Diffeomorphism
In mathematics , especially in the areas of analysis , differential geometry and differential topology , a diffeomorphism is a bijective , continuously differentiable mapping , the inverse mapping of which is also continuously differentiable.
The definition and target areas of the mapping can be open sets of the finite-dimensional real vector space or, more generally, differentiable manifolds. Depending on the differentiability class, one speaks of -diffeomorphisms ( ).
definition
In vector space
A mapping between open subsets of the real vector space is called a diffeomorphism if:
- is bijective ,
- is continuously differentiable everywhere ,
- the inverse mapping is continuously differentiable everywhere.
If and even times continuously differentiable ("from the class " ), then one calls a -diffeomorphism. Are and infinitely differentiable ( "the Class "), this is referred as the diffeomorphism. If and both are real-analytic (“of the class ”), one calls a -diffeomorphism.
A mapping between open subsets is called local diffeomorphism if every point has an open neighborhood , so that its image is open and the restriction of to is a diffeomorphism.
On differentiable manifolds
The term is defined analogously on differentiable manifolds:
A mapping between two differentiable manifolds and is called diffeomorphism if it is bijective and both the inverse mapping and the inverse mapping are continuously differentiable . As above, the terms -, - and -diffeomorphism and local diffeomorphism are defined.
Two manifolds and are called diffeomorphic if there is a diffeomorphism from to . Manifolds that are diffeomorphic do not differ in terms of their differentiable structure .
Thus the diffeomorphism is precisely the isomorphism in the category of differentiable manifolds.
properties
- A diffeomorphism is always also a homeomorphism , but the reverse is not true.
- From the differentiability of the inverse mapping it follows that at each point the derivative of (as a linear mapping from to or from the tangential space to ) is invertible (bijective, regular, of maximum rank).
- Conversely, if the mapping is bijective and ( times) continuously differentiable and its derivation is invertible at every point, then it is a ( ) -diffeomorphism.
The sentence about the inverse mapping contains a stronger statement:
Theorem about the reverse mapping
A differentiable mapping with an invertible differential is locally a diffeomorphism. More precisely formulated:
Let be continuously differentiable and let the derivative of be invertible at this point . Then there exists an open neighborhood of in such that open and the restriction is a diffeomorphism.
This statement applies both to mappings between open sets of and to mappings between manifolds.
Examples
- The mapping , where , is a diffeomorphism between the open set and the set of real numbers . Thus the open interval is diffeomorphic to .
- The figure , is bijective and differentiable. But it is not a diffeomorphism, because it is not differentiable at the point 0.
Diffeomorphism and homeomorphism
For differentiable manifolds in dimension less than 4, homeomorphism always implies diffeomorphism: two differentiable manifolds of dimension less than or equal to 3 that are homeomorphic are also diffeomorphic. That is, if there is a homeomorphism, then there is also a diffeomorphism. This does not mean that every homeomorphism is a diffeomorphism.
This is not necessarily the case in higher dimensions. A prominent example are the Milnor spheres, according to John Willard Milnor : They are homeomorphic to the normal 7-dimensional sphere , but not diffeomorphic. For this discovery, Milnor received the Fields Medal in 1962 .
literature
- Klaus Jänich : Vector analysis. 5th edition. Springer Verlag, Berlin et al. 2005, ISBN 3-540-23741-0 ( Springer textbook ).
- DK Arrowsmith, CM Place: An Introduction to Dynamical Systems. Cambridge University Press, Cambridge et al. 1990, ISBN 0-521-30362-1 .