Diffeomorphism

In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth. In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B In Mathematics and related technical fields the term map or mapping is often a Synonym for function. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability

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

1 Definition
2 Examples
3 Local description
4 Diffeomorphism group
5 Homeomorphism and diffeomorphism
6 See also
7 References

Definition

Given two manifolds M and N, a bijective map $f$ from M to N is called a diffeomorphism if both

$f:M\to N$

and its inverse

$f^{-1}:N\to M$

are differentiable (if these functions are r times continuously differentiable, f is called a $C r$ -diffeomorphism). In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property

Two manifolds M and N are diffeomorphic (symbol being usually $\simeq$ ) if there is a diffeomorphism $f$ from M to N.

Examples

$\mathbb{R}/\mathbb{Z} \simeq S^1.$

That is, the quotient group of the real numbers modulo the integers is again a smooth manifold, which is diffeomorphic to the 1-sphere, usually known as the circle. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics, the real numbers may be described informally in several different ways The word modulo (Latin with respect to a modulus of ___ is the Latin Ablative of Modulus which itself means "a small measure The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe The diffeomorphism is given by

$x\mapsto e^{2\pi ix}.$

This map provides not only a diffeomorphism, but also an isomorphism of Lie groups between the two spaces. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group

Local description

Model example: if $U$ and $V$ are two open subsets of $\mathbb{R}^n$ , a differentiable map $f$ from $U$ to $V$ is a diffeomorphism if

it is a bijection,
its derivative $D f$ is invertible (as the matrix of all $\partial f_i/\partial x_j$ , $1 \leq i,j \leq n$ ), which means the same as having non-zero Jacobian determinant. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property Suppose that &phi: M → N is a smooth map between smooth manifolds then the differential of &phi at a point x is in some In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant.

Remarks:

Condition 2 excludes diffeomorphisms going from dimension $n$ to a different dimension $k$ (the matrix $D f$ would not be square hence certainly not invertible). In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it
A differentiable bijection is not necessarily a diffeomorphism, e. g. $f (x) = x 3$ is not a diffeomorphism from $\mathbb{R}$ to itself because its derivative vanishes at 0 (and hence its inverse is not differentiable at 0).
$f$ also happens to be a homeomorphism, as easily follows from the inverse function theorem

Now, $f$ from M to N is called a diffeomorphism if in coordinates charts it satisfies the definition above. Topological equivalence redirects here see also Topological equivalence (dynamical systems. In Mathematics, the inverse function theorem gives sufficient conditions for a Vector-valued function to be Invertible on an Open region containing A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be More precisely, pick any cover of M by compatible coordinate charts, and do the same for N. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be 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 $ψ f φ - 1$ from $U$ to $V$ is a diffeomorphism as in the definition above (whenever it makes sense). One has to check that for every couple of charts $φ$ , $ψ$ of two given atlases, but once checked, it will be true for any other compatible chart. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be Again we see that dimensions have to agree.

Diffeomorphism group

The diffeomorphism group of a manifold is the group of all its automorphisms (diffeomorphisms to itself). For dimension greater than or equal to one this is a large group. For a connected manifold M the diffeomorphisms act transitively on M: this is true locally because it is true in Euclidean space and then a topological argument shows that given any p and q there is a diffeomorphism taking p to q. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Mathematics, a phenomenon is sometimes said to occur locally if roughly speaking it occurs on sufficiently small or arbitrarily small Neighborhoods That is, all points of M in effect look the same, intrinsically. The same is true for finite configurations of points, so that the diffeomorphism group is k- fold multiply transitive for any integer k ≥ 1, provided the dimension is at least two (it is not true for the case of the circle or real line). In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a This group can be given the structure of an infinite dimensional Lie group, modeled on the space of vector fields on the manifold. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. In general, this will not be a Banach Lie group, and the exponential map will not be a local diffeomorphism.

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. Tibor Radó ( June 2 1895 &ndash December 29 1965) was a Hungarian mathematician who moved to the USA after World In Mathematics, the Poisson integral formula gives an explicit solution to the Dirichlet problem for Laplace's equation in a ball in Euclidean In Mathematics, a unit circle is In Mathematics, the open unit disk around P (where P is a given point in the plane) is the set of points whose distance from P is 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. Hellmuth Kneser ( April 16, 1898 - August 23, 1973) was a German Mathematician, who made notable contributions to Group Gustave Choquet ( 1 March 1915 &ndash 14 November 2006) was a French Mathematician.

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). Alexander's trick, also known as the Alexander trick, is a basic result in Geometric topology, named after J

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. René Thom ( September 2, 1923 – October 25, 2002) was a French Mathematician. John Willard Milnor (b February 20, 1931 in Orange New Jersey) is an American Mathematician known for his work in Differential Stephen Smale (born July 15, 1930) is an American Mathematician from Flint Michigan. 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 Bⁿ. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, an exotic sphere is a Differentiable manifold that is Homeomorphic to the standard Euclidean n - Sphere, but not In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics, the identity component of a Topological group G is the connected component G 0 that contains the Identity

For oriented manifolds of dimension >1, the diffeomorphism group is usually not connected. Its component group is called the mapping class group. In Mathematics, in the sub-field of Geometric topology, the mapping class group is an important algebraic invariant of a Topological space. In dimension 2, i. e. for surfaces, the mapping class group is a finitely presented group, generated by Dehn twists (Dehn, Lickorish, Hatcher). In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. In Mathematics, one method of defining a group is by a presentation. In Geometric topology, a branch of Mathematics, a Dehn twist is a certain type of self-homeomorphism of a Surface (two-dimensional Max Dehn ( November 13, 1878, Hamburg, Germany – June 27, 1952, Black Mountain, North Carolina, William Bernard Raymond Lickorish is a Mathematician. He is Emeritus professor of Geometric topology in DPMMS, University of Cambridge Allen Edward Hatcher is an American Topologist and also a noted author Max Dehn and Jakob Nielsen showed that it can be identified with the outer automorphism group of the fundamental group of the surface. Max Dehn ( November 13, 1878, Hamburg, Germany – June 27, 1952, Black Mountain, North Carolina, For other people with similar names see Jakob Nielsen. Jakob Nielsen ( October 15, 1890 – August 3, In Mathematics, the outer automorphism group of a group G is the quotient of the Automorphism group Aut( G) by its Inner In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology.

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. William Paul Thurston (born October 30, 1946) is an American Mathematician. In Mathematics, Thurston's classification theorem characterizes Homeomorphisms of a Compact surface. In Mathematics, a periodic function is a function that repeats its values after some definite period has been added to its Independent variable In Mathematics, specifically in Topology, a pseudo-Anosov map is a type of Homeomorphism of a Surface to itself In the case of the torus S¹ x S¹ = R²/Z², 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. In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar In Mathematics, the modular group Γ is a fundamental object of study in Number theory, Geometry, algebra, and many other areas of advanced Möbius transformations should not be confused with the Möbius transform or the Möbius function. Möbius transformations should not be confused with the Möbius transform or the Möbius function. Möbius transformations should not be confused with the Möbius transform or the Möbius function. Thurston accomplished his classification by observing that the mapping class group acted naturally on a compactification of Teichmuller space; since this enlarged space was homeomorphic to a closed ball, the Brouwer fixed point theorem became applicable. In Mathematics, given a Riemann surface X, the Teichmüller space of X, notated TX or Teich( X) is a complex In Mathematics, the Brouwer fixed point theorem is an important Fixed point theorem that applies to finite-dimensional spaces and which forms the basis for several

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. In Mathematics, the identity component of a Topological group G is the connected component G 0 that contains the Identity This had first been proved for a product of circles by Michel Herman; it was proved in full generality by Thurston.

Homeomorphism and diffeomorphism

It is easy to find a homeomorphism which is not a diffeomorphism, but it is more difficult to find a pair of homeomorphic manifolds that are not diffeomorphic. Topological equivalence redirects here see also Topological equivalence (dynamical systems. 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) which is homeomorphic to the standard 7-sphere but not diffeomorphic to it. John Willard Milnor (b February 20, 1931 in Orange New Jersey) is an American Mathematician known for his work in Differential In Mathematics, an exotic sphere is a Differentiable manifold that is Homeomorphic to the standard Euclidean n - Sphere, but not There are in fact 28 oriented diffeomorphism classes of manifolds homeomorphic to the 7-sphere (each of them is a fiber bundle over the 4-sphere with fiber the 3-sphere). In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space. In Mathematics, a 3-sphere is a higher-dimensional analogue of a Sphere.

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 $\mathbb{R}^4$ each of which is homeomorphic to $\mathbb{R}^4$ , and also there are uncountably many pairwise non-diffeomorphic differentiable manifolds homeomorphic to $\mathbb{R}^4$ which do not embed smoothly in $\mathbb{R}^4$ . In Mathematics, 4-manifold is a 4-dimensional Topological manifold. Simon Kirwan Donaldson (born August 20 1957 in Cambridge, England) is an English mathematician famous for his work on the Topology of Michael Hartley Freedman (born 21 April 1951 in Los Angeles, California, U In Mathematics, an exotic R 4 is a Differentiable manifold that is Homeomorphic to the Euclidean space R 4

References

Banyaga, Augustin (1997), The structure of classical diffeomorphism groups, Mathematics and its Applications, 400, Kluwer Academic, ISBN 0-7923-4475-8
Omori, Hideki (1997), Infinite-dimensional Lie groups, Translations of Mathematical Monographs, 158, American Mathematical Society, ISBN 0-8218-4575-6

Kriegl, Andreas & Michor, Peter (1997), The convenient setting of global analysis, Mathematical Surveys and Monographs, 53, American Mathematical Society, ISBN 0-8218-0780-3

Duren, Peter L. In Mathematics, a local diffeomorphism is a Smooth map f: M &rarr N between Smooth manifolds such that for every point Augustin Banyaga (b March 31, 1947 in Kigali, Rwanda) is a Rwandan-born American Mathematician whose research fields include (2004), Harmonic Mappings in the Plane, Cambridge Mathematical Tracts, 156, Cambridge University Press, ISBN 0521641217

Milnor, John W. (2007), Collected Works Vol. John Willard Milnor (b February 20, 1931 in Orange New Jersey) is an American Mathematician known for his work in Differential III, Differential Topology, American Mathematical Society, ISBN 0821842307

Dictionary

-noun

A differentiable homeomorphism (with differentiable inverse) between differentiable manifolds.

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