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)
Given two -times differentiable manifolds (or smooth manifolds), then a diffeomorphism
is a differentiable function such that there exists an inverse differentiabe function (a function which is an inverse function on the underlying sets and is itself differentiable to the given degree).
Diffeomorphisms are the isomorphisms in the corrresponding category Diff of differentiable manifolds/smooth manifolds.
It is clear that
Every diffeomorphism is in particular a homeomorphism between the underlying topological spaces.
The converse in general fails. There exist differentiable maps with only continuous inverse. There are also differentiable bijections whose inverse is not even continuous.
The function given by is a homeomorphism but not a diffeomorphism. The diffeomorphism property fails at the origin, where the differential is not onto.
But there is a rich collection of theorems about cases when the converse is true after all.
For , the open n-ball is the open subset
of the Cartesian space of all points of distance lower than 1 from the origin. This inherits the structure of a smooth manifold from the embedding into .
In dimension for we have:
every open subset of which is homeomorphic to is also diffeomorphic to it.
See the first page of (Ozols) for a list of references.
What’s a good/canonical textbook reference for this?
In dimension 4 the analog statement fails due to the existence of exotic smooth structures on .
For and smooth manifolds of dimension , or we have:
if there is a homeomorphism from to , then there is also a diffeomorphism.
See the corollary on p. 2 of (Munkres).
For the following kinds of manifolds it is true that every homotopy equivalence
(hence every equivalence of their fundamental infinity-groupoids) is homotopic to a diffeomorphism
i.e. that given there is with
for any surface (Zieschang-Vogt-Coldeway)
for a Haken 3-manifold (Waldhausen)
for any hyperbolic manifold of finite volume and of dimension (by Mostow rigidity theorem) (check)
A review of results and relevant literature is also on the first page of (Hass-Scott 92)-
V. Ozols, Largest normal neighbourhoods Proceedings of the American Mathematical Society Vol. 61, No. 1 (Nov., 1976), pp. 99-101 (jstor) (AMS: pdf)
James Munkres, Obstructions to the smoothing of piecewise-differentiable homeomorphisms Bull. Amer. Math. Soc. Volume 65, Number 5 (1959), 332-334. (Euclid)(AMS: pdf)
Zieschang, Vogt and Coldeway, Surfaces and planar discontinuous groups
Friedhelm Waldhausen, On Irreducible 3-Manifolds Which are Sufficiently Large, Annals of Mathematics
Second Series, Vol. 87, No. 1 (Jan., 1968), pp. 56-88 (JSTOR)
Joel Hass, Peter Scott, Homotopy equivalence and homeomoprhism of 3-manifolds, Topology, Vol. 31, No. 3 (1992) pp. 493-517 (pdf)
Last revised on June 28, 2017 at 14:46:58. See the history of this page for a list of all contributions to it.