# 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 ( ). ${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle C ^ {k}}$${\ displaystyle k \ in \ {1,2, \ dots, \ infty, \ omega \}}$

Image of a right angle mesh on a square under a diffeomorphism from the square on itself.

## definition

### In vector space

A mapping between open subsets of the real vector space is called a diffeomorphism if: ${\ displaystyle f \ colon U \ to V}$ ${\ displaystyle U, V}$${\ displaystyle \ mathbb {R} ^ {n}}$

• ${\ displaystyle f}$ is bijective ,
• ${\ displaystyle f}$ is continuously differentiable everywhere ,
• the inverse mapping  is continuously differentiable everywhere.${\ displaystyle f ^ {- 1}}$

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. ${\ displaystyle f}$${\ displaystyle f ^ {- 1}}$${\ displaystyle k}$${\ displaystyle C ^ {k}}$${\ displaystyle k = 1,2,3, \ dotsc}$${\ displaystyle f}$${\ displaystyle C ^ {k}}$${\ displaystyle f}$${\ displaystyle f ^ {- 1}}$${\ displaystyle C ^ {\ infty}}$${\ displaystyle f}$${\ displaystyle C ^ {\ infty}}$${\ displaystyle f}$${\ displaystyle f ^ {- 1}}$${\ displaystyle C ^ {\ omega}}$${\ displaystyle f}$${\ displaystyle C ^ {\ omega}}$

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. ${\ displaystyle f \ colon U \ to V}$ ${\ displaystyle U, V \ subset \ mathbb {R} ^ {n}}$${\ displaystyle p \ in U}$ ${\ displaystyle W \ subset U}$${\ displaystyle f (W) \ subset V}$ ${\ displaystyle f | _ {W} \ colon W \ to f (W)}$${\ displaystyle f}$${\ displaystyle W}$

### 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. ${\ displaystyle f \ colon M \ to N}$ ${\ displaystyle M}$${\ displaystyle N}$${\ displaystyle f}$${\ displaystyle C ^ {k}}$${\ displaystyle C ^ {\ infty}}$${\ displaystyle C ^ {\ omega}}$

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 . ${\ displaystyle M}$${\ displaystyle N}$${\ displaystyle f}$${\ displaystyle M}$${\ displaystyle N}$

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).${\ displaystyle p}$${\ displaystyle f}$${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle T_ {p} M}$${\ displaystyle T_ {f (p)} N}$
• Conversely, if the mapping is bijective and ( times) continuously differentiable and its derivation is invertible at every point, then it is a ( ) -diffeomorphism.${\ displaystyle f}$${\ displaystyle k}$${\ displaystyle f}$${\ displaystyle C ^ {k}}$

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. ${\ displaystyle f \ colon U \ to V}$${\ displaystyle f}$${\ displaystyle p \ in U}$${\ displaystyle W}$${\ displaystyle p}$${\ displaystyle U}$${\ displaystyle f (W)}$${\ displaystyle f | _ {W} \ colon W \ to f (W)}$

This statement applies both to mappings between open sets of and to mappings between manifolds. ${\ displaystyle \ mathbb {R} ^ {n}}$

## Examples

• The mapping , where , is a diffeomorphism between the open set and the set of real numbers . Thus the open interval is diffeomorphic to .${\ displaystyle f \ colon (-1,1) \ to \ mathbb {R}}$${\ displaystyle f (t) = \ tan \ left (t \ cdot \ pi / 2 \ right)}$${\ displaystyle (-1.1)}$${\ displaystyle \ mathbb {R}}$${\ displaystyle (-1.1)}$${\ displaystyle \ mathbb {R}}$
• The figure , is bijective and differentiable. But it is not a diffeomorphism, because it is not differentiable at the point 0.${\ displaystyle f \ colon \ mathbb {R} \ to \ mathbb {R}}$${\ displaystyle f (x) = x ^ {3}}$${\ displaystyle f ^ {- 1}}$

## 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 .