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 tangent vector field on a differentiable manifold $X$ then its flow is the group of diffeomorphisms of $X$ that lets the points of the manifold “flow along the vector field” hence which sends them along flow lines (integral curvs) that are tangent to the vector field.
Throughout, let $X$ be a differentiable manifold and let $v \in \Gamma(T X)$ be a continuously differentiable vector field on $X$ (i.e. of class $C^1$).
(integral curves/flow lines)
An integral curve or flow line of the vector field $v$ is a differentiable function of the form
for $U \subset \mathbb{R}$ an open interval with the property that its tangent vector at any $t \in \mathbb{R}$ equals the value of the vector field $v$ at the point $\gamma(t)$:
(flow of a vector field)
A global flow of $v$ is a function of the form
such that for each $x \in X$ the function $\phi(x,-) \colon \mathbb{R} \to X$ is an integral curve of $v$ (def. 1).
A flow domain is an open subset $O \subset X \times \mathbb{R}$ such that for all $x \in X$ the intersection $O \cap \{x\} \times \mathbb{R}$ is an open interval containing $0$.
A flow of $v$ on a flow domain $O \subset X \times \mathbb{R}$ is a differentiable function
such that for all $x \in X$ the function $\phi(x,-)$ is an integral curve of $v$ (def. 1).
(complete vector field)
The vector field $v$ is called a complete vector field if it admits a global flow (def. 2).
In synthetic differential geometry a tangent vector field is a morphism $v \colon X \to X^D$ such that
The internal hom-adjunct of such a morphism is of the form
If $X$ is sufficiently nice (a microlinear space should be sufficient) then this morphism factors through the internal automorphism group $\mathbf{Aut}(X)$ inside the internal endomorphisms $X^X$
Then a group homomorphism
with the property that restricted along any of the affine inclusions $D \hookrightarrow \mathbb{R}$ it equals $\tilde v$
is a flow for $v$.
Let $\phi$ be a global flow of a vector field $v$ (def. 2). This yields an action of the additive group $(\mathbb{R},+)$ of real numbers on the differentiable manifold $X$ by diffeomorphisms, in that
$\phi_v(-,0) = id_X$;
$\phi_n(-,t_2) \circ \phi_v(-,t_1) = \phi_v(-, t_1 + t_2)$;
$\phi_v(-,-t) = \phi_v(-,t)^{-1}$.
(fundamental theorem of flows)
Let $X$ be a smooth manifold and $v \in \Gamma(T X)$ a smooth vector field. Then $v$ has a unique maximal flow (def. 2).
This unique flow is often denoted $\phi_v$ or $\exp(v)$ (see also at exponential map).
e.g. Lee, theorem 12.9
Let $X$ be a compact smooth manifold. Then every smooth vector field $v \in \Gamma(T X)$ is a complete vector field (def. 3) hence has a global flow (def. 2).
e.g. Lee, theorem 12.12