# Diffeomorphism

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 $\displaystyle f$ from M to N is called a diffeomorphism if both $\displaystyle f:M\to N$ and its inverse $\displaystyle f^{-1}:N\to M$ are smooth.

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

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

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

$\displaystyle x\mapsto e^{ix}.$

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

## Local description

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

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

Remarks:

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

Now, $\displaystyle f$ 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 $\displaystyle \phi$ and $\displaystyle \psi$ be charts on M and N respectively, with $\displaystyle U$ being the image of $\displaystyle \phi$ and $\displaystyle V$ the image of $\displaystyle \psi$ . Then the conditions says that the map $\displaystyle \psi f \phi^{-1}$ from $\displaystyle U$ to $\displaystyle V$ is a diffeomorphism as in the definition above (whenever it makes sense). One has to check that for every couple of charts $\displaystyle \phi$ , $\displaystyle \psi$ 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 $\displaystyle \mathbb{R}^4$ each of which is homeomorphic to $\displaystyle \mathbb{R}^4$ , and also that there are uncountably many pairwise non-diffeomorphic differentiable manifolds homeomorphic to $\displaystyle \mathbb{R}^4$ which do not embed smoothly in $\displaystyle \mathbb{R}^4$ .