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
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Given a vector space $V$ and an element $\phi$ in the exterior product $\wedge^p V^\ast$ (a $p$-covector), then a differential p-form $\omega$ on a smooth manifold $X$ whose tangent spaces look like $V$ is called definite on $\phi$ (Bryant 05, section 3.1.1) or stable at each point (Hitchin, p. 3) if at each point $x \in X$ the restriction $\omega|_x \in \wedge^p T^\ast X \simeq \wedge^p V^\ast$ is equal to $\phi$, up to a general linear transformation.
The existence of a definite form implies a G-structure on $X$ for $G$ the stabilizer subgroup of $\phi$.
A class of examples of definite forms are the 3-forms on G2-manifolds, these are definite on the “associative 3-form” on $\mathbb{R}^7$.
The higher prequantization of a definition form is a definite globalization of a WZW term.
Given a vector space $V$ and a stable form $\phi \in \wedge^p V^\ast$ (hence a form whose orbit under the general linear group $GL(V)$ is an open subspace in $\wedeg^p V$), and given a smooth manifold modeled on the vector space $V$, then a differential form $\omega \in \Omega^p(X)$ is definite on $\phi$ if at each point it is in this open orbit.
See at G2-manifold – Definite forms
Nigel Hitchin, Special holonomy and beyond, Clay Mathematics Proceedings (pdf)
Robert Bryant, Some remarks on $G_2$-structures, Proceedings of the 12th Gökova Geometry-Topology Conference 2005, pp. 75-109 pdf