synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
The concept of jet group is the generalization of general linear group from first order to higher order jets.
In terms of synthetic differential geometry/differential cohesion a general linear group is the automorphism group of a first-order infinitesimal disk, while a jet group is the automorphism group of a higher order infinitesimal disk. See also at differential cohesion – Frame bundles.
For all $k \in \mathbb{N}$, the homotopy type of the orientation preserving jet group $GL^k_p(n)$ is that of the ordinary orientation-preserving general linear group $GL(n)$, and the canonical projection
is, on the level of the underlying topological spaces, a homotopy equivalence, indeed it preserves the maximal compact subgroup, which is the special orthogonal group $SO(n)$ on both sides (recalled e.g. in Dartnell 94, section 1).
The canonical projection $GL^k_+(n) \longrightarrow GL_+(n)$ also induces an isomorphism on group homology with constant integer coefficients
Original discussion (in the context of integrability of G-structures) is due to
Textbook accounts and lecture notes include
C.L. Terng, Natural vector bundles and natural differential operators, Amer. J. Math. 100 (1978) 775-828.
Demeter Krupka, Josef Janyška, Lectures on differential invariants, Univerzita JEP, Brno, 1990.
Ivan Kolář, Peter Michor, Jan Slovák, section 13 of Natural operators in differential geometry (pdf)
See also
Wikipedia, Jet group
Discussion of the group homology of jet groups includes
Pablo Dartnell, On the homology of groups of jets, Journal of Pure and Applied Algebra Volume 92, Issue 2, 7 March 1994, Pages 109–121 (publisher)
Dror Farjoun, Jekel, Suciu, Homology of jet groups (pdf)
Last revised on January 16, 2015 at 20:09:28. See the history of this page for a list of all contributions to it.