Algebraic structure → Group theoryGroup theory |
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Lie groups |
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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 differentiable.

Given two manifolds and , a differentiable map is called a **diffeomorphism** if it is a bijection and its inverse is differentiable as well. If these functions are times continuously differentiable, is called a -diffeomorphism**.**

Two manifolds and are **diffeomorphic** (usually denoted ) if there is a diffeomorphism from to . They are -**diffeomorphic** if there is an times continuously differentiable bijective map between them whose inverse is also times continuously differentiable.

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*). The function *f* is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth.

- Hadamard-Caccioppoli Theorem
^{[1]}

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}* :

- First remark
It is essential for

Vto be simply connected for the functionfto be globally invertible (under the sole condition that its derivative be a bijective map at each point). For example, consider the "realification" of the complex square function

Then

fis surjective and it satisfies

Thus, though

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

- Second remark
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 . This so-called Jacobian matrix is often used for explicit computations.

- Third remark
Diffeomorphisms are necessarily between manifolds of the same dimension. Imagine

fgoing from dimensionnto dimensionk. Ifn<kthenDfcould never be surjective, and if_{x}n>kthenDfcould never be injective. In both cases, therefore,_{x}Dffails to be a bijection._{x}

- Fourth remark
If

Dfis a bijection at_{x}xthenfis said to be a local diffeomorphism (since, by continuity,Dfwill also be bijective for all_{y}ysufficiently close tox).

- Fifth remark
Given a smooth map from dimension

nto dimensionk, ifDf(or, locally,Df) is surjective,_{x}fis said to be a submersion (or, locally, a "local submersion"); and ifDf(or, locally,Df) is injective,_{x}fis said to be an immersion (or, locally, a "local immersion").

- Sixth remark
A differentiable bijection is

notnecessarily a diffeomorphism.f(x) =x^{3}, for example, 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.

- Seventh remark
When

fis a map betweendifferentiablemanifolds, a diffeomorphicfis a stronger condition than a homeomorphicf. For a diffeomorphism,fand its inverse need to be differentiable; for a homeomorphism,fand its inverse need only be continuous. Every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism.

*f* : *M* → *N* is called a **diffeomorphism** if, in coordinate 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, respectively, *M* and *N*, with *U* and *V* as, respectively, the images of φ and ψ. The map ψ*f*φ^{−1} : *U* → *V* is then a diffeomorphism as in the definition above, whenever *f*(φ^{−1}(U)) ⊆ ψ^{−1}(V).

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*could only be a diffeomorphism away from the*x*-axis and the*y*-axis. However,*f*is not bijective since*f*(*x*,*y*) =*f*(-*x*,*y*), and thus it cannot be a diffeomorphism.

- 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.

In mechanics, a stress-induced transformation is called a deformation and may be described by a diffeomorphism.
A diffeomorphism *f* : *U* → *V* between two surfaces *U* and *V* has a Jacobian matrix *Df* that is an invertible matrix. In fact, it is required that for *p* in *U*, there is a neighborhood of *p* in which the Jacobian *Df* stays non-singular. Suppose that in a chart of the surface,

The total differential of *u* is

- , and similarly for
*v*.

Then the image is a linear transformation, fixing the origin, and expressible as the action of a complex number of a particular type. When (*dx*, *dy*) is also interpreted as that type of complex number, the action is of complex multiplication in the appropriate complex number plane. As such, there is a type of angle (Euclidean, hyperbolic, or slope) that is preserved in such a multiplication. Due to *Df* being invertible, the type of complex number is uniform over the surface. Consequently, a surface deformation or diffeomorphism of surfaces has the **conformal property** of preserving (the appropriate type of) angles.

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

The diffeomorphism group has two natural topologies: *weak* and *strong* (Hirsch 1997). 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:

The diffeomorphism group equipped with its weak topology is locally homeomorphic to the space of *C ^{r}* vector fields (Leslie 1967). Over a compact subset of

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 at each point in space:

so the infinitesimal generators are the vector fields

- 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 the subgroup Diff(**R**^{n}, 0) of diffeomorphisms fixing the origin under the map*f*(*x*) ↦*f*(*tx*) /*t*,*t*∈ (0,1]. In particular, the general linear group is also a deformation retract of the full diffeomorphism group. - 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.

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

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. In 1945, Gustave Choquet, apparently unaware of this result, produced a completely different proof.

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 (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}.

For manifolds, the diffeomorphism group is usually not connected. Its component group is called the mapping class group. In dimension 2 (i.e. surfaces), the mapping class group is a finitely presented group generated by Dehn twists (Dehn, Lickorish, Hatcher).^{[citation needed]} 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 simply the modular group SL(2, **Z**) and the classification becomes classical 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; as this enlarged space was homeomorphic to a closed ball, the Brouwer fixed-point theorem became applicable. Smale conjectured that if *M* is an oriented smooth closed manifold, 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.

- The diffeomorphism group of
**S**^{2}has the homotopy-type of the subgroup O(3). This was proven by Steve Smale.^{[2]} - 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 understood. For example, it is an open problem whether or not Diff(**S**^{4}) has more than two components. Via Milnor, Kahn and Antonelli, however, it is known that provided*n*> 6, Diff(**S**^{n}) does not have the homotopy-type of a finite CW-complex.

Since every diffeomorphism is a homeomorphism, every diffeomorphic manifolds are homeomorphic, but the converse is not true. While it is easy to find homeomorphisms that are non-diffeomorphisms, it is more difficult to find a pair of homeomorphic manifolds that are not diffeomorphic. In dimensions 1, 2 and 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).

More unusual 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 **R**^{4}s: 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}.