For a surjective submersion of smooth manifolds and , the order - jet bundle 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 derivatives coincide.
We discuss a general abstract definition of jet bundles.
be a cohesive (∞,1)-topos
equipped with infinitesimal 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). See (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 infinitesimal and local structures
|first order infinitesimal||formal = arbitrary order infinitesimal||local = stalkwise||finite|
|derivative||Taylor series||germ||smooth function|
|tangent vector||jet||germ of curve||curve|
|square-0 ring extension||nilpotent ring extension||ring extension|
|Lie algebra||formal group||local Lie group||Lie group|
|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