notes on the classical definition of a manifold
Classical Definition
Historically, the data for a manifold was specified as a collection^{} of coordinate domains related by changes of coordinates. The manifold itself could be obtained by gluing the domains in accordance with the transition functions^{}, provided the changes of coordinates were free of inconsistencies.
In this formulation, a ${\mathcal{C}}^{k}$ manifold is specified by two types of information. The first item of information is a collection of open sets
$${V}_{\alpha}\subset {\mathbb{R}}^{n},\alpha \in \mathcal{A},$$ 
indexed by some set $\mathcal{A}$. The second item is a collection of transition functions, that is to say ${\mathcal{C}}^{k}$ diffeomorphisms
$${\sigma}_{\alpha \beta}:{V}_{\alpha \beta}\to {\mathbb{R}}^{n},{V}_{\alpha \beta}\subset {V}_{\alpha},\text{open},\alpha ,\beta \in \mathcal{A},$$ 
obeying certain consistency and topological conditions.
We call a pair
$$(\alpha ,x),\alpha \in \mathcal{A},x\in {V}_{\alpha}$$ 
the coordinates of a point relative to chart $\alpha $, and define the manifold $M$ to be the set of equivalence classes^{} of such pairs modulo the relation^{}
$$(\alpha ,x)\simeq (\beta ,{\sigma}_{\alpha \beta}(x)).$$ 
To ensure that the above is an equivalence relation we impose the following hypotheses.

•
For $\alpha \in \mathcal{A}$, the transition function ${\sigma}_{\alpha \alpha}$ is the identity on ${V}_{\alpha}$.

•
For $\alpha ,\beta \in \mathcal{A}$ the transition functions ${\sigma}_{\alpha \beta}$ and ${\sigma}_{\beta \alpha}$ are inverses^{}.

•
For $\alpha ,\beta ,\gamma \in \mathcal{A}$ we have for a suitably restricted domain
$${\sigma}_{\beta \gamma}\circ {\sigma}_{\alpha \beta}={\sigma}_{\alpha \gamma}$$
We topologize $M$ with the least coarse topology that will make the mappings from each ${V}_{\alpha}$ to $M$ continuous^{}. Finally, we demand that the resulting topological space^{} be paracompact and Hausdorff^{}.
0.0.1 Notes
To understand the role played by the notion of a differential manifold, one has to go back to classical differential geometry, which dealt with geometric objects such as curves and surface only in reference to some ambient geometric setting — typically a 2dimensional plane or 3dimensional space. Roughly speaking, the concept of a manifold was created in order to treat the intrinsic geometry^{} of such an object, independent of any embedding^{}. The motivation for a theory of intrinsic geometry can be seen in results such as Gauss’s famous Theorema Egregium, that showed that a certain geometric property of a surface, namely the scalar curvature, was fully determined by intrinsic metric properties of the surface, and was independent of any particular embedding. Riemann [1] took this idea further in his habilitation lecture by describing intrinsic metric geometry of $n$dimensional space without recourse to an ambient Euclidean^{} setting. The modern notion of manifold, as a general setting for geometry involving differential properties evolved early in the twentieth century from works of mathematicians such as Hermann Weyl [3], who introduced the ideas of an atlas and transition functions, and Elie Cartan, who investigation global properties and geometric structures^{} on differential manifolds. The modern definition of a manifold was introduced by Hassler Whitney [4] (For more foundational information, follow http://web.archive.org/web/20041010165022/http://www.math.uchicago.edu/ mfrank/founddiffgeom3.htmlthis link to some old notes by http://web.archive.org/web/20040511092724/www.math.uchicago.edu/ mfrank/Matthew Frank ).
References
 1 Riemann, B., “Über die Hypothesen welche der Geometrie zu Grunde liegen (On the hypotheses that lie at the foundations of geometry)” in M. Spivak, A comprehensive introduction to differential geometry^{}, vol. II.
 2 Spivak, M., A comprehensive introduction to differential geometry, vols I & II.
 3 Weyl, H., The concept of a Riemann surface, 1913
 4 Whitney, H., Differentiable Manifolds, Annals of Mathematics, 1936.
Title  notes on the classical definition of a manifold 

Canonical name  NotesOnTheClassicalDefinitionOfAManifold 
Date of creation  20130322 14:14:47 
Last modified on  20130322 14:14:47 
Owner  rmilson (146) 
Last modified by  rmilson (146) 
Numerical id  11 
Author  rmilson (146) 
Entry type  Topic 
Classification  msc 5303 
Related topic  Manifold 