aarjaneiro/planetmath huggingface dataset:
K-distance set

K-distance set

Let

X

be a set with metric

d

,

Y

⊆

X

, and

L

=

{

d

⁢

(

x

,

y

)

:

x

,

y

∈

Y

,

x

≠

y

}

. If

K

:=

#

⁢

(

L

)

is finite,

Y

is said to be a

K

-

distance

set

.

Y

is called a

maximal

K

-distance set

if and only if for all

x

∈

X

∖

Y

, there exists

y

∈

Y

such that

d

⁢

(

x

,

y

)

∉

L

. That is, if anything is added to

Y

, it is no longer a

K

-distance set.

Y

is called a

spherical

K

-distance set

if and only if

Y

is a

K

-distance set and every element of

Y

is a fixed distance

r

from some element

c

, so

Y

is a subset of the sphere (

http://planetmath.org/SphereMetricSpace

) centered at

c

with radius

r

.

For example, let

X

=

ℝ

2

with

d

=

the box metric:

d

⁢

(

x

,

y

)

=

max

⁡

{

|

x

1

-

y

1

|

,

|

x

2

-

y

2

|

}

with

x

i

,

y

i

components

of

x

,

y

, respectively. Let

Y

=

{

(

0

,

0

)

,

(

1

,

0

)

,

(

2

,

0

)

,

(

0

,

1

)

,

(

1

,

1

)

,

(

2

,

1

)

,

(

0

,

2

)

,

(

1

,

2

)

,

(

2

,

2

)

}

. Then

L

=

{

1

,

2

}

, so

K

=

2

, so

Y

is a 2-distance set.

Note: please do not confuse this

definition

of

K

-distance set with

Δ

K

⁢

(

Y

)

, the

K

-distance set of

Y

.

Title

K-distance set

Canonical

name

KdistanceSet

Date of creation

2013-03-22 14:19:17

Last modified on

2013-03-22 14:19:17

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 52C35

Generated on Fri Feb 9 22:17:01 2018 by

LaTeXML
Sylvester’s theorem

Sylvester’s theorem

For every finite

collection

of non-collinear points in

Euclidean space

, there is a line that

passes through

exactly two of them.

Proof.

Consider all lines passing through two or more points in the collection.
Since not all points lie on the same line, among pairs of points and lines that are non-incident we can find a point

A

and a line

l

such that the

distance

d

⁢

(

A

,

l

)

between them is

minimal

. Suppose the line

l

contained

more than two points. Then at least two of them, say

B

and

C

, would lie on the same side of the

perpendicular

from

p

to

l

. But then either

d

⁢

(

A

⁢

B

,

C

)

or

d

⁢

(

A

⁢

C

,

B

)

would be smaller than the distance

d

⁢

(

A

,

l

)

which contradicts the minimality of

d

⁢

(

A

,

l

)

.
∎

Title

Sylvester’s

theorem

Canonical name

SylvestersTheorem

Date of creation

2013-03-22 13:59:36

Last modified on

2013-03-22 13:59:36

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

5

Author

bbukh (348)

Entry type

Theorem

Classification

msc 52C35

Classification

msc 51M04

Generated on Fri Feb 9 22:17:03 2018 by

LaTeXML
closed differential forms on a simply connected domain

closed differential forms on a simply connected domain

Let

D

⊂

ℝ

2

be an

open set

and let

ω

be a

differential form

defined on

D

.

Theorem 1

If

D

is simply connected and

ω

is a

closed differential form

,
then

ω

is an

exact differential form

.

The proof of this result is a

consequence

of the following useful lemmas.

Lemma 1

Let

ω

be a closed differential form
and suppose that

γ

0

and

γ

1

are two

regular

homotopic curves in

D

(with the same

end points

). Then

∫

γ

0

ω

=

∫

γ

1

ω

.

Lemma 2

Let

ω

be a

continuous

differential form.
If given any two curves

γ

0

,

γ

1

in

D

with the same end-points,
it holds

∫

γ

0

ω

=

∫

γ

1

ω

,

then

ω

is exact.

See the Poincaré Lemma for a

generalization

of this result on

n

-dimensional

manifolds

.

Title

closed differential forms on a simply connected domain

Canonical name

ClosedDifferentialFormsOnASimplyConnectedDomain

Date of creation

2013-03-22 13:32:46

Last modified on

2013-03-22 13:32:46

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

14

Author

paolini (1187)

Entry type

Theorem

Classification

msc 53-00

Related topic

ClosedCurveTheorem

Related topic

PoincareLemma

Generated on Fri Feb 9 22:17:27 2018 by

LaTeXML
commuting vector fields

commuting vector fields

Vector fields

X

,

Y

on a

manifold

are

commuting

at

p

∈

M

if

[

X

,

Y

]

p

=

0

where

[

⋅

,

⋅

]

is the

Lie bracket

.

If

S

is a subset of

M

, then we say that vector fields

X

and

Y

commute on

S

if they commute at every pont of

S

. In the case where

S

=

M

, i.e. when the vector fields commute at every point of the manifold, then we simply say that

X

and

Y

are commute.

A set

V

of vector fields on a manifold is said to be commuting on a set

S

if, for every pair of vector fields

A

∈

V

and

B

∈

V

, it is the case that

A

and

B

commute.

If

S

is an open set and

V

is a set of

commuting vector fields

on

S

, then the cardinality of

V

is not greater than the

dimension

of the manifold and one can find a local coordinate system about any point of

S

for which these vector fields are

coordinate vector

fields.

Title

commuting vector fields

Canonical

name

CommutingVectorFields

Date of creation

2013-03-22 15:22:37

Last modified on

2013-03-22 15:22:37

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

6

Author

matte (1858)

Entry type

Definition

Classification

msc 53-00

Generated on Fri Feb 9 22:17:29 2018 by

LaTeXML
distribution

distribution

In the following we will

C

∞

when we say smooth.

Definition

.

Let

M

be a

smooth manifold

of

dimension

m

. Let

n

≤

m

and for each

x

∈

M

, we assign an

n

-dimensional

subspace

Δ

x

⊂

T

x

⁢

(

M

)

of the

tangent space

in such a way that for a

neighbourhood

N

x

⊂

M

of

x

there exist

n

linearly independent

smooth

vector fields

X

1

,

…

,

X

n

such that for any point

y

∈

N

x

,

X

1

⁢

(

y

)

,

…

,

X

n

⁢

(

y

)

span

Δ

y

. We let

Δ

refer to the

collection

of all the

Δ

x

for all

x

∈

M

and we then call

Δ

a

distribution

of dimension

n

on

M

, or sometimes a

C

∞

n

-plane distribution

on

M

. The set of smooth
vector fields

{

X

1

,

…

,

X

n

}

is called a

local basis

of

Δ

.

Note: The naming is unfortunate here as these distributions have nothing
to do with distributions in the sense of

analysis

(

http://planetmath.org/Distribution

).
However the naming is in wide use.

Definition

.

We say that a distribution

Δ

on

M

is

involutive

if for every point

x

∈

M

there exists a local basis

{

X

1

,

…

,

X

n

}

in a neighbourhood of

x

such that for all

1

≤

i

,

j

≤

n

,

[

X

i

,

X

j

]

(the commutator of two vector fields) is in the span of

{

X

1

,

…

,

X

n

}

. That is, if

[

X

i

,

X

j

]

is a

linear combination

of

{

X

1

,

…

,

X

n

}

.
Normally this is written as

[

Δ

,

Δ

]

⊂

Δ

.

References

1

William M. Boothby.
,
Academic Press, San Diego, California, 2003.

Title

distribution

Canonical name

Distribution1

Date of creation

2013-03-22 14:51:57

Last modified on

2013-03-22 14:51:57

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Definition

Classification

msc 53-00

Synonym

C^∞n-plane distribution

Related topic

FrobeniussTheorem

Defines

involutive

Defines

involutive distribution

Defines

local basis

Generated on Fri Feb 9 22:17:34 2018 by

LaTeXML
exact (differential form)

exact (differential form)

A

differential form

η

is called exact if there exists another form

ξ

such that

d

⁢

ξ

=

η

.
Exact forms are always closed (

http://planetmath.org/ClosedDifferentialForm

), since

d

2

=

0

.

Title

exact (differential form)

Canonical

name

ExactdifferentialForm

Date of creation

2013-03-22 13:12:24

Last modified on

2013-03-22 13:12:24

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

4

Author

bwebste (988)

Entry type

Definition

Classification

msc 53-00

Synonym

exact form

Synonym

exact differential form

Related topic

PoincareLemma

Related topic

TotalDifferential

Generated on Fri Feb 9 22:17:35 2018 by

LaTeXML
exhaustion by compact sets

exhaustion by compact sets

Let

U

be an

open set

in

ℝ

n

(or a

manifold

with countable base).
Then there exists a sequence of

compact sets

K

1

,

K

2

,

…

such that

K

i

⊆

int

⁡

K

i

+

1

,

i

=

1

,

2

,

…

,

U

=

∪

i

=

1

∞

K

i

,

where “

int

” denotes the topological

interior

.
Such a sequence is called an

exhaustion by compact sets

for

U

.

Title

exhaustion by compact sets

Canonical name

ExhaustionByCompactSets

Date of creation

2013-03-22 15:18:13

Last modified on

2013-03-22 15:18:13

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

6

Author

matte (1858)

Entry type

Theorem

Classification

msc 53-00

Related topic

MethodOfExhaustion

Generated on Fri Feb 9 22:17:37 2018 by

LaTeXML
fundamental concepts in differential geometry

fundamental concepts in differential geometry

The following is an index of fundamental

concepts

in differental

geometry

. It only deals with basic concepts which are common to all branches of

differential geometry

. For concepts pertinent to specific branches of differential geometry, please see

concepts in symplectic geometry

and concepts in

Riemannian geometry

0.1

Manifolds

•

manifold

•

smooth manifold

•

manifold with boundary

•

boundary manifold

•

Riemannian manifold

•

chart

•

coordinate

function

•

coordinate map

•

coordinate neighborhood

•

coordinate system

•

submanifold

•

immersion

•

differential structure

•

diffeomorphism

0.2

Vector and Tensor Fields

•

vector field

•

tensor product

•

contraction

•

exterior product

•

differential form

•

exterior derivative

•

tensor field

•

jet

•

spinor field

•

Lie bracket

•

Lie derivative

•

flow

•

integrable

•

Pfaffian system

•

frame fields

•

Lie groups and

algebras

•

fields as

section

on bundles

0.3

Bundles and Connections

•

fibre bundle

•

vector bundle

•

tangent bundle

•

cotangent bundle

•

jet bundle

•

connection

•

affine connection

•

Riemannian metric

•

curvature

•

torsion

•

geodesic

•

homotopic

classification and

characteristic classes

•

space of connections

0.4

Gauge theory

•

non-Abelian

gauge theory

Title

fundamental concepts in differential geometry

Canonical name

FundamentalConceptsInDifferentialGeometry

Date of creation

2013-03-22 14:47:23

Last modified on

2013-03-22 14:47:23

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

15

Author

rspuzio (6075)

Entry type

Topic

Classification

msc 53-00

Generated on Fri Feb 9 22:17:39 2018 by

LaTeXML
integral curve

integral curve

Definition

Suppose

M

is a

smooth manifold

, and

X

is a
smooth vector field on

M

. Then an

integral curve

of

X

through
a point

x

∈

M

is a curve

c

:

I

→

M

, such that

c

′

⁢

(

t

)

=

(

X

∘

c

)

⁢

(

t

)

,

for all

t

in

I

c

⁢

(

0

)

=

x

.

Here

I

⊂

ℝ

is some

open interval

of

0

, and

c

′

⁢

(

t

)

is
the

tangent vector

in

T

c

⁢

(

t

)

⁢

M

represented by the curve.

Suppose

x

i

are local coordinates for

M

,

c

i

are

functions

representing

c

in these local coordinates, and

X

=

X

i

⁢

∂

∂

⁡

x

i

. Then the condition on

c

is

d

⁢

c

i

d

⁢

t

⁢

(

t

)

=

X

i

∘

c

⁢

(

t

)

,

for all

t

.

Title

integral curve

Canonical name

IntegralCurve

Date of creation

2013-03-22 15:16:31

Last modified on

2013-03-22 15:16:31

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

5

Author

matte (1858)

Entry type

Definition

Classification

msc 53-00

Generated on Fri Feb 9 22:17:41 2018 by

LaTeXML
Lie bracket

Lie bracket

The

Lie bracket

is an

anticommutative

,

bilinear

, first order

differential operator

on

vector fields

. It may be defined either in terms of

local coordinates

or in a global, coordinate-free fashion. Though both defintions are prevalent, it is perhaps easier to formulate the Lie Bracket without the use of

coordinates

at all, as a commutator:

Definition

(Global, coordinate-free) Suppose

X

and

Y

are vector fields on a smooth manifold

M

. Regarding these vector fields as

operators

on functions, the Lie bracket is their commutator:

[

X

,

Y

]

⁢

(

f

)

=

X

⁢

(

Y

⁢

(

f

)

)

-

Y

⁢

(

X

⁢

(

f

)

)

.

Definition

(Local coordinates) Suppose

X

and

Y

are vector fields on a smooth

n

-dimensional manifold

M

,
suppose

(

x

1

,

…

,

x

n

)

are local coordinates around some point

x

∈

M

,
and suppose that in these local coordinates

X

⁢

(

x

)

=

X

i

⁢

(

x

)

⁢

∂

∂

⁡

x

i

|

x

,

Y

⁢

(

x

)

=

Y

i

⁢

(

x

)

⁢

∂

∂

⁡

x

i

|

x

.

Then the

Lie bracket

of the above vector fields is the locally defined vector field

[

X

,

Y

]

⁢

(

x

)

=

X

i

⁢

∂

⁡

Y

j

∂

⁡

x

i

⁢

∂

∂

⁡

x

j

|

x

-

Y

i

⁢

∂

⁡

X

j

∂

⁡

x

i

⁢

∂

∂

⁡

x

j

|

x

.

(The

Einstein summation convention

employed in the above equations —
repeated indices are to be summed from the range 1 to

n

.)

Properties

Suppose

X

,

Y

,

Z

are smooth vector fields on a smooth manifold

M

.

•

[

X

,

Y

]

=

ℒ

X

⁢

Y

where

ℒ

X

⁢

Y

is the

Lie derivative

.

•

[

⋅

,

⋅

]

is

anti-symmetric

and bi-linear.

•

Vector fields on

M

with the Lie bracket is a Lie algebra. That is to say, the Lie bracket satisfies the

Jacobi identity

:

[

X

,

[

Y

,

Z

]

]

+

[

Y

,

[

Z

,

X

]

]

+

[

Z

,

[

X

,

Y

]

]

=

0

.

•

The Lie bracket is covariant with respect to changes of coordinates.

Title

Lie bracket

Canonical name

LieBracket

Date of creation

2013-03-22 14:10:02

Last modified on

2013-03-22 14:10:02

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 53-00

Related topic

HamiltonianAlgebroids

Generated on Fri Feb 9 22:17:44 2018 by

LaTeXML
Lie derivative

Lie derivative

Let

M

be a

smooth manifold

,

X

a

vector field

on

M

, and

T

a tensor on

M

. Then the

Lie derivative

ℒ

X

⁢

T

of

T

along

X

is a tensor of the same rank as

T

defined as

ℒ

X

⁢

T

=

d

d

⁢

t

⁢

(

ρ

t

*

⁢

(

T

)

)

|

t

=

0

where

ρ

is the flow of

X

, and

ρ

t

*

is pullback by

ρ

t

.

The Lie derivative is a notion of

directional derivative

for tensors.
Intuitively, this is the change in

T

in the

direction

of

X

.

If

X

and

Y

are vector fields, then

ℒ

X

⁢

Y

=

[

X

,

Y

]

, the standard Lie bracket of vector fields.

Title

Lie derivative

Canonical name

LieDerivative

Date of creation

2013-03-22 13:14:10

Last modified on

2013-03-22 13:14:10

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 53-00

Related topic

LeibnizNotationForVectorFields

Related topic

CartanCalculus

Generated on Fri Feb 9 22:17:46 2018 by

LaTeXML
Lie derivative (for vector fields)

Lie derivative (for vector fields)

Let

M

be a

smooth manifold

, and

X

,

Y

∈

𝒯

⁢

(

M

)

smooth vector fields
on

M

. Let

Θ

:

𝒰

→

M

be the flow of

X

, where

𝒰

⊆

ℝ

×

M

is an open neighborhood of

{

0

}

×

M

. We make use of the following notation:

𝒰

p

=

{

t

∈

ℝ

|

(

t

,

p

)

∈

𝒰

}

,

∀

p

∈

M

,

𝒰

t

=

{

p

∈

M

|

(

t

,

p

)

∈

𝒰

}

,

∀

t

∈

ℝ

,

and we introduce the
auxiliary maps

θ

t

:

𝒰

t

→

M

and

θ

p

:

𝒰

p

→

M

defined as

Θ

⁢

(

t

,

p

)

=

θ

t

⁢

(

p

)

=

θ

p

⁢

(

t

)

,

∀

(

t

,

p

)

∈

𝒰

.

The

Lie derivative

of

Y

along

X

is the vector field

ℒ

X

⁢

Y

∈

𝒯

⁢

(

M

)

defined by

(

ℒ

X

⁢

Y

)

p

=

d

d

⁢

t

⁢

(

d

⁢

(

θ

-

t

)

θ

t

⁢

(

p

)

⁢

(

Y

θ

t

⁢

(

p

)

)

)

|

t

=

0

=

lim

t

→

0

⁡

d

⁢

(

θ

-

t

)

θ

t

⁢

(

p

)

⁢

(

Y

θ

t

⁢

(

p

)

)

-

Y

p

t

,

∀

p

∈

M

,

where

d

⁢

(

θ

-

t

)

θ

t

⁢

(

p

)

∈

Hom

⁢

(

T

θ

t

⁢

(

p

)

⁢

M

,

T

p

⁢

M

)

if the push-forward of

θ

-

t

, i.e.

d

⁢

(

θ

-

t

)

θ

t

⁢

(

p

)

⁢

(

v

)

⁢

(

f

)

=

v

⁢

(

f

∘

θ

-

t

)

,

∀

v

∈

T

θ

-

t

⁢

(

p

)

⁢

M

,

f

∈

C

∞

⁢

(

p

)

.

The following result is not immediate at all.

Theorem 1

ℒ

X

⁢

Y

=

[

X

,

Y

]

, where

[

X

,

Y

]

=

X

⁢

Y

-

Y

⁢

X

is the

Lie bracket

of

X

and

Y

.

Title

Lie derivative (for vector fields)

Canonical

name

LieDerivativeforVectorFields

Date of creation

2013-03-22 14:09:59

Last modified on

2013-03-22 14:09:59

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

9

Author

matte (1858)

Entry type

Definition

Classification

msc 53-00

Defines

Lie derivative

Generated on Fri Feb 9 22:17:48 2018 by

LaTeXML
local dimension of a locally Euclidean space

local dimension of a locally Euclidean space

Let

X

be a locally Euclidean space. Recall that the local

dimension

of

X

in

y

∈

X

is a

natural number

n

∈

ℕ

such that there is an open

neighbourhood

U

⊆

X

of

y

homeomorphic

to

ℝ

n

. This number is well defined (please, see parent object for more details) and we will denote it by

dim

y

⁢

X

.

Proposition

.

Function

f

:

X

→

ℕ

defined by

f

⁢

(

y

)

=

dim

y

⁢

X

is

continuous

(where on

ℕ

we have

discrete topology

).

Proof.

It is enough to show that

preimage

of a point is open. Assume that

n

∈

ℕ

and

y

∈

X

is such that

f

⁢

(

y

)

=

n

. Then there is an open neighbourhood

U

⊆

X

of

y

such that

U

is homeomorphic to

ℝ

n

. Obviously for any

x

∈

U

we have that

U

is an open neighbourhood of

x

homeomorphic to

ℝ

n

. Therefore

f

⁢

(

x

)

=

n

, so

U

⊆

f

-

1

⁢

(

n

)

. Thus (since

y

was arbitrary) we’ve shown that around every point in

f

-

1

⁢

(

n

)

there is an open neighbourhood of that point

contained

in

f

-

1

⁢

(

n

)

. This shows that

f

-

1

⁢

(

n

)

is open, which

completes

the proof.

□

Corollary.

Assume that

X

is a

connected

, locally Euclidean space. Then local dimension is

constant

, i.e. there exists natural number

n

∈

ℕ

such that for any

y

∈

X

we have

dim

y

⁢

X

=

n

.

Proof.

Consider the

mapping

f

:

X

→

ℕ

such that

f

⁢

(

y

)

=

dim

y

⁢

X

. Proposition shows that

f

is continuous. Therefore

f

⁢

(

X

)

is connected, because

X

is. But

ℕ

has discrete topology, so there are no other connected subsets then points. Thus there is

n

∈

ℕ

such that

f

⁢

(

X

)

=

{

n

}

, which completes the proof.

□

Remark.

Generally, local dimension need not be constant. For example consider

X

1

,

X

2

⊆

ℝ

3

such that

X

1

=

{

(

x

,

0

,

0

)

|

x

∈

ℝ

}

X

2

=

{

(

x

,

y

,

1

)

|

x

,

y

∈

ℝ

}

.

One can easily show that

X

=

X

1

∪

X

2

(with

topology

inherited from

ℝ

3

) is locally Euclidean, but

dim

(

0

,

0

,

0

)

⁢

X

=

1

and

dim

(

1

,

1

,

1

)

⁢

X

=

2

.

Title

local dimension of a locally Euclidean space

Canonical

name

LocalDimensionOfALocallyEuclideanSpace

Date of creation

2013-03-22 18:55:34

Last modified on

2013-03-22 18:55:34

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

7

Author

joking (16130)

Entry type

Theorem

Classification

msc 53-00

Generated on Fri Feb 9 22:17:51 2018 by

LaTeXML
locally Euclidean

locally Euclidean

A

locally Euclidean space

X

is a

topological space

that locally
“looks” like

ℝ

n

.
This makes it possible to talk about

coordinate

axes around

X

. It also gives some topological structure
to the space: for example, since

ℝ

n

is

locally compact

, so is

X

.
However, the

restriction

does not induce any

geometry

onto

X

.

Definition

Suppose

X

is a topological space. Then

X

is
called

locally Euclidean

if for each

x

∈

X

there is a

neighbourhood

U

⊆

X

, a

V

⊆

ℝ

n

, and
a

homeomorphism

ϕ

:

U

→

V

. Then the triple

(

U

,

ϕ

,

n

)

is called a

chart

for

X

.

Here,

ℝ

is the set of real numbers, and for

n

=

0

we define

ℝ

0

as set with a single point equipped with the

discrete topology

.

Local dimension

Suppose

X

is a locally Euclidean space with

x

∈

X

. Further,
suppose

(

U

,

ϕ

,

n

)

is a chart of

X

such that

x

∈

U

.
Then we define the

local

of

X

at

x

is

n

.
This is well defined, that is, the local

dimension

does not
depend on the chosen chart. If

(

U

′

,

ϕ

′

,

n

′

)

is another chart with

x

∈

U

′

, then

ψ

∘

ϕ

-

1

:

ϕ

⁢

(

U

∩

U

′

)

→

ψ

⁢

(

U

∩

U

′

)

is a homeomorphism between

ϕ

⁢

(

U

∩

U

′

)

⊆

ℝ

n

and

ψ

⁢

(

U

∩

U

′

)

⊆

ℝ

n

′

. By Brouwer’s

theorem

for the

invariance of dimension

(which is nontrivial),
it follows that

n

=

n

′

.

If the local dimension is constant, say

n

, we say that the dimension
of

X

is

n

, and write

dim

⁡

X

=

n

.

Examples

•

Any set with the discrete topology, is a locally
Euclidean of dimension

0

.

•

Any

open subset

of

ℝ

n

is locally Euclidean.

•

Any

manifold

is locally Euclidean. For example,
using a

stereographic projection

, one can show that the sphere

S

n

is locally Euclidean.

•

The

long line

is locally Euclidean of dimension one. Note that the long line is not Hausforff.

[

1

]

.

Notes

The concept locally Euclidean has a different meaning in the
setting of

Riemannian manifolds

.

References

1

L. Conlon,

Differentiable Manifolds: A first course

,
Birkhäuser, 1993.

Title

locally Euclidean

Canonical

name

LocallyEuclidean

Date of creation

2013-03-22 14:14:49

Last modified on

2013-03-22 14:14:49

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

14

Author

matte (1858)

Entry type

Definition

Classification

msc 53-00

Related topic

Manifold

Related topic

LocallyHomeomorphic

Related topic

EmptyProduct

Defines

locally Euclidean space

Defines

chart

Generated on Fri Feb 9 22:17:53 2018 by

LaTeXML
Poincaré lemma

Poincaré lemma

The Poincaré lemma states that every

closed differential form

is locally exact (

http://planetmath.org/ExactDifferentialForm

).

Theorem

.

(Poincaré Lemma)

[

1

]

Suppose

X

is a

smooth
manifold

,

Ω

k

⁢

(

X

)

is the set of smooth

differential

k

-forms on

X

, and suppose

ω

is a closed form
in

Ω

k

⁢

(

X

)

for some

k

>

0

.

•

Then for every

x

∈

X

there is a neighbourhood

U

⊂

X

, and a

(

k

-

1

)

-form

η

∈

Ω

k

-

1

⁢

(

U

)

, such that

d

⁢

η

=

ι

∗

⁢

ω

,

where

ι

is the inclusion

ι

:

U

↪

X

.

•

If

X

is

contractible

, this

η

exists globally; there exists a

(

k

-

1

)

-form

η

∈

Ω

k

-

1

⁢

(

X

)

such that

d

⁢

η

=

ω

.

Notes

Despite the name, the Poincaré lemma is an
extremely important result. For instance, in

algebraic topology

,
the definition of the

k

th

de Rham cohomology group

H

k

⁢

(

X

)

=

Ker

⁡

{

d

:

Ω

k

⁢

(

X

)

→

Ω

k

+

1

⁢

(

X

)

}

Im

⁡

{

d

:

Ω

k

-

1

⁢

(

X

)

→

Ω

k

⁢

(

X

)

}

can be seen as a measure of the degree in which the Poincaré lemma fails.
If

H

k

⁢

(

X

)

=

0

, then every

k

form is exact, but if

H

k

⁢

(

X

)

is non-zero, then

X

has a non-trivial

topology

(or “holes”) such that

k

-forms are not
globally exact. For instance, in

X

=

ℝ

2

∖

{

0

}

with

polar coordinates

(

r

,

ϕ

)

,
the

1

-form

ω

=

d

⁢

ϕ

is not globally exact.

References

1

L. Conlon,

Differentiable Manifolds: A first course

,
Birkhäuser, 1993.

Title

Poincaré lemma

Canonical name

PoincareLemma

Date of creation

2013-03-22 14:06:28

Last modified on

2013-03-22 14:06:28

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

12

Author

matte (1858)

Entry type

Theorem

Classification

msc 53-00

Related topic

ExactDifferentialForm

Related topic

ClosedDifferentialFormsOnASimpleConnectedDomain

Related topic

LaminarField

Generated on Fri Feb 9 22:17:57 2018 by

LaTeXML