# nLab jet space

Jet spaces

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Jet spaces

## 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.

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