nLab jet space

Jet spaces


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Jet spaces


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-11 tangency at a given point in the target, jet spaces are equivalence classes of germs of smooth maps with respect to (finite) order-kk 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

Last revised on January 4, 2021 at 18:26:14. See the history of this page for a list of all contributions to it.