notes on the classical definition of a manifold

Classical Definition

Historically, the data for amanifold was specified as a collection of coordinate domains relatedby changes of coordinates. The manifold itself could be obtained bygluing the domains in accordance with the transition functions,provided the changes of coordinates were free of inconsistencies.

In this formulation, a ${𝒞}^k$ manifold is specified by two types ofinformation. The first item of information is a collection of opensets

$$V_{\alpha }\subset {ℝ}^n,\alpha \in 𝒜,$$

indexed by some set$𝒜$. The second item is a collection of transition functions, thatis to say ${𝒞}^k$ diffeomorphisms

$${\sigma }_{\alpha \beta }:V_{\alpha \beta }\to {ℝ}^n,V_{\alpha \beta }\subset V_{\alpha },\text{open},\alpha ,\beta \in 𝒜,$$

obeying certainconsistency and topological conditions.

We call a pair

$$(\alpha ,x),\alpha \in 𝒜,x\in V_{\alpha }$$

the coordinates ofa 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 anequivalence relation we impose the following hypotheses.

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

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

• For $\alpha ,\beta ,\gamma \in 𝒜$ we have for a suitablyrestricted domain

  $${\sigma }_{\beta \gamma }\circ {\sigma }_{\alpha \beta }={\sigma }_{\alpha \gamma }$$

We topologize $M$ with the least coarse topology that will makethemappings from each $V_{\alpha }$ to $M$ continuous. Finally, wedemandthat the resulting topological space be paracompact and Hausdorff.

0.0.1 Notes

To understand the role played by the notion of adifferential manifold, one has to go back to classical differentialgeometry, which dealt with geometric objects such as curves andsurface only in reference to some ambient geometric setting —typically a 2-dimensional plane or 3-dimensional space. Roughlyspeaking, the concept of a manifold was created in order to treat theintrinsic geometry of such an object, independent of any embedding.The motivation for a theory of intrinsic geometry can be seen inresults such as Gauss’s famous Theorema Egregium, that showed that acertain geometric property of a surface, namely the scalarcurvature, was fully determined by intrinsic metric properties of thesurface, and was independent of any particular embedding. Riemann[1]took this idea further in his habilitation lecture by describingintrinsic metric geometry of $n$-dimensional space without recourse toan ambient Euclidean setting. The modern notion of manifold, as ageneral setting for geometry involving differential properties evolvedearly in the twentieth century from works of mathematicians such asHermann Weyl [3], who introduced the ideas of an atlas and transitionfunctions, and Elie Cartan, who investigation global properties andgeometric structures on differential manifolds. The modern definitionof 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 ).