sections of $p$, where two germs are considered equivalent if their first $k$ partial derivatives at $x$ coincide.

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

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

$$J^\infty E \coloneqq \underset{\longleftarrow}{\lim}_k J^k E = \underset{\longleftarrow}{\lim} \left( \cdots J^3 E \to J^2 E \to J^1 E \to E \right)$$

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.

The Atiyah exact sequence

When $(X,\mathcal{O}_X)$ is a complex-analytic manifold with the structure sheaf of holomorphic functions, and $E$ a locally free sheaf of $\mathcal{O}_X$-modules, we can be even more explicit. The first jet bundle $J^1(E)$ fits into a short exact sequence, called the Atiyah exact sequence:

$$0\to E\otimes_{\mathcal{O}_X}\Omega_X^1\to J^1(E)\to E\to 0$$

where $J^1(E) = (E\otimes\Omega_X^1)\oplus E$ as a $\mathbb{C}$-module, but with an $\mathcal{O}_X$-action given by

$$f(s\otimes\omega,t) = (f s\otimes\omega+t\otimes\mathrm{d}f, f t).$$

The extension class $[J^1(E)]\in\mathrm{Ext}_{\mathcal{O}_X}^1(E,E\otimes\Omega_X^1)$ of this exact sequence is called the Atiyah class of $E$, and is somewhat equivalent to the first Chern class of $E$. Note that the Atiyah class is exactly the obstruction to the Atiyah exact sequence admitting a splitting, and a (holomorphic) splitting of the Atiyah exact sequence is exactly a Koszul connection.

General abstract

We discuss a general abstract definition of jet bundles.

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

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

Notice that we have the canonical morphism, the $X$-component of the unit of the $\Im$-monad

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

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

The corresponding base change geometric morphism is

$$(i^\ast \dashv i_\ast) \;\colon\; \mathbf{H}_{/X} \stackrel{\overset{i^*}{\longleftarrow}}{\underset{Jet := i_*}{\longrightarrow}} \mathbf{H}_{/\Im(X)}$$

Definition

The jet comonad is the (∞,1)-comonad

$$i^\ast i_\ast \;\colon\; \mathbf{H}_{/X} \longrightarrow \mathbf{H}_{/X}$$

Remark

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

$$T_{inf}X \times_X (-) \;\dashv\; Jet$$

where $T_{inf} X$ is the infinitesimal disk bundle of $X$, 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 in Paugam, 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 $X$ is the dependent product operation along the unit of the infinitesimal shape modality

$$jet(E) \coloneqq \underset{X \to \Im X}{\prod} E \,.$$

Definition

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

$$X \to Mod \,.$$

We write

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

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

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

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

We write

$$DQC(X) := Hom(\Im (X), Mod)$$

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

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

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

Application

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

Related concepts

• jet prolongation

• jet space

• jet group, jet groupoid

• differential equation

• evolutionary vector field

• variational bicomplex

  • local Lagrangian

  • source form, Lepage form

• metric jet

• h-principle

• arithmetic jet space

Examples of sequences of local structures

geometry	point	first order infinitesimal	$\subset$	formal = arbitrary order infinitesimal	$\subset$	local = stalkwise	$\subset$	finite
			$\leftarrow$ differentiation		integration $\to$
smooth functions		derivative		Taylor series		germ		smooth function
curve (path)		tangent vector		jet		germ of curve		curve
smooth space		infinitesimal neighbourhood		formal neighbourhood		germ of a space		open neighbourhood
function algebra		square-0 ring extension		nilpotent ring extension/formal completion				ring extension
arithmetic geometry	$\mathbb{F}_p$ finite field			$\mathbb{Z}_p$ p-adic integers		$\mathbb{Z}_{(p)}$ localization at (p)		$\mathbb{Z}$ integers
Lie theory		Lie algebra		formal group		local Lie group		Lie group
symplectic geometry		Poisson manifold		formal deformation quantization		local strict deformation quantization		strict deformation quantization

• in homotopy theory/Goodwillie calculus: jet (∞,1)-category

References

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

• Jana Musilová, Stanislav Hronek, The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories, Communications in Mathematics, Volume 24, Issue 2 (Dec 2016) (doi.org/10.1515/cm-2016-0012)

Lecture notes and textbook accounts:

• 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.

• Demeter Krupka, Josef Janyška, Part 1 of: Lectures on differential invariants, Univerzita J. E. Purkyně, Brno (1990) [ISBN:80-210-165-8, researchgate]

• Jean-François Pommaret, Chapter II in: Partial Differential Equations and Group Theory, Springer (1994) [doi:10.1007/978-94-017-2539-2]

• 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

• Michal MarvanOn 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)

• Igor Khavkine, Urs Schreiber, Synthetic geometry of differential equations: I. Jets and comonad structure (arXiv:1701.06238)

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 in

• Jacob Lurie, p. 6 of: Notes on crystals and algebraic $\mathcal{D}$-modules (2010) [pdf]

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

• Alexander Beilinson, Vladimir Drinfeld, Chiral Algebras

Exposition:

• Frédéric Paugam, Section 2.3 of: Homotopical Poisson Reduction of gauge theories (pdf)

A discussion of jet bundles with an eye towards discussion of the variational bicomplex on them :

• Ian Anderson, chapter 1, section A of: The variational bicomplex [pdf]

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 all this in the convenient context of smooth sets:

• Grigorios Giotopoulos, Hisham Sati, §4 in: Field Theory via Higher Geometry I: Smooth Sets of Fields [arXiv:2312.16301]

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 $\mathbb{Z}_2$-graded zero-curvature representations and its applications (arXiv:1301.7143)

• Gennadi Sardanashvily, Graded infinite order jet manifolds, Int. J. Geom. Methods Mod. Phys. v.4 (2007) 1335-1362 (arXiv:0708.2434)

See also

• Joseph Krasil'shchik, Alexander Verbovetsky, Homological Methods in Equations of Mathematical Physics (arXiv:math/9808130)

On jet bundles of $\mathbb{Z}$-graded manifolds and their vector bundles:

• Jan Vysoky. Graded Jet Geometry. (2023). (arXiv:2311.15754)

On jet bundles in noncommutative geometry:

• Shahn Majid, Francisco Simão, Quantum jet bundles, Lett. Math. Phys. 113 120 (2023) [arXiv:2202.03067, doi:10.1007/s11005-023-01738-z]