# Diffeomorphism

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In mathematics, a **diffeomorphism** is an 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.

## Contents

## Definition

Given two manifolds *M* and *N*, a differentiable map *f* : *M* → *N* is called a **diffeomorphism** if it is a bijection and its inverse *f*^{−1} : *N* → *M* is differentiable as well. If these functions are *r* times continuously differentiable, *f* is called a ** C^{r}-diffeomorphism**).

Two manifolds *M* and *N* are **diffeomorphic** (symbol usually being ≃) if there is a diffeomorphism *f* from *M* to *N*. They are ** C^{r} diffeomorphic** if there is an

*r*times continuously differentiable bijective map between them whose inverse is also

*r*times continuously differentiable.

## Diffeomorphisms of subsets of manifolds

Given a subset *X* of a manifold *M* and a subset *Y* of a manifold *N*, a function *f* : *X* → *Y* is said to be smooth if for all *p* in *X* there is a neighborhood *U* ⊂ *M* of *p* and a smooth function *g* : *U* → *N* such that the restrictions agree (note that *g* is an extension of *f*). We say that *f* is a diffeomorphism if it is bijective, smooth and its inverse is smooth.

## Local description

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**Model Example.** If *U*, *V* are connected open subsets of **R**^{n} such that *V* is simply connected, a differentiable map *f* : *U* → *V* is a **diffeomorphism**, if it is proper and if the differential *Df _{x}* :

**R**

^{n}→

**R**

^{n}is bijective at each point

*x*in

*U*.

Remark 1.It is essential forVto be simply connected for the functionfto be globally invertible (under the sole condition that its derivative is a bijective map at each point). For example, consider the "realification" of the complex square functionThen

fis surjective and its satisfiesthus

Dfis bijective at each point yet_{x}fis not invertible, because it fails to be injective, e.g.,f(1,0) = (1,0) =f(−1,0).

Remark 2.Since the differential at a point (for a differentiable function)is a linear map it has a well defined inverse if, and only if,

Dfis a bijection. The matrix representation of_{x}Dfis the_{x}n×nmatrix of first order partial derivatives whose entry in thei-th row andj-th column is . We often use this so-called Jacobian matrix for explicit computations.

Remark 3.Diffeomorphisms are necessarily between manifolds of the same dimension. Imagine thatfwere going from dimensionnto dimensionk. Ifn<kthenDfcould never be surjective, and if_{x}n>kthenDfcould never be injective. So in both cases_{x}Dffails to be a bijection._{x}

Remark 4.IfDfis a bijection at_{x}xthen we say thatfis a local diffeomorphism (since by continuityDfwill also be bijective for all_{y}ysufficiently close tox).

Remark 5.Given a smooth map from dimensionnto dimensionk, ifDf(resp.Df) is surjective then we say that_{x}fis a submersion (resp. local submersion), and ifDf(resp.Df) is injective we say that_{x}fis an immersion (resp. local immersion).

Remark 6.A differentiable bijection isnotnecessarily a diffeomorphism, e.g.f(x) =x^{3}is not a diffeomorphism fromRto 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.

Remark 7.fbeing a diffeomorphism is a stronger condition thanfbeing a homeomorphism (whenfis a map betweendifferentiablemanifolds). For a diffeomorphism we needfand its inverse to be differentiable. For a homeomorphism we only require thatfand its inverse be continuous. Thus every diffeomorphism is a homeomorphism, but the converse is false: not every homeomorphism is a diffeomorphism.

Now, *f* : *M* → *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} : *U* → *V* is a diffeomorphism as in the definition above (whenever it makes sense). One has to check that for every pair 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.

## Examples

Since any manifold can be locally parametrised, we can consider some explicit maps from **R**^{2} into **R**^{2}.

- Let

- We can calculate the Jacobian matrix:
- The Jacobian matrix has zero determinant if, and only if
*xy*= 0. We see that*f*is a diffeomorphism away from the*x*-axis and the*y*-axis.

- Let

- where the and are arbitrary real numbers, and the omitted terms are of degree at least two in
*x*and*y*. We can calculate the Jacobian matrix at**0**: - We see that
*g*is a local diffeomorphism at**0**if, and only if, - i.e. the linear terms in the components of
*g*are linearly independent as polynomials.

- Let

- We can calculate the Jacobian matrix:
- The Jacobian matrix has zero determinant everywhere! In fact we see that the image of
*h*is the unit circle.

## Diffeomorphism group

Let *M* be a differentiable manifold that is second-countable and Hausdorff. The **diffeomorphism group** of *M* is the group of all *C ^{r}* diffeomorphisms of

*M*to itself, and is denoted by Diff

^{r}(

*M*) or Diff(

*M*) when

*r*is understood. This is a 'large' group, in the sense that it is not locally compact (provided

*M*is not zero-dimensional).

### Topology

The diffeomorphism group has two natural topologies, called the *weak* and *strong* topology Template:Harv. When the manifold is compact, these two topologies agree. The weak topology is always metrizable. When the manifold is not compact, the strong topology captures the behavior of functions "at infinity", and is not metrizable. It is, however, still Baire.

Fixing a Riemannian metric on *M*, the weak topology is the topology induced by the family of metrics

as *K* varies over compact subsets of *M*. Indeed, since *M* is σ-compact, there is a sequence of compact subsets *K*_{n} whose union is *M*. Then, define

The diffeomorphism group equipped with its weak topology is locally homeomorphic to the space of *C ^{r}* vector fields Template:Harv. Over a compact subset of

*M*, this follows by fixing a Riemannian metric on

*M*and using the exponential map for that metric. If

*r*is finite and the manifold is compact, the space of vector fields is a Banach space. Moreover, the transition maps from one chart of this atlas to another are smooth, making the diffeomorphism group into a Banach manifold. If

*r*= ∞ or if the manifold is σ-compact, the space of vector fields is a Fréchet space. Moreover, the transition maps are smooth, making the diffeomorphism group into a Fréchet manifold.

### Lie algebra

In particular, the Lie algebra of the diffeomorphism group of *M* consists of all vector fields on *M*, equipped with the Lie bracket of vector fields. Somewhat formally, this is seen by making a small change to the coordinate x at each point in space:

so the infinitesimal generators are the vector fields

### Examples

- When
*M*=*G*is a Lie group, there is a natural inclusion of*G*in its own diffeomorphism group via left-translation. Let Diff(*G*) denote the diffeomorphism group of*G*, then there is a splitting Diff(*G*) ≃*G*× Diff(*G*,*e*) where Diff(*G*,*e*) is the subgroup of Diff(*G*) that fixes the identity element of the group.

- The diffeomorphism group of Euclidean space
**R**^{n}consists of two components, consisting of the orientation preserving and orientation reversing diffeomorphisms. In fact, the general linear group is a deformation retract of subgroup Diff(**R**^{n}, 0) of diffeomorphisms fixing the origin under the map*f*(*x*) Template:Mapsto*f*(*tx*)/*t*,*t*∈ (0,1]. Hence, in particular, the general linear group is also a deformation retract of the full diffeomorphism group as well.

- For a finite set of points, the diffeomorphism group is simply the symmetric group. Similarly, if
*M*is any manifold there is a group extension 0 → Diff_{0}(*M*) → Diff(*M*) → Σ(π_{0}(*M*)). Here Diff_{0}(*M*)is the subgroup of Diff(*M*) that preserves all the components of*M*, and Σ(π_{0}(*M*)) is the permutation group of the set π_{0}(*M*) (the components of*M*). Moreover, the image of the map Diff(*M*) → Σ(π_{0}(*M*)) is the bijections of π_{0}(*M*) that preserve diffeomorphism classes.

### Transitivity

For a connected manifold *M* the diffeomorphism group acts transitively on *M*. More generally, the diffeomorphism group acts transitively on the configuration space *C _{k}M*. If the dimension of

*M*is at least two the diffeomorphism group acts transitively on the configuration space

*F*: the action on

_{k}M*M*is multiply transitive Template:Harv.

### Extensions of diffeomorphisms

In 1926, Tibor Radó asked whether the harmonic extension of any homeomorphism (or diffeomorphism) of the unit circle to the unit disc yields a diffeomorphism on the open disc. An elegant proof was provided shortly afterwards by Hellmuth Kneser and a completely different proof was discovered in 1945 by Gustave Choquet, apparently unaware that the theorem was already known.

The (orientation-preserving) diffeomorphism group of the circle is pathwise connected. This can be seen by noting that any such diffeomorphism can be lifted to a diffeomorphism *f* of the reals satisfying *f*(*x*+1) = *f*(*x*) + 1; this space is convex and hence path connected. A smooth eventually constant path to the identity gives a second more elementary way of extending a diffeomorphism from the circle to the open unit disc (this is a special case of the Alexander trick). Moreover, the diffeomorphism group of the circle has the homotopy-type of the orthogonal group O(2).

The corresponding extension problem for diffeomorphisms of higher-dimensional spheres **S**^{n−1} was much studied in the 1950s and 1960s, with notable contributions from René Thom, John Milnor and Stephen Smale. An obstruction to such extensions is given by the finite Abelian group Γ_{n}, the "group of twisted spheres", defined as the quotient of the Abelian component group of the diffeomorphism group by the subgroup of classes extending to diffeomorphisms of the ball *B*^{n}.

### Connectedness

For manifolds the diffeomorphism group is usually not connected. Its component group is called the mapping class group. In dimension 2, i.e. for surfaces, the mapping class group is a finitely presented group, generated by Dehn twists (Dehn, Lickorish, Hatcher).{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
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}} Max Dehn and Jakob Nielsen showed that it can be identified with the outer automorphism group of the fundamental group of the surface.

William Thurston refined this analysis by classifying elements of the mapping class group into three types: those equivalent to a periodic diffeomorphism; those equivalent to a diffeomorphism leaving a simple closed curve invariant; and those equivalent to pseudo-Anosov diffeomorphisms. In the case of the torus **S**^{1} × **S**^{1} = **R**^{2}/**Z**^{2}, the mapping class group is just the modular group SL(2, **Z**) and the classification reduces to the classical one in terms of elliptic, parabolic and hyperbolic matrices. Thurston accomplished his classification by observing that the mapping class group acted naturally on a compactification of Teichmüller space; since this enlarged space was homeomorphic to a closed ball, the Brouwer fixed-point theorem became applicable.

If *M* is an oriented smooth closed manifold, it was conjectured by Smale that the identity component of the group of orientation-preserving diffeomorphisms is simple. This had first been proved for a product of circles by Michel Herman; it was proved in full generality by Thurston.

### Homotopy types

- The diffeomorphism group of
**S**^{2}has the homotopy-type of the subgroup O(3). This was proved by Steve Smale.^{[1]}

- The diffeomorphism group of the torus has the homotopy-type of its linear automorphisms:
**S**^{1}×**S**^{1}× GL(2,**Z**).

- The diffeomorphism groups of orientable surfaces of genus
*g*> 1 have the homotopy-type of their mapping class groups—i.e.: the components are contractible.

- The homotopy-type of the diffeomorphism groups of 3-manifolds are fairly well-understood via the work of Ivanov, Hatcher, Gabai and Rubinstein although there are a few outstanding open cases, primarily 3-manifolds with finite fundamental groups.

- The homotopy-type of diffeomorphism groups of
*n*-manifolds for*n*> 3 are poorly undersood. For example, it is an open problem whether or not Diff(**S**^{4}) has more than two components. But via the work of Milnor, Kahn and Antonelli it's known that Diff(**S**^{n}) does not have the homotopy-type of a finite CW-complex provided*n*> 6.

## Homeomorphism and diffeomorphism

It is easy to find a homeomorphism that 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) that 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 the total space of a fiber bundle over the 4-sphere with the 3-sphere as the fiber).

Much more extreme phenomena occur for 4-manifolds: in the early 1980s, a combination of results due to Simon Donaldson and Michael Freedman led to the discovery of exotic R4s: there are uncountably many pairwise non-diffeomorphic open subsets of **R**^{4} each of which is homeomorphic to **R**^{4}, and also there are uncountably many pairwise non-diffeomorphic differentiable manifolds homeomorphic to **R**^{4} that do not embed smoothly in **R**^{4}.

## See also

## Notes

- ↑ Smale, Diffeomorphisms of the 2-sphere, Proc. Amer. Math. Soc. 10 (1959) 621–626.

## References

Chaudhuri, Shyamoli, Hakuru Kawai and S.-H Henry Tye. "Path-integral formulation of closed strings," Phys. Rev. D, 36: 1148, 1987.

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