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
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
manifolds and cobordisms
cobordism theory, Introduction
Definitions
Genera and invariants
Classification
Theorems
A differential structure on a topological space is the extra structure of a differential manifold on . A smooth structure on is the extra structure of a smooth manifold.
(smooth structure)
Let be a topological manifold and let
be two atlases, both making into a smooth manifold (this def.).
Then there is a diffeomorphism of the form
precisely if the identity function on the underlying set of constitutes such a diffeomorphism. (Because if is a diffeomorphism, then also is a diffeomorphism.)
That the identity function is a diffeomorphism between 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 . An equivalence class with respect to this equivalence relation is called a smooth structure on .
(uniqueness of smooth structure on Euclidean space in )
For a natural number with , there is a unique (up to isomorphism) smooth structure on the Cartesian space .
This was shown in (Stallings 62).
In the analog of this statement is false. One says that on there exist exotic smooth structures.
Many topological spaces have canonical or “obvious” smooth structures. For instance a Cartesian space 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 -sphere may naturally be thought of as the collection of points
given by and this induces a smooth structure of .
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
Last revised on June 21, 2017 at 20:12:34. See the history of this page for a list of all contributions to it.