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)
A differential structure on a topological space $X$ is the extra structure of a differential manifold on $X$. A smooth structure on $X$ is the extra structure of a smooth manifold.
(smooth structure)
Let $X$ be a topological manifold and let
be two atlases, both making $X$ into a smooth manifold (this def.).
Then there is a diffeomorphism of the form
precisely if the identity function on the underlying set of $X$ constitutes such a diffeomorphism. (Because if $f$ is a diffeomorphism, then also $f^{-1}\circ f = id_X$ is a diffeomorphism.)
That the identity function is a diffeomorphism between $X$ equipped with these two atlases means (by definition) that
Hence diffeomorphsm induces an equivalence relation on the set of smooth atlases that exist on a given topological manifold $X$. An equivalence class with respect to this equivalence relation is called a smooth structure on $X$.
(uniqueness of smooth structure on Euclidean space in $d \neq 4$)
For $n \in \mathbb{N}$ a natural number with $n \neq 4$, there is a unique (up to isomorphism) smooth structure on the Cartesian space $\mathbb{R}^n$.
This was shown in (Stallings 62).
In $d = 4$ the analog of this statement is false. One says that on $\mathbb{R}^4$ there exist exotic smooth structures.
Many topological spaces have canonical or “obvious” smooth structures. For instance a Cartesian space $\mathbb{R}^n$ has the evident smooth structure induced from the fact that it can be covered by a single chart – itself.
From this example, various topological spaces inherit a canonical smooth structure by embedding. For instance the $n$-sphere may naturally be thought of as the collection of points
given by $S^n = \{\vec x \in \mathbb{R}^n | \sum_i (x^i)^2 = 1\}$ and this induces a smooth structure of $\mathbb{S}^n$.
But there may be other, non-equivalent smooth structures than these canonical ones. These are called exotic smooth structures. See there for more details.
See also