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.

The image of a rectangular grid on a square under a diffeomorphism from the square onto itself.

Definition[edit]

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 Cr-diffeomorphism).

Two manifolds M and N are diffeomorphic (symbol usually being ≃) if there is a diffeomorphism f from M to N. They are Cr 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[edit]

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 g_{|U \cap X} = f_{|U \cap X} (note that g is an extension of f). f is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth.

Local description[edit]

Model example

If U, V are connected open subsets of Rn such that V is simply connected, a differentiable map f : U → V is a diffeomorphism if it is proper and if the differential Dfx : Rn → Rn is bijective at each point x in U.

First remark

It is essential for V to be simply connected for the function f to 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 function

\begin{cases}
f : \mathbf{R}^2 \setminus \{(0,0)\} \to \mathbf{R}^2 \setminus \{(0,0)\} \\
(x,y)\mapsto(x^2-y^2,2xy)
\end{cases}

Then f is surjective and it satisfies

\det Df_x=4(x^2+y^2)\neq0

Thus, though Dfx is bijective at each point, f is 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)

Df_x : T_xU \to T_{f(x)}V

is a linear map, it has a well-defined inverse if and only if Dfx is a bijection. The matrix representation of Dfx is the n × n matrix of first-order partial derivatives whose entry in the i-th row and j-th column is \partial f_i / \partial x_j. This so-called Jacobian matrix is often used for explicit computations.

Third remark

Diffeomorphisms are necessarily between manifolds of the same dimension. Imagine f going from dimension n to dimension k. If n < k then Dfx could never be surjective; and if n > k then Dfx could never be injective. In both cases, therefore, Dfx fails to be a bijection.

Fourth remark

If Dfx is a bijection at x then f is said to be a local diffeomorphism (since, by continuity, Dfy will also be bijective for all y sufficiently close to x).

Fifth remark

Given a smooth map from dimension n to dimension k, if Df (or, locally, Dfx) is surjective, f is said to be a submersion (or, locally, a "local submersion"); and if Df (or, locally, Dfx) is injective, f is said to be an immersion (or, locally, a "local immersion").

Sixth remark

A differentiable bijection is not necessarily a diffeomorphism. f(x) = x3, for example, is not a diffeomorphism from R to 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 f is a map between differentiable manifolds, a diffeomorphic f is a stronger condition than a homeomorphic f. For a diffeomorphism, f and its inverse need to be differentiable; for a homeomorphism, f and 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 ψfφ−1 : U → V is then a diffeomorphism as in the definition above (whenever it makes sense).[clarify]

Examples[edit]

Since any manifold can be locally parametrised, we can consider some explicit maps from R2 into R2.

  • Let
f(x,y) = \left (x^2 + y^3, x^2 - y^3 \right ).
We can calculate the Jacobian matrix:
 J_f = \begin{pmatrix} 2x & 3y^2 \\ 2x & -3y^2 \end{pmatrix} .
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
g(x,y) = \left (a_0 + a_{1,0}x + a_{0,1}y + \cdots, \ b_0 + b_{1,0}x + b_{0,1}y + \cdots \right )
where the a_{i,j} and b_{i,j} 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:
 J_g(0,0) = \begin{pmatrix} a_{1,0} & a_{0,1} \\ b_{1,0} & b_{0,1} \end{pmatrix}.
We see that g is a local diffeomorphism at 0 if, and only if,
a_{1,0}b_{0,1} - a_{0,1}b_{1,0} \neq 0,
i.e. the linear terms in the components of g are linearly independent as polynomials.
  • Let
h(x,y) = \left (\sin(x^2 + y^2), \cos(x^2 + y^2) \right ).
We can calculate the Jacobian matrix:
 J_h = \begin{pmatrix} 2x\cos(x^2 + y^2) & 2y\cos(x^2 + y^2) \\ -2x\sin(x^2+y^2) & -2y\sin(x^2 + y^2) \end{pmatrix} .
The Jacobian matrix has zero determinant everywhere! In fact we see that the image of h is the unit circle.

Diffeomorphism group[edit]

Let M be a differentiable manifold that is second-countable and Hausdorff. The diffeomorphism group of M is the group of all Cr diffeomorphisms of M to itself, denoted by Diffr(M) or, when r is understood, Diff(M). This is a "large" group, in the sense that – provided M is not zero-dimensional – it is not locally compact.

Topology[edit]

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

d_K(f,g) = \sup\nolimits_{x\in K} d(f(x),g(x)) + \sum\nolimits_{1\le p\le r} \sup\nolimits_{x\in K} \left \|D^pf(x) - D^pg(x) \right \|

as K varies over compact subsets of M. Indeed, since M is σ-compact, there is a sequence of compact subsets Kn whose union is M. Then:

d(f,g) = \sum\nolimits_n 2^{-n}\frac{d_{K_n}(f,g)}{1+d_{K_n}(f,g)}.

The diffeomorphism group equipped with its weak topology is locally homeomorphic to the space of Cr vector fields (Leslie 1967). 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[edit]

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:

x^{\mu} \to x^{\mu} + \varepsilon h^{\mu}(x)

so the infinitesimal generators are the vector fields

 L_{h} = h^{\mu}(x)\frac{\partial}{\partial x_\mu}.

Examples[edit]

  • 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(Ge), where Diff(Ge) is the subgroup of Diff(G) that fixes the identity element of the group.
  • The diffeomorphism group of Euclidean space Rn 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(Rn, 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 → Diff0(M) → Diff(M) → Σ(π0(M)). Here Diff0(M)is the subgroup of Diff(M) that preserves all the components of M, and Σ(π0(M)) is the permutation group of the set π0(M) (the components of M). Moreover, the image of the map Diff(M) → Σ(π0(M)) is the bijections of π0(M) that preserve diffeomorphism classes.

Transitivity[edit]

For a connected manifold M, the diffeomorphism group acts transitively on M. More generally, the diffeomorphism group acts transitively on the configuration space CkM. If M is at least two-dimensional, the diffeomorphism group acts transitively on the configuration space FkM and the action on M is multiply transitive (Banyaga 1997, p. 29).

Extensions of diffeomorphisms[edit]

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 Sn−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 Γn, 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 Bn.

Connectedness[edit]

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 S1 × S1 = R2/Z2, 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.

Homotopy types[edit]

  • The diffeomorphism group of S2 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: S1 × S1 × 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(S4) has more than two components. Via Milnor, Kahn and Antonelli, however, it is known that provided n > 6, Diff(Sn) does not have the homotopy-type of a finite CW-complex.

Homeomorphism and diffeomorphism[edit]

Unlike non-diffeomorphic homeomorphisms, it is relatively 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).

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 R4s: there are uncountably many pairwise non-diffeomorphic open subsets of R4 each of which is homeomorphic to R4, and also there are uncountably many pairwise non-diffeomorphic differentiable manifolds homeomorphic to R4 that do not embed smoothly in R4.

See also[edit]

Notes[edit]

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

References[edit]