nLab jet space

Idea

The notion of jet space or jet bundle is a generalization of the notion of tangent spaces and tangent bundles, respectively. While a tangent vector is an equivalence class of germs of curves with order-$1$ tangency at a given point in the target, jet spaces are equivalence classes of germs of smooth maps with respect to (finite) order-$k$ tangency at some point in the target.

One version in algebraic geometry is jet scheme.

From the nPOV, the map from a bundle to its jet bundle can be understood in terms of the jet comonad.

Related pages

• jet group, jet groupoid

• jet bundle

• metric jet

• h-principle

• in algebraic geometry: jet scheme

• ín arithmetic geometry: arithmetic jet space

References

Jet bundles were first introduced by Charles Ehresmann.

• wikipedia: jet, jet bundle

• Ivan Kolar, Jan Slovak, Peter Michor, Natural operations in differential geometry, book 1993, 1999, pdf, hyper-dvi, ps

• G. Sardanashvily, Fibre bundles, jet manifolds and Lagrangian theory, Lectures for theoreticians, arXiv:0908.1886

• D. J. Saunders, The geometry of jet bundles, London Mathematical Society Lecture Note Series 142, Cambridge Univ. Press 1989.

• Arthemy Kiselev, The twelve lectures in the (non)commutative geometry of differential equations, preprint IHES M/12/13 pdf

In the language of schemes

• R. Moosa, T. Scanlon, Jet and prolongation spaces, J. Inst. Math. Jussieu 9(02) 391 (2010) doi
• Shihoko Ishii, Jet schemes, arc spaces and the Nash problem, arXiv:math.AG/0704.3327