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.
For $p \coloneq P \to X$ a surjective submersion of smooth manifolds and $k \in \mathbb{N}$, the bundle $J^k P \to X$ of $k$- order 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.
For $k \to \infty$ one takes $J^\infty P$ to the the formal projective limit over the finite-order jet bundles, i.e. the pro-manifold presented by the sequence $\cdots \to J^2 P \to J^1 P \to J^0 P = P$.
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 $J^k(X,Y)$.
We discuss a general abstract definition of jet bundles.
Let
$\mathbf{H}$ be an (∞,1)-topos equipped with differential cohesion with infinitesimal shape modality $\Im$;
and equipped with an (∞,2)-sheaf
Mod $\colon \; \mathbf{H}^{op} \to$ Stab(∞,1)Cat
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
(“inclusion of constant paths into all infinitesimal paths”).
The corresponding base change geometric morphism is
The jet comonad is the (∞,1)-comonad
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. 1,
where $T_{inf} X$ is the infinitesimal disk bundle of $X$, see at differential cohesion – infinitesimal disk bundle – relation to jet bundles
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. 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.
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
A quasicoherent (∞,1)-sheaf on $X$ is a morphism of (∞,2)-sheaves
We write
for the stable (∞,1)-category of quasicoherent (∞,1)-sheaves.
A D-module on $X$ is a morphism of (∞,2)-sheaves
We write
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 $\mathcal{O}(X)$
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
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 |
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
Discussion of the Frechet manifold structure on jet bundles includes
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)
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.
Andrew Lewis, The bundle of infinite jets 2006 (pdf)
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)
Anders Kock, section 2.7 of Synthetic geometry of manfiolds, Cambridge Tracts in Mathematics 180 (2010). (pdf)
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
The de Rham cohomology of jet bundles is discussed in
Discussion of jet-restriction of the Haefliger groupoid is in
Discussion of jet bundles in supergeometry includes