Diffeomorphism

The image of a square of a rectangular grid with some diffeomorphism of this square into itself.

A diffeomorphism is a mapping of a certain type between smooth manifolds.

Content

Definition

Diffeomorphism is a one-to-one and smooth mapping $f\colon M\to N$ ${\ displaystyle f \ colon M \ to N}$ smooth variety $M$ ${\ displaystyle M}$ into smooth variety $N$ ${\ displaystyle N}$ , the converse of which is also smooth.

Usually, smoothness is understood as $C^{\infty }$ ${\ displaystyle C ^ {\ infty}}$ -smoothness, but diffeomorphisms with a different type of smoothness, in particular, of the class $C^{k}$ ${\ displaystyle C ^ {k}}$ at any kind $k$ ${\ displaystyle k}$ .

Examples

The simplest examples of diffeomorphisms are non-degenerate linear (affine) transformations of vector (resp. Affine) spaces of the same dimension.

Related Definitions

If for $M$ {\ displaystyle M} and $N$ {\ displaystyle N} there is a diffeomorphism $f$ $:$ $M$ $\to$ $N$ {\ displaystyle f \ colon M \ to N} then they say that $M$ {\ displaystyle M} and $N$ {\ displaystyle N} diffeomorphic .
- This relationship is usually denoted by $M\cong N$ ${\ displaystyle M \ cong N}$ .
- Note that only manifolds of the same dimension can be diffeomorphic.

The set of diffeomorphisms of a manifold $M$ ${\ displaystyle M}$ forms a group called a group of diffeomorphisms $M$ ${\ displaystyle M}$ and denoted by $\operatorname {Diff} \,M$ ${\ displaystyle \ operatorname {Diff} \, M}$ .

Display $f\colon M\to N$ ${\ displaystyle f \ colon M \ to N}$ called a local diffeomorphism at the point $x\in M$ ${\ displaystyle x \ in M}$ if its restriction to some neighborhood of a point $x$ ${\ displaystyle x}$ is a diffeomorphism to some neighborhood of the point $y=f(x)\in N$ ${\ displaystyle y = f (x) \ in N}$ .

Properties

Any diffeomorphism is a homeomorphism.
- The converse is not true. Moreover, there exist homeomorphic but not diffeomorphic smooth manifolds.
One-to-one mapping $f\colon M\to N$ ${\ displaystyle f \ colon M \ to N}$ is a diffeomorphism if and only if $f$ ${\ displaystyle f}$ - A smooth mapping and its Jacobian is nowhere equal to zero.

Literature

Zorich V.A. Mathematical analysis. - M .: Fizmatlit , 1984. - 544 p.
Milnor J., Wallace A. Differential topology (elementary course), - Any publication.
Hirsch M. Differential Topology, - Any publication.
Spivak M. Mathematical analysis on manifolds. - M .: Mir, 1968.