Diffeomorphisms

In the same way that spaces or topological manifolds are equivalent if they are related by a homeomorphism, differentiable manifolds are equivalent if they are related by a diffeomorphism, a homeomorphism that is differentiable along with its inverse. As usual we define differentiability by moving the mapping to \({\mathbb{R}^{n}}\), e.g. \({\Phi\colon M\to N}\) is differentiable if \({\alpha_{N}\circ\Phi\circ\alpha_{M}^{-1}\colon\mathbb{\mathbb{R}}^{m}\to\mathbb{R}^{n}}\) is, where \({\alpha_{M}}\) and \({\alpha_{N}}\) are charts for \({M}\) and \({N}\). Intuitively, a diffeomorphism can still be thought of as arbitrary stretching and bending, but it is “nicer” in that it preserves the differentiable structure.

Δ It is important to distinguish between coordinate transformations, which are locally defined and so may have singularities outside of a given region; and diffeomorphisms, which are globally defined and form a group. One can define a coordinate transformation on a region of a manifold that avoids any resulting singularities, but a diffeomorphism must be smooth on the entire manifold.

 

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