Editing Diffeomorphism (section)

== Local description ==
Testing whether a differentiable map is a diffeomorphism can be made locally under some mild restrictions. This is the Hadamard-Caccioppoli theorem:<ref>{{cite book |author1=Steven G. Krantz |author2=Harold R. Parks |title=The implicit function theorem: history, theory, and applications |date=2013 |isbn=978-1-4614-5980-4 |page=Theorem 6.2.4|publisher=Springer }}</ref>

If <math>U</math>, <math>V</math> are [[Connected space|connected]] [[Open set|open subsets]] of <math>\R^n</math> such that <math>V</math> is [[simply connected]], a differentiable map <math>f:U\to V</math> is a diffeomorphism if it is [[Proper map|proper]] and if the [[Pushforward (differential)|differential]] <math>Df_x:\R^n\to\R^n</math> is bijective (and hence a [[linear isomorphism]]) at each point <math>x</math> in <math>U</math>.

Some remarks:

It is essential for <math>V</math> to be [[simply connected]] for the function <math>f</math> to be globally invertible (under the sole condition that its derivative be a bijective map at each point). For example, consider the "realification" of the [[Complex number|complex]] square function 
: <math>\begin{cases}
f : \R^2 \setminus \{(0,0)\} \to \R^2 \setminus \{(0,0)\} \\
(x,y)\mapsto(x^2-y^2,2xy).
\end{cases}</math> 
Then <math>f</math> is [[surjective]] and it satisfies 
: <math>\det Df_x = 4(x^2+y^2) \neq 0.</math> 
Thus, though <math>Df_x</math> is bijective at each point, <math>f</math> is not invertible because it fails to be [[injective]] (e.g. <math>f(1,0)=(1,0)=f(-1,0)</math>).

Since the differential at a point (for a differentiable function) 
: <math>Df_x : T_xU \to T_{f(x)}V</math> 
is a [[linear map]], it has a well-defined inverse if and only if <math>Df_x</math> is a bijection. The [[Matrix (mathematics)|matrix]] representation of <math>Df_x</math> is the <math>n\times n</math> matrix of first-order [[partial derivative]]s whose entry in the <math>i</math>-th row and <math>j</math>-th column is <math>\partial f_i / \partial x_j</math>. This so-called [[Jacobian matrix]] is often used for explicit computations.

Diffeomorphisms are necessarily between manifolds of the same [[dimension]]. Imagine <math>f</math> going from dimension <math>n</math> to dimension <math>k</math>. If <math>n<k</math> then <math>Df_x</math> could never be surjective, and if <math>n>k</math> then <math>Df_x</math> could never be injective. In both cases, therefore, <math>Df_x</math> fails to be a bijection.

If <math>Df_x</math> is a bijection at <math>x</math> then <math>f</math> is said to be a [[local diffeomorphism]] (since, by continuity, <math>Df_y</math> will also be bijective for all <math>y</math> sufficiently close to <math>x</math>).

Given a smooth map from dimension <math>n</math> to dimension <math>k</math>, if <math>Df</math> (or, locally, <math>Df_x</math>) is surjective, <math>f</math> is said to be a [[Submersion (mathematics)|submersion]] (or, locally, a "local submersion"); and if <math>Df</math> (or, locally, <math>Df_x</math>) is injective, <math>f</math> is said to be an [[Immersion (mathematics)|immersion]] (or, locally, a "local immersion").

A differentiable bijection is ''not'' necessarily a diffeomorphism. <math>f(x)=x^3</math>, for example, is not a diffeomorphism from <math>\R</math> to itself because its derivative vanishes at 0 (and hence its inverse is not differentiable at 0). This is an example of a [[homeomorphism]] that is not a diffeomorphism.

When <math>f</math> is a map between differentiable manifolds, a diffeomorphic <math>f</math> is a stronger condition than a homeomorphic <math>f</math>. For a diffeomorphism, <math>f</math> and its inverse need to be [[Differentiable manifold#Differentiable functions|differentiable]]; for a homeomorphism, <math>f</math> and its inverse need only be [[Continuous function|continuous]]. Every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism.

<math>f:M\to N</math> is a diffeomorphism if, in [[Manifold#Differentiable manifolds|coordinate charts]], it satisfies the definition above. More precisely: Pick any cover of <math>M</math> by compatible [[Manifold#Differentiable manifolds|coordinate charts]] and do the same for <math>N</math>. Let <math>\phi</math> and <math>\psi</math> be charts on, respectively, <math>M</math> and <math>N</math>, with <math>U</math> and <math>V</math> as, respectively, the images of <math>\phi</math> and <math>\psi</math>. The map <math>\psi f\phi^{-1}:U\to V</math> is then a diffeomorphism as in the definition above, whenever <math>f(\phi^{-1}(U))\subseteq\psi^{-1}(V)</math>.
<!--Huh?: One has to check that for every pair of charts φ, ψ of two given [[Manifold#Differentiable manifolds|atlases]], but once checked, it will be true for any other compatible chart. Again we see that dimensions have to agree.-->