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.
be a cohesive (∞,1)-topos
equipped with differential cohesion
and equipped with an (∞,2)-sheaf
For , write for the corresponding de Rham space object.
Notice that we have the canonical morphism
(“inclusion of constant paths into all infinitesimal paths”).
for the corresponding base change geometric morphism.
In the context of D-schemes this is (BeilinsonDrinfeld, 2.3.2). The abstract formulation as used here appears in (Lurie, prop. 0.9). See also (Paugam, section 2.3) for a review. There this is expressed dually in terms of algebras in D-modules. We indicate how the translation works
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|
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
Standard textbook references include
G. Sardanashvily, Fibre bundles, jet manifolds and Lagrangian theory, Lectures for theoreticians, arXiv:0908.1886
Shihoko Ishii, Jet schemes, arc spaces and the Nash problem, arXiv:math.AG/0704.3327
D. J. Saunders, The geometry of jet bundles, London Mathematical Society Lecture Note Series 142, Cambridge Univ. Press 1989.
A discussion of jet bundles with an eye towards discussion of the variational bicomplex on them is in chapter 1, section A of
Discussion of jet-restriction of the Haefliger groupoid is in