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Wikipedia / Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable.

Definition

, a continuously differentiable map is a diffeomorphism if it is a bijection-diffeomorphism.

are diffeomorphic-diffeomorphism is a homeomorphism.

Diffeomorphisms of subsets of manifolds

Given a subset there is a neighborhood such that the restrictions is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth.

Local description

Testing whether a differentiable map is a diffeomorphism can be made locally under some mild restrictions. This is the Hadamard-Caccioppoli theorem:

are connected open subsets is simply connected is a diffeomorphism if it is proper and if the differential is bijective (and hence a linear isomorphism.

Some remarks:

to be simply connected to 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

is surjective and it satisfies

is not invertible because it fails to be injective).

Since the differential at a point (for a differentiable function)

is a linear map is a bijection. The matrix matrix of first-order partial derivatives. This so-called Jacobian matrix is often used for explicit computations.

Diffeomorphisms are necessarily between manifolds of the same dimension fails to be a bijection.

is said to be a local diffeomorphism).

is said to be a submersion is said to be an immersion (or, locally, a "local immersion").

A differentiable bijection is not 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.

and its inverse need to be differentiable and its inverse need only be continuous. Every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism.

is a diffeomorphism if, in coordinate charts by compatible coordinate charts.

Examples

.

  • Let

  • We can calculate the Jacobian matrix:

  • The Jacobian matrix has zero determinant, and thus it cannot be a diffeomorphism.

  • Let

  • 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.

Surface deformations

In mechanics, a stress-induced transformation is called a deformation between two surfaces that is an invertible matrix, there is a neighborhood stays non-singular

The total differential of u is

  1. , and similarly for v.

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.

Diffeomorphism group

be a differentiable manifold that is second-countable and Hausdorff. The diffeomorphism group is the group is not zero-dimensional—it is not locally compact.

Topology

The diffeomorphism group has two natural topologies: weak and strong . 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, the weak topology is the topology induced by the family of metrics

whose union. Then:

vector fields and using the exponential map 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, the space of vector fields is a Fréchet space. Moreover, the transition maps are smooth, making the diffeomorphism group into a Fréchet manifold and even into a regular Fréchet Lie group-compact and not compact the full diffeomorphism group is not locally contractible for any of the two topologies. One has to restrict the group by controlling the deviation from the identity near infinity to obtain a diffeomorphism group which is a manifold; see .

Lie algebra

The Lie algebra consists of all vector fields equipped with the Lie bracket of vector fields at each point in space:

so the infinitesimal generators are the vector fields

Examples

Transitivity

, the diffeomorphism group acts transitively. More generally, the diffeomorphism group acts transitively on the configuration space is multiply transitive .

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

; 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.

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, the "group of twisted spheres", defined as the quotient of the abelian component group.

Connectedness

For manifolds, the diffeomorphism group is usually not connected. Its component group is called the mapping class group. In dimension 2 (i.e. surfacess), the mapping class group is a finitely presented group generated by Dehn twists; this has been proved by Max Dehn, W. B. R. Lickorish, and Allen Hatcher). 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, the mapping class group is simply the modular group 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 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

  • . This was proven by Stephen Smale.

  • The diffeomorphism group of the torus has the homotopy-type of its linear automorphisms.

  • The diffeomorphism groups of orientable surfaces of genus 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).

  • does not have the homotopy-type of a finite CW-complex.

Homeomorphism and diffeomorphism

Since every diffeomorphism is a homeomorphism, given a pair of manifolds which are diffeomorphic to each other they are in particular homeomorphic to each other. The converse is not true in general.

While it is easy to find homeomorphisms that are not 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 exist. 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: there are uncountably many that do not embed smoothly.

See also

Notes

References