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 manifold is specified by two types of information. The first item of information is a collection of open sets
indexed by some set . The second item is a collection of transition functions, that is to say diffeomorphisms
obeying certain consistency and topological conditions.
We call a pair
the coordinates of
a point relative to chart , and define the manifold
to be the set of equivalence classes of such pairs modulo the relation
To ensure that the above is an equivalence relation we impose the following hypotheses.
-
•
For , the transition function is the identity on .
-
•
For the transition functions and are inverses
.
-
•
For we have for a suitably restricted domain
We topologize with the least coarse topology that will make
the
mappings from each to 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 2-dimensional plane or 3-dimensional 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 -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 | 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 |