notes on the classical definition of a manifold

Classical Definition

Historically, the data for a manifold was specified as a collectionMathworldPlanetmath of coordinate domains related by changes of coordinates. The manifold itself could be obtained by gluing the domains in accordance with the transition functionsMathworldPlanetmath, provided the changes of coordinates were free of inconsistencies.

In this formulation, a 𝒞k manifold is specified by two types of information. The first item of information is a collection of open sets


indexed by some set 𝒜. The second item is a collection of transition functions, that is to say 𝒞k diffeomorphisms


obeying certain consistency and topological conditions.

We call a pair


the coordinates of a point relative to chart α, and define the manifold M to be the set of equivalence classesMathworldPlanetmathPlanetmath of such pairs modulo the relationMathworldPlanetmath


To ensure that the above is an equivalence relation we impose the following hypotheses.

  • For α𝒜, the transition function σαα is the identity on Vα.

  • For α,β𝒜 the transition functions σαβ and σβα are inversesPlanetmathPlanetmath.

  • For α,β,γ𝒜 we have for a suitably restricted domain


We topologize M with the least coarse topology that will make the mappings from each Vα to M continuousPlanetmathPlanetmath. Finally, we demand that the resulting topological spaceMathworldPlanetmath be paracompact and HausdorffPlanetmathPlanetmath.

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 2-dimensional plane or 3-dimensional space. Roughly speaking, the concept of a manifold was created in order to treat the intrinsic geometryMathworldPlanetmath of such an object, independent of any embeddingMathworldPlanetmathPlanetmath. 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 EuclideanPlanetmathPlanetmath 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 structuresMathworldPlanetmath on differential manifolds. The modern definition of a manifold was introduced by Hassler Whitney [4] (For more foundational information, follow mfrank/founddiffgeom3.htmlthis link to some old notes by mfrank/Matthew Frank ).


  • 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 geometryMathworldPlanetmath, 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 2013-03-22 14:14:47
Last modified on 2013-03-22 14:14:47
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 11
Author rmilson (146)
Entry type Topic
Classification msc 53-03
Related topic Manifold