notes on the classical definition of a manifold

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

𝒞

k

manifold is specified by two types of
information. The first item of information is a collection of

open
sets

V

α

⊂

ℝ

n

,

α

∈

𝒜

,

indexed by

some set

𝒜

. The second item is a collection of transition functions, that
is to say

𝒞

k

diffeomorphisms

σ

α

⁢

β

:

V

α

⁢

β

→

ℝ

n

,

V

α

⁢

β

⊂

V

α

,

open

,

α

,

β

∈

𝒜

,

obeying certain

consistency

and topological conditions.

We call a pair

(

α

,

x

)

,

α

∈

𝒜

,

x

∈

V

α

the coordinates of
a point relative to chart

α

, and define the manifold

M

to be the set of

equivalence classes

of such pairs modulo the

relation

(

α

,

x

)

≃

(

β

,

σ

α

⁢

β

⁢

(

x

)

)

.

To ensure that the above is an

equivalence relation

we impose the following hypotheses.

•

For

α

∈

𝒜

, the transition function

σ

α

⁢

α

is the

identity

on

V

α

.

•

For

α

,

β

∈

𝒜

the transition functions

σ

α

⁢

β

and

σ

β

⁢

α

are

inverses

.

•

For

α

,

β

,

γ

∈

𝒜

we have for a suitably
restricted domain

σ

β

⁢

γ

∘

σ

α

⁢

β

=

σ

α

⁢

γ

We topologize

M

with the least

coarse topology

that will make
the

mappings

from each

V

α

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 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

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.html

this 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

Generated on Fri Feb 9 22:18:37 2018 by

LaTeXML