For a surjective submersion of smooth manifolds and , the bundle of - order jets of sections of is the bundle whose fiber over a point is the space of equivalence classes of germs of sections of , where two germs are considered equivalent if their first partial derivatives at coincide.
In the case when is a trivial bundle its sections are canonically in bijection with maps from to and two sections have the same partial derivatives iff the partial derivatives of the corresponding maps from to agree. So in this case the jet space is called the space of jets of maps from to and commonly denoted with .
We discuss a general abstract definition of jet bundles.
and equipped with an (∞,2)-sheaf
For , write for the corresponding de Rham space object.
(“inclusion of constant paths into all infinitesimal paths”).
where is the infinitesimal disk bundle of , 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 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).
for the stable (∞,1)-category of D-modules.
Examples of sequences of local structures
|geometry||point||first order infinitesimal||formal = arbitrary order infinitesimal||local = stalkwise||finite|
|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||finite field||p-adic integers||localization at (p)||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.
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
Early accounts include
Discussion of the Frechet manifold structure on jet bundles includes
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.
Discussion in tems of synthetic differential geometry is in
The comonad structure on the jet operation in the context of differential geometry is made explicit in
(notice that prop. 1.3 there is wrong, the correct version is in the thesis of the author)
with further developments in
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.
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
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