Diffeomorphism

In mathematics, particularly in the areas of analysis, differential geometry and differential topology, a diffeomorphism is a bijective, continuous differentiable mapping whose inverse mapping is also continuously differentiable.

It may be open sets of finite dimensional real vector space, or more generally differentiable manifolds the definition and target areas of the image. Depending on Differenzierbarkeitsklasse spoken of - diffeomorphisms ().

• 2.1 theorem on the inverse mapping

Definition

In the vector space

A mapping between open subsets of the real vector space is called a diffeomorphism if it is bijective and both the inverse mapping are continuously differentiable everywhere.

Are and even - times continuously differentiable ( " the class " ) it is called a diffeomorphism. Are and infinitely differentiable ( " the Class"), this is known as diffeomorphism, and both are real - analytic ( " the Class") it is called a diffeomorphism.

A mapping between open subsets is called a local diffeomorphism, if every point has an open environment, so that their image is open and the restriction of a diffeomorphism.

On differentiable manifolds

On differentiable manifolds, the term is defined analogously:

A map between two differentiable manifolds and is called a diffeomorphism if it is bijective and both the inverse mapping are continuously differentiable. - - As above, the terms and diffeomorphism and local diffeomorphism defined.

Two manifolds M and N are diffeomorphic if there is a diffeomorphism f from M to N. Manifolds that are diffeomorphic, do not differ with respect to their differentiable structure.

Thus the Diffeomorphie is just the isomorphism in the category of differentiable manifolds.

Properties

• A diffeomorphism is always a homeomorphism, but the converse is not true.

• From the differentiability of the inverse mapping follows that at each point is the derivation of ( as a linear map from to or from the tangent space after ) is invertible ( bijective, regularly, of maximal rank ).

• Conversely, if the picture is bijective and ( times) is continuously differentiable and its derivative at any point invertible, so is a () - diffeomorphism.

A stronger statement contains the theorem on the inverse map:

Theorem on the inverse mapping

A differentiable map with invertierbarem differential is locally a diffeomorphism. Specifically formulated:

Be continuously differentiable and the derivative of is invertible on the site. Then there exists an open neighborhood of in, so open, and the restriction is a diffeomorphism.

This statement applies to both maps between open sets as well as for the maps between manifolds.

Examples

• The figure, which is a diffeomorphism between the open set (-1.1 ) and the set of real numbers. Thus, the open interval ( -1,1 ) is diffeomorphic to.
• The figure is bijective and differentiable. But it is not a diffeomorphism, because is not differentiable at the point 0.

Diffeomorphie and homeomorphism

For differentiable manifolds in dimension less than 4 homeomorphism always implies Diffeomorphie: two differentiable manifolds of dimension less than or equal to 3, which are homeomorphic are also diffeomorphic. That is, if there is a homeomorphism, then there is a diffeomorphism. This does not mean that every homeomorphism would be a diffeomorphism.

In higher dimensions this is not necessarily the case. A prominent example is the Milnor spheres, by John Willard Milnor: You are homeomorphic to the normal 7 -dimensional sphere but not diffeomorphic. For this discovery Milnor received the Fields Medal in 1962.