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 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 curves) that are tangent to the vector field.
Integral curves generalize to integral sections for multivector fields. See there for more.
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 $t \in U$ 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. ).
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. ).
(complete vector field)
The vector field $v$ is called a complete vector field if it admits a global flow (def. ).
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. ). 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. ).
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. ) hence has a global flow (def. ).
e.g. Lee, theorem 12.12
Last revised on April 18, 2024 at 16:47:45. See the history of this page for a list of all contributions to it.