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)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(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 higher order frame bundle is higher order jet version of the frame bundle.
Its structure group is the jet group $GL^k$ of the given order.
In synthetic differential geometry/differential cohesion the higher order frame bundle is the bundle of infinitesimal disks of the given order. It is the principal bundle (principal infinity-bundle) to which the infinitesimal disk bundle is the associated bundle (associated infinity-bundle)
See at differential cohesion – Frame bundles.
Demeter Krupka, Josef Janyška, Lectures on differential invariants, Univerzita JEP, Brno, 1990.
Ivan Kolář, Peter Michor, Jan Slovák, section 12.12 of Natural operators in differential geometry (pdf)
Ivan Kolář, Connections on higher order frame bundles and their gauge analogies pdf
Last revised on May 13, 2015 at 12:44:21. See the history of this page for a list of all contributions to it.