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Encyclopedia > Diffeomorphism

In mathematics, a diffeomorphism is a kind of 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. Euclid, detail from The School of Athens by Raphael. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ...


Formally, given two manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both f:Mto N and its inverse f^{-1}:Nto M are differentiable (if these functions are r times continuously differentiable, f is called a Cr-diffeomorphism). In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...


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

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. The diffeomorphism is given by In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ... Please refer to Real vs. ... The word modulo is the Latin ablative of modulus. ... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... A sphere (< Greek σφαίρα) is a perfectly symmetrical geometrical object. ...

xmapsto e^{ix}.

This map provides not only a diffeomorphism, but also an isomorphism of Lie groups between the two spaces. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ...

Contents


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 In mathematics, the derivative is defined as the instantaneous rate of change of a function. ...

  1. it is a bijection,
  2. its derivative Df 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.

Remarks: A bijective function. ... In mathematics, the push forward (or pushforward) of a smooth map F : M → N between smooth manifolds at a point p is, in some sense, the best linear approximation of F near p. ... In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...

  • Condition 2 excludes diffeomorphisms going from dimension n to a different dimension k (the matrix Df would not be square hence certainly not invertible).
  • A differentiable bijection is not necessarily a diffeomorphism, e.g. f(x) = x3 is not a diffeomorphism from mathbb{R} to itself because its derivative vanishes at 0.
  • f also happens to be a homeomorphism.

Now, f from M to N is called a diffeomorphism if in coordinates 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 M and N respectively, with U being the image of φ and V the image of ψ. Then the conditions says that the map ψfφ − 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. Again we see that dimensions have to agree. 2-dimensional renderings (ie. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...


Diffeomorphism group

The diffeomorphism group of a manifold is the group of all its automorphisms (diffeomormorphisms 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. 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). This group can be given the structure of an infinite dimensional Lie group, modeled on the space of vector fields on the manifold. In general, this will not be a Banach Lie group, and the exponential map will not be a local diffeomorphism. Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, something is said to occur locally in the category of topological spaces if it occurs on small enough open sets. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, a symmetry group describes all symmetries of objects. ... In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ... In mathematics, the real line is simply the set of real numbers. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...


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. 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. 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). This word should not be confused with homomorphism. ... John Willard Milnor (b. ... In mathematics, an exotic sphere is a differentiable manifold M, which is homeomorphic to the ordinary sphere, but not diffeomorphic. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. ...


Much more extreme phenomena occur: in the early 1980s, a combination of results due to Fields Medal winners Simon Donaldson and Michael Freedman led to the discoveries that there are uncountably many pairwise non-diffeomorphic open subsets of mathbb{R}^4 each of which is homeomorphic to mathbb{R}^4, and also that there are uncountably many pairwise non-diffeomorphic differentiable manifolds homeomorphic to mathbb{R}^4 which do not embed smoothly in mathbb{R}^4. The Fields Medal is a prize awarded to up to four mathematicians not over forty years of age at each International Congress of the International Mathematical Union (therefore once every four years). ... Simon Kirwan Donaldson, born in Cambridge in 1957, is a mathematician famous for his work on exotic four-dimensional spaces in differential geometry using instantons, and the discovery of new differential invariants. ... Michael Hartley Freedman (born 21 April 1951 in Los Angeles, California, USA) is a mathematician at Microsoft Research. ...


See also


  Results from FactBites:
 
Diffeomorphism - Wikipedia, the free encyclopedia (718 words)
In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds.
A differentiable bijection is not necessarily a diffeomorphism, e.g.
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.
Anosov diffeomorphism - Wikipedia, the free encyclopedia (981 words)
If a flow on a manifold splits the tangent bundle into three invariant subbundles, with one subbundle that is exponentially contracting, and one that is exponentially expanding, and a third, non-expanding, non-contracting one-dimensional sub-bundle, then the flow is called an Anosov flow.
It was proved that any other Anosov diffeomorphism on a torus is topologically conjugate to one of this kind.
The problem of classifying manifolds that admit Anosov diffeomorphisms turned out to be very difficult, and still as of 2005 has no answer.
  More results at FactBites »

 
 

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