manifold

manifold

Summary.

A

manifold

is a space that is
locally like

ℝ

n

, however lacking a preferred system of
coordinates. Furthermore, a manifold can have global topological
properties, such as non-contractible loops (

http://planetmath.org/Curve

), that distinguish it from
the topologically

trivial

ℝ

n

.

Standard Definition.

An

n

-dimensional

topological manifold

M

is a

second countable

,

Hausdorff
topological space

1

1

For

connected

manifolds, the

assumption

that

M

is
second-countable is

logically equivalent

to

M

being

paracompact

, or
equivalently to

M

being

metrizable

. The topological hypotheses in the definition of a manifold are needed
to exclude certain counter-intuitive pathologies. Standard
illustrations of these pathologies are given by the

long line

(lack of paracompactness) and the

forked line

(points cannot be

separated

). These pathologies are fully described in Spivak.
See this page (

http://planetmath.org/BibliographyForDifferentialGeometry

).

that is

locally homeomorphic

to

open subsets

of

ℝ

n

.

A differential manifold is a topological manifold with some additional

structure

information. A

chart

, also known as a

system
of local coordinates

, is a

mapping

α

:

U

→

ℝ

n

, such that the domain

U

⊂

M

is an open set, and such that

U

is

homeomorphic

to the image

α

⁢

(

U

)

. Let

α

:

U

α

→

ℝ

n

,

and

β

:

U

β

→

ℝ

n

be two charts with overlapping
domains (

http://planetmath.org/

Function

). The

continuous

injection

β

∘

α

-

1

:

α

⁢

(

U

α

∩

U

β

)

→

ℝ

n

is called a

transition function

,
and also called a

a change of coordinates

. An

atlas

𝒜

is a

collection

of charts

α

:

U

α

→

ℝ

n

whose domains cover

M

, i.e.

M

=

⋃

α

U

α

.

Note that each transition function
is really just

n

real-valued functions of

n

real variables, and so
we can ask whether these are

continuously differentiable

. The atlas

𝒜

defines a differential structure on

M

, if every transition function
is continuously differentiable.

More generally, for

k

=

1

,

2

,

…

,

∞

,

ω

, the atlas

𝒜

is
said to define a

𝒞

k

differential structure, and

M

is said to be
of class

𝒞

k

, if all the transition functions are

k

-times
continuously differentiable, or

real analytic

in the case of

𝒞

ω

. Two differential structures of class

𝒞

k

on

M

are
said to be

isomorphic

if the union of the corresponding atlases is
also a

𝒞

k

atlas, i.e. if all the new transition functions arising
from the merger of the two atlases remain of class

𝒞

k

. More
generally, two

𝒞

k

manifolds

M

and

N

are said to be

diffeomorphic

, i.e. have

equivalent

differential structure, if there
exists a

homeomorphism

ϕ

:

M

→

N

such that the atlas of

M

is
equivalent to the atlas obtained as

ϕ

-

pullbacks

of charts on

N

.

The atlas allows us to define differentiable mappings to and from a
manifold. Let

f

:

U

→

ℝ

,

U

⊂

M

be a

continuous function

. For each

α

∈

𝒜

we define

f

α

:

V

→

ℝ

,

V

⊂

ℝ

n

,

called the
representation of

f

relative to chart

α

, as the suitably
restricted

composition

f

α

=

f

∘

α

-

1

.

We judge

f

to be

differentiable

if all
the representations

f

α

are differentiable. A path

γ

:

I

→

M

,

I

⊂

ℝ

is judged to be differentiable, if for all differentiable
functions

f

, the suitably restricted composition

f

∘

γ

is a
differentiable function from

ℝ

to

ℝ

. Finally, given
manifolds

M

,

N

, we judge a continuous mapping

ϕ

:

M

→

N

between them to be differentiable if for all differentiable
functions

f

on

N

, the suitably restricted composition

f

∘

ϕ

is a
differentiable function on

M

.

Title

manifold

Canonical name

Manifold

Date of creation

2013-03-22 12:20:22

Last modified on

2013-03-22 12:20:22

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

35

Author

matte (1858)

Entry type

Definition

Classification

msc 53-00

Classification

msc 57R50

Classification

msc 58A05

Classification

msc 58A07

Synonym

differentiable manifold

Synonym

differential manifold

Synonym

smooth manifold

Related topic

NotesOnTheClassicalDefinitionOfAManifold

Related topic

LocallyEuclidean

Related topic

3Manifolds

Related topic

Surface

Related topic

TopologicalManifold

Related topic

ProofOfLagrangeMultiplierMethodOnManifolds

Related topic

Submanifold

Defines

coordinate chart

Defines

chart

Defines

local coordinates

Defines

atlas

Defines

change of coordinates

Defines

differential structure

Defines

transition function

Defines

smooth structure

Defines

diffeomorphism

Defines

diffeomorphic

Defines

topological manifold

Defines

real-analytic manifold

Generated on Fri Feb 9 22:17:55 2018 by

LaTeXML