Editing Differential structure (section)

==Definition==
For a natural number ''n'' and some ''k'' which may be a non-negative integer or infinity, an '''''n''-dimensional ''C''<sup>''k''</sup> differential structure'''<ref>[[Morris Hirsch|Hirsch, Morris]], ''Differential Topology'', Springer (1997), {{ISBN|0-387-90148-5}}. for a general mathematical account of differential structures</ref> is defined using a '''''C''<sup>''k''</sup>-[[atlas (topology)|atlas]]''', which is a set of [[bijections]] called '''charts''' between subsets of ''M'' (whose union is the whole of ''M'') and open subsets of <math>\mathbb{R}^{n}</math>:

:<math>\varphi_{i}:M\supset W_{i}\rightarrow U_{i}\subset\mathbb{R}^{n}</math>

which are '''''C''<sup>''k''</sup>-compatible''' (in the sense defined below):

Each chart allows a subset of the manifold to be viewed as an open subset of <math>\mathbb{R}^{n}</math>, but the usefulness of this depends on how much the charts agree when their domains overlap.

Consider two charts:

:<math>\varphi_{i}:W_{i}\rightarrow U_{i},</math>
:<math>\varphi_{j}:W_{j}\rightarrow U_{j}.</math>

The intersection of their domains is

:<math>W_{ij}=W_{i}\cap W_{j}</math>

whose images under the two charts are 

:<math>U_{ij}=\varphi_{i}\left(W_{ij}\right),</math>
:<math>U_{ji}=\varphi_{j}\left(W_{ij}\right).</math>

The [[transition map]] between the two charts translates between their images on their shared domain:

:<math>\varphi_{ij}:U_{ij}\rightarrow U_{ji}</math>
:<math>\varphi_{ij}(x)=\varphi_{j}\left(\varphi_{i}^{-1}\left(x\right)\right).</math>

Two charts <math>\varphi_{i},\,\varphi_{j}</math> are '''''C''<sup>''k''</sup>-compatible''' if

:<math>U_{ij},\, U_{ji}</math>

are open, and the transition maps

:<math>\varphi_{ij},\,\varphi_{ji}</math>

have [[smoothness|continuous partial derivatives of order ''k'']]. If ''k''&nbsp;=&nbsp;0, we only require that the transition maps are continuous, consequently a ''C''<sup>0</sup>-atlas is simply another way to define a topological manifold. If ''k''&nbsp;=&nbsp;∞, derivatives of all orders must be continuous. A family of ''C''<sup>''k''</sup>-compatible  charts covering the whole manifold is a ''C''<sup>''k''</sup>-atlas defining a ''C''<sup>''k''</sup> differential manifold. Two atlases are '''''C''<sup>''k''</sup>-equivalent''' if the union of their sets of charts forms a ''C''<sup>''k''</sup>-atlas. In particular, a ''C''<sup>''k''</sup>-atlas that is ''C''<sup>0</sup>-compatible with a ''C''<sup>0</sup>-atlas that defines a topological manifold is said to determine a '''''C''<sup>''k''</sup> differential structure''' on the topological manifold. The ''C''<sup>''k''</sup> [[equivalence classes]] of such atlases are the '''distinct ''C''<sup>''k''</sup> differential structures''' of the [[manifold]]. Each distinct differential structure is determined by a unique maximal atlas, which is simply the union of all atlases in the equivalence class.

For simplification of language, without any loss of precision, one might just call a maximal ''C''<sup>''k''</sup>−atlas on a given set a ''C''<sup>''k''</sup>−manifold. This maximal atlas then uniquely determines both the topology and the underlying set, the latter being the union of the domains of all charts, and the former having the set of all these domains as a basis.