Diffeomorphism

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In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. The mathematical definition is as follows. Given two differentiable manifolds M and N, a bijective map [itex]f[itex] from M to N is called a diffeomorphism if both [itex]f:M\to N[itex] and its inverse [itex]f^{-1}:N\to M[itex] are smooth.

Two manifolds M and N are diffeomorphic (symbol being usually [itex]\simeq[itex]) if there is a diffeomorphism [itex]f[itex] from M to N. For example

[itex]\mathbb{R}/\mathbb{Z} \simeq S^1.[itex]

That is, the quotient group of the real numbers modulo the integers is again a smooth manifold, which is diffeomorphic to the 1-sphere, usually known as the circle. The diffeomorphism is given by

[itex]x\mapsto e^{ix}.[itex]

This map provides not only a diffeomorphism, but also an isomorphism of Lie groups between the two spaces.

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Local description

Model example: if [itex]U[itex] and [itex]V[itex] are two open subsets of [itex]\mathbb{R}^n[itex], a differentiable map [itex]f[itex] from [itex]U[itex] to [itex]V[itex] is a diffeomorphism if

1. it is a bijection,
2. its differential [itex]df[itex] is invertible (as the matrix of all [itex]\partial f_i/\partial x_j[itex], [itex]1 \leq i,j \leq n[itex]).

Remarks:

• Condition 2 excludes diffeomorphisms going from dimension [itex]n[itex] to a different dimension [itex]k[itex] (the matrix of [itex]df[itex] would not be square hence certainly not invertible).
• A differentiable bijection is not necessarily a diffeomorphism, e.g. [itex]f(x)=x^3[itex] is not a diffeomorphism from [itex]\mathbb{R}[itex] to itself because its derivative vanishes at 0.
• [itex]f[itex] also happens to be a homeomorphism.

Now, [itex]f[itex] from M to N is called a diffeomorphism if in coordinates charts it satisfies the definition above. More precisely, pick any cover of M by compatible coordinate charts, and do the same for N. Let [itex]\phi[itex] and [itex]\psi[itex] be charts on M and N respectively, with [itex]U[itex] being the image of [itex]\phi[itex] and [itex]V[itex] the image of [itex]\psi[itex]. Then the conditions says that the map [itex]\psi f \phi^{-1}[itex] from [itex]U[itex] to [itex]V[itex] is a diffeomorphism as in the definition above (whenever it makes sense). One has to check that for every couple of charts [itex]\phi[itex], [itex]\psi[itex] of two given atlases, but once checked, it will be true for any other compatible chart. Again we see that dimensions have to agree.

Diffeomorphism group

The diffeomorphism group of a manifold is the group of all its self-diffeomorphisms. For dimension ≥ 1 this is a large group (too big to be a Lie group). For a connected manifold M the diffeomorphisms act transitively on M: this is true locally because it is true in Euclidean space and then a topological argument shows that given any p and q there is a diffeomorphism taking p to q. That is, all points of M in effect look the same, intrinsically. The same is true for finite configurations of points, so that the diffeomorphism group is k- fold multiply transitive for any integer k ≥ 1, provided the dimension is at least two (it is not true for the case of the circle or real line).

Homeomorphism and diffeomorphism

It is easy to find a homeomorphism which is not a diffeomorphism, but it is more difficult to find a pair of homeomorphic manifolds that are not diffeomorphic. In dimensions 1, 2, 3, any pair of homeomorphic smooth manifolds are diffeomorphic. In dimension 4 or greater, examples of homeomorphic but not diffeomorphic pairs have been found. The first such example was constructed by John Milnor in dimension 7, he constructed a smooth 7-dimensional manifold (called now Milnor's sphere) which is homeomorphic to the standard 7-sphere but not diffeomorphic to it. There are in fact 28 oriented diffeomorphism classes of manifolds homeomorphic to the 7-sphere (each of them is a fiber bundle over the 4-sphere with fiber the 3-sphere).

Much more extreme phenomena occur: in the early 1980s, a combination of results due to Fields Medal winners Simon Donaldson and Michael Freedman led to the discoveries that there are uncountably many pairwise non-diffeomorphic open subsets of [itex]\mathbb{R}^4[itex] each of which is homeomorphic to [itex]\mathbb{R}^4[itex], and also that there are uncountably many pairwise non-diffeomorphic differentiable manifolds homeomorphic to [itex]\mathbb{R}^4[itex] which do not embed smoothly in [itex]\mathbb{R}^4[itex].

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