Diffeomorphism
In
Definition
Given two differentiable manifolds and , a differentiable map is 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 . Two -differentiable manifolds are -diffeomorphic if there is an times continuously differentiable bijective map between them whose inverse is also times continuously differentiable.
Diffeomorphisms of subsets of manifolds
Given a subset of a manifold and a subset of a manifold , a function is said to be smooth if for all in there is a
Local description
Testing whether a differentiable map is a diffeomorphism can be made locally under some mild restrictions. This is the Hadamard-Caccioppoli theorem:[1]
If , are connected open subsets of such that is
Some remarks:
It is essential for to be
Then is
Thus, though is bijective at each point, is not invertible because it fails to be
Since the differential at a point (for a differentiable function)
is a linear map, it has a well-defined inverse if and only if is a bijection. The matrix representation of is the matrix of first-order partial derivatives whose entry in the -th row and -th column is . This so-called
Diffeomorphisms are necessarily between manifolds of the same dimension. Imagine going from dimension to dimension . If then could never be surjective, and if then could never be injective. In both cases, therefore, fails to be a bijection.
If is a bijection at then is said to be a local diffeomorphism (since, by continuity, will also be bijective for all sufficiently close to ).
Given a smooth map from dimension to dimension , if (or, locally, ) is surjective, is said to be a submersion (or, locally, a "local submersion"); and if (or, locally, ) is injective, is said to be an immersion (or, locally, a "local immersion").
A differentiable bijection is not necessarily a diffeomorphism. , for example, is not a diffeomorphism from 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.
When is a map between differentiable manifolds, a diffeomorphic is a stronger condition than a homeomorphic . For a diffeomorphism, and its inverse need to be differentiable; for a homeomorphism, 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, it satisfies the definition above. More precisely: Pick any cover of by compatible coordinate charts and do the same for . Let and be charts on, respectively, and , with and as, respectively, the images of and . The map is then a diffeomorphism as in the definition above, whenever .
Examples
Since any manifold can be locally parametrised, we can consider some explicit maps from into .
- Let
- We can calculate the Jacobian matrix:
- The Jacobian matrix has zero determinant if and only if . We see that could only be a diffeomorphism away from the -axis and the -axis. However, is not bijective since , 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.
Surface deformations
In
The
- , and similarly for v.
Then the image is a
Diffeomorphism group
Let be a differentiable manifold that is
Topology
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
as varies over compact subsets of . Indeed, since is -compact, there is a sequence of compact subsets whose union is . Then:
The diffeomorphism group equipped with its weak topology is locally homeomorphic to the space of vector fields (Leslie 1967). Over a compact subset of , this follows by fixing a Riemannian metric on and using the exponential map for that metric. If 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 with smooth right translations; left translations and inversion are only continuous. If , 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. If the manifold is -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 (Michor & Mumford 2013).
Lie algebra
The Lie algebra of the diffeomorphism group of consists of all vector fields on 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
Examples
- When is a Lie group, there is a natural inclusion of in its own diffeomorphism group via left-translation. Let denote the diffeomorphism group of , then there is a splitting , where is the subgroup of that fixes the identity element of the group.
- The diffeomorphism group of Euclidean space consists of two components, consisting of the orientation-preserving and orientation-reversing diffeomorphisms. In fact, the deformation retractof the subgroup of diffeomorphisms fixing the origin under the map . 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 is any manifold there is a group extension . Here is the subgroup of that preserves all the components of , and is the permutation group of the set (the components of ). Moreover, the image of the map is the bijections of that preserve diffeomorphism classes.
Transitivity
For a connected manifold , the diffeomorphism group
Extensions of diffeomorphisms
In 1926,
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 of the reals satisfying ; 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
The corresponding extension problem for diffeomorphisms of higher-dimensional spheres 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 "
Connectedness
For manifolds, the diffeomorphism group is usually not connected. Its component group is called the
Homotopy types
- The diffeomorphism group of has the homotopy-type of the subgroup . This was proven by Steve Smale.[2]
- 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).
- The homotopy-type of diffeomorphism groups of -manifolds for are poorly understood. For example, it is an open problem whether or not has more than two components. Via Milnor, Kahn and Antonelli, however, it is known that provided , 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
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 pairwise non-diffeomorphic open subsets of each of which is homeomorphic to , and also there are uncountably many pairwise non-diffeomorphic differentiable manifolds homeomorphic to that do not embed smoothly in .
See also
- Anosov diffeomorphism such as Arnold's cat map
- Diffeology, smooth parameterizations on a set, which makes a diffeological space
- Diffeomorphometry, metric study of shape and form in computational anatomy
- Étale morphism
- Large diffeomorphism
- Local diffeomorphism
- Superdiffeomorphism
Notes
References
- Krantz, Steven G.; Parks, Harold R. (2013). The implicit function theorem: history, theory, and applications. Modern Birkhäuser classics. Boston. ISBN 978-1-4614-5980-4.)
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: CS1 maint: location missing publisher (link - Chaudhuri, Shyamoli; Kawai, Hikaru; Tye, S.-H. Henry (1987-08-15). "Path-integral formulation of closed strings" (PDF). Physical Review D. 36 (4): 1148–1168. (PDF) from the original on 2018-07-21.
- ISBN 0-7923-4475-8
- Duren, Peter L. (2004), Harmonic Mappings in the Plane, Cambridge Mathematical Tracts, vol. 156, Cambridge University Press, ISBN 0-521-64121-7
- "Diffeomorphism", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Hirsch, Morris (1997), Differential Topology, Berlin, New York: ISBN 978-0-387-90148-0
- Kriegl, Andreas; Michor, Peter (1997), The convenient setting of global analysis, Mathematical Surveys and Monographs, vol. 53, American Mathematical Society, ISBN 0-8218-0780-3
- Leslie, J. A. (1967), "On a differential structure for the group of diffeomorphisms", MR 0210147
- Michor, Peter W.; Mumford, David (2013), "A zoo of diffeomorphism groups on Rn.", Annals of Global Analysis and Geometry, 44 (4): 529–540, S2CID 118624866
- ISBN 978-0-8218-4230-0
- Omori, Hideki (1997), Infinite-dimensional Lie groups, Translations of Mathematical Monographs, vol. 158, American Mathematical Society, ISBN 0-8218-4575-6
- Kneser, Hellmuth (1926), "Lösung der Aufgabe 41.", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 35 (2): 123