Diffeomorphism

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In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.

Formally, given two manifolds M and N, a bijective map $f$ from M to N is called a diffeomorphism if both

$f:M\to N$

and its inverse

$f^{-1}:N\to M$

are differentiable (if these functions are r times continuously differentiable, f is called a $C r$ -diffeomorphism).

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

$\mathbb{R}/\mathbb{Z} \simeq S^1.$

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

$x\mapsto e^{2\pi ix}.$

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

1 Local description
2 Diffeomorphism group
3 Homeomorphism and diffeomorphism
4 See also

[edit] Local description

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

it is a bijection,
its derivative $D f$ is invertible (as the matrix of all $\partial f_i/\partial x_j$ , $1 \leq i,j \leq n$ ), which means the same as having non-zero Jacobian determinant.

Remarks:

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

Now, $f$ 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 $φ$ and $ψ$ be charts on M and N respectively, with $U$ being the image of $φ$ and $V$ the image of $ψ$ . Then the conditions says that the map $ψ f φ - 1$ from $U$ to $V$ is a diffeomorphism as in the definition above (whenever it makes sense). One has to check that for every couple of charts $φ$ , $ψ$ of two given atlases, but once checked, it will be true for any other compatible chart. Again we see that dimensions have to agree.

[edit] Diffeomorphism group

The diffeomorphism group of a manifold is the group of all its automorphisms (diffeomorphisms to itself). For dimension greater than or equal to one this is a large 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). This group can be given the structure of an infinite dimensional Lie group, modeled on the space of vector fields on the manifold. In general, this will not be a Banach Lie group, and the exponential map will not be a local diffeomorphism.

[edit] 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 $\mathbb{R}^4$ each of which is homeomorphic to $\mathbb{R}^4$ , and also that there are uncountably many pairwise non-diffeomorphic differentiable manifolds homeomorphic to $\mathbb{R}^4$ which do not embed smoothly in $\mathbb{R}^4$ .