Diffeomorphism

Template:Stub A diffeomorphism is an infinitely differentiable homeomorphism with an infinitely differentiable inverse. Diffeomorphisms provide the right notion for two smooth manifolds to be "the same": if there exists a diffeomorphism, between two manifolds, then they will have all the same topological properties.

Put another way, a diffeomorphism is a type of function that satisfies the following condition: both the function and its inverse have continuous mixed partial derivatives of all orders in neighborhoods of every possible point.

Examples

Let <math>\mathbb R^+</math> denote the positive real numbers. The map <math>\phi : \R \to \R^+</math> defined by <math>\phi(x)=e^x</math> is a diffeomorphism.

The quotient space <math>\mathbb R/\mathbb Z</math> (that is, the real numbers modulo 1) is diffeomorphic to the unit circle <math>S^1</math> in the complex plane. <math>\phi : \mathbb R/\mathbb Z \to S^1</math> defined by <math>[x] \mapsto e^{2\pi i x}</math> provides a diffeomorphism. Note that this map is well-defined: if a and b represent the same class in <math>\mathbb R/\mathbb Z</math>, then <math>a=b+n</math> for some integer n. Then <math>\phi(a) = e^{2 \pi ia} = e^{2 \pi ia+2\pi n i} = e^{2 \pi i b} = \phi(b)</math>.

Distinct from Homeomorphism

Two topological spaces are said to be homeomorphic if there exists a homeomorphism between them. In this case, the two spaces are topologically the same. Similarly, two smooth manifolds (which carry the structure of topological spaces) are said to be diffeomorphic if there exists a diffeomorphism between them. Since a diffeomorphism is a special kind of homeomorphism, any two smooth manifolds which are diffeomorphic are in fact homeomorphic. One might expect that if two smooth manifolds are homeomorphic (i.e., equivalent as topological spaces) then they are diffeomorphic (i.e., equivalent as manifolds). However, this is not the case! It is possible to give the 7-dimensional sphere <math>S^7</math> the structure of a smooth manifold in such a way that it is not diffeomorphic to <math>S^7</math> with the standard smooth structure, though it is homeomorphic! Such examples are called exotic spheres.