In mathematics, the idea of a diffeomorphism is to be able to have a notion of isomorphism of smooth manifolds. That is, the whole point of diffeomorphisms is to have a concept of whether two differentiable manifolds are diffeomorphic (symbol being usually ), i.e. mapped one to the other by a diffeomorphism. Should that be the case, then as far as differential topologyis concerned, they are identical. For example
- .
Model example: if and are two open subsets of , a differentiable map from to is a diffeomorphism if
- it is a bijection,
- its differential is invertible (as the matrix of all , ).
Remarks:
- Condition 2 excludes diffeomorphisms going from dimension to a different dimension (the matrix of would not be square hence certainly not invertible).
- A differentiable bijection is not necessarily a diffeomorphism, e.g. is not a diffeomorphism from to itself because its derivative vanishes at 0.
- also happens to be a homeomorphism.
Definition: given two differentiable manifolds
M and N (they could be the same), a bijective map from M to N is called a diffeomorphism if both and its inverse are smooth. Equivalently, in coordinates chartss it satisfies the definition above.
More precisely, pick any cover of M by compatible coordinate chartss, and do the same for N. Let and be charts on M and N respectively, with being the image of and the image of . Then the conditions says that the map from to is a diffeomorphism as in the definition above (whenever it makes sense). One has to check that for every couple of charts , of two given atlases, but once checked, it will be true for any other compatible chart. Again we see that dimensions have to agree.
Remark: the interesting problem is global. Indeed two manifolds of the same dimension are always locally diffeomorphic (since they are locally diffeomorphic to an open set of euclidean space). More to the point, a differentiable map whose differential is invertible is always a local diffeomorphism, and in particular locally a bijection, by virtue of the inverse function theorem.
