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 <math>f<math> from M to N is called a diffeomorphism if both <math>f:M\to N<math> and its inverse <math>f^{-1}:N\to M<math> are smooth.
Two manifolds M and N are diffeomorphic (symbol being usually <math>\simeq<math>) if there is a diffeomorphism <math>f<math> from M to N. For example
- <math>\mathbb{R}/\mathbb{Z} \simeq S^1.<math>
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
- <math>x\mapsto e^{ix}.<math>
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 <math>U<math> and <math>V<math> are two open subsets of <math>\mathbb{R}^n<math>, a differentiable map <math>f<math> from <math>U<math> to <math>V<math> is a diffeomorphism if
- it is a bijection,
- its differential <math>df<math> is invertible (as the matrix of all <math>\partial f_i/\partial x_j<math>, <math>1 \leq i,j \leq n<math>).
Remarks:
- Condition 2 excludes diffeomorphisms going from dimension <math>n<math> to a different dimension <math>k<math> (the matrix of <math>df<math> would not be square hence certainly not invertible).
- A differentiable bijection is not necessarily a diffeomorphism, e.g. <math>f(x)=x^3<math> is not a diffeomorphism from <math>\mathbb{R}<math> to itself because its derivative vanishes at 0.
- <math>f<math> also happens to be a homeomorphism.
Now, <math>f<math> 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 <math>\phi<math> and <math>\psi<math> be charts on M and N respectively, with <math>U<math> being the image of <math>\phi<math> and <math>V<math> the image of <math>\psi<math>. Then the conditions says that the map <math>\psi f \phi^{-1}<math> from <math>U<math> to <math>V<math> is a diffeomorphism as in the definition above (whenever it makes sense). One has to check that for every couple of charts <math>\phi<math>, <math>\psi<math> 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 <math>\mathbb{R}^4<math> each of which is homeomorphic to <math>\mathbb{R}^4<math>, and also that there are uncountably many pairwise non-diffeomorphic differentiable manifolds homeomorphic to <math>\mathbb{R}^4<math> which do not embed smoothly in <math>\mathbb{R}^4<math>.