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
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Given a differentiable map $\phi \colon X_1 \xrightarrow{\;} X_2$ between differentiable manifolds (e.g. a smooth map between smooth manifolds) and thinking of vector fields as infinitesimal approximations to differentiable curves
then the postcomposition of these curves with $\phi$ induces maps of equivalence classes
alternatively denoted “$\phi_\ast$” or “$\mathrm{d}\phi$” (cf. differentiation as a functor) and called the pushforward of vector fields along $\phi$.
Most texts on differential geometry will discuss pushforward of vector fields.
See also
Created on June 21, 2024 at 09:45:41. See the history of this page for a list of all contributions to it.