^^ [[Schutz - 2 - Differentiable Manifolds and Tensors|2. Differentiable Manifolds and Tensors]] <<[[Schutz - 2.3 - Other Examples of Manifolds|2.3 Other Examples of Manifolds]] | [[Schutz - 2.5 - Curves|2.5 Curves]]>> # Global Considerations Because every manifold is locally the same as some $\mathbb{R}^n$, any two manifolds of the same dimension and differentiability class are locally indistinguishable. Thus, manifolds are divided into classes depending on their global structure. For example, $S^2$ and the surface of a crayon have the same global structure. There is a perfectly good bijection from one onto the other, even though you need multiple charts to cover them completely. ![[Pasted image 20210123185625.png]] > **Diffeomorphism** > A *diffeomorphism* is a $C^\infty$ [[Schutz - 1.2 - Mappings#^6c8b58|bijection]] from one [[Schutz - 2.1 - Definition of a Manifold#^31cc42|manifold]] $M$ into another $N$, and whose inverse is also $C^\infty$. This is called a *diffeomorphism* of $M$ onto $N$ and $M$ and $N$ are said to be *diffeomorphic*. Recall that a map is $C^\infty$ if the coordinates of a point in $N$ are infinitely differentiable functions of the coordinates of the inverse image of the point in $M$. > **Homeomorphism** > A homeomorphism is a diffeomrophism without the differentiability requirement, i.e. a bijection from one space onto another which is continuous and whose inverse is also continuous.