aarjaneiro/planetmath huggingface dataset:
transversality

transversality

Transversality

is a fundamental concept in

differential topology

. We say that two

smooth submanifolds

A

,

B

of a

smooth manifold

M

intersect

transversely

, if at any point

x

∈

A

∩

B

, we have

T

x

⁢

A

+

T

x

⁢

B

=

T

x

⁢

X

,

where

T

x

denotes the

tangent space

at

x

, and we naturally identify

T

x

⁢

A

and

T

x

⁢

B

with

subspaces

of

T

x

⁢

X

.

In this case,

A

and

B

intersect properly in the sense that

A

∩

B

is a

submanifold

of

M

, and

codim

⁢

(

A

∩

B

)

=

codim

⁢

(

A

)

+

codim

⁢

(

B

)

.

A useful generalization is obtained if we replace the inclusion

A

↪

M

with a

smooth map

f

:

A

→

M

. In this case we say that

f

is transverse to

B

⊂

M

, if for each point

a

∈

f

-

1

⁢

(

B

)

, we have

d

⁢

f

a

⁢

(

T

a

⁢

A

)

+

T

f

⁢

(

a

)

⁢

B

=

T

f

⁢

(

a

)

⁢

M

.

In this case it turns out, that

f

-

1

⁢

(

B

)

is a submanifold of

A

, and

codim

⁢

(

f

-

1

⁢

(

B

)

)

=

codim

⁢

(

B

)

.

Note that if

B

is a single point

b

, then the condition of

f

being transverse to

B

is precisely that

b

is a

regular value

for

f

. The result is that

f

-

1

⁢

(

b

)

is a submanifold of

A

. A further generalization can be obtained by replacing the inclusion of

B

by a smooth function as well. We leave the details to the reader.

The importance of transversality is that it’s a stable and

generic

condition. This means, in broad terms that if

f

:

A

→

M

is transverse to

B

⊂

M

, then

small perturbations

of

f

are also transverse to

B

. Also, given any smooth map

A

→

M

, it can be perturbed slightly to obtain a smooth map which is transverse to a given submanifold

B

⊂

M

.

Title

transversality

Canonical name

Transversality

Date of creation

2013-03-22 13:29:46

Last modified on

2013-03-22 13:29:46

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Definition

Classification

msc 57R99

Defines

transversal

Defines

transverse

Defines

transversally

Defines

transversely

Generated on Sat Feb 10 11:49:19 2018 by

LaTeXML
isomorphism of the group PSL_2(C) with the group of Möbius transformations

isomorphism of the group PSL_2(C) with the group of Möbius transformations

We identify the group

G

of Möbius

transformations

with the

projective special linear group

P

⁢

S

⁢

L

2

⁢

(

ℂ

)

. The

isomorphism

Ψ

(of topological groups) is given by

Ψ

:

[

(

a

b

c

d

)

]

↦

a

⁢

z

+

b

c

⁢

z

+

d

. (Here, the notation

[

M

]

means the

equivalence class

[

M

]

=

{

M

⁢

t

∣

t

∈

ℂ

}

)

This

mapping

is:

Well-defined:

If

[

(

a

b

c

d

)

]

=

[

(

a

′

b

′

c

′

d

′

)

]

then

(

a

′

,

b

′

,

c

′

,

d

′

)

=

t

⁢

(

a

,

b

,

c

,

d

)

for some

t

, so

z

↦

a

⁢

z

+

b

c

⁢

z

+

d

is the same transformation as

z

↦

a

′

⁢

z

+

b

′

c

′

⁢

z

+

d

′

.

A

homomorphism

:

Calculating the

composition

a

⁢

z

+

b

c

⁢

z

+

d

|

z

=

e

⁢

w

+

f

g

⁢

w

+

h

=

a

⁢

e

⁢

w

+

f

g

⁢

w

+

h

+

b

c

⁢

e

⁢

w

+

f

g

⁢

w

+

h

+

d

=

(

a

⁢

e

+

b

⁢

g

)

⁢

w

+

(

a

⁢

f

+

b

⁢

h

)

(

c

⁢

e

+

d

⁢

g

)

⁢

w

+

(

c

⁢

f

+

d

⁢

h

)

we see that

Ψ

⁢

(

[

(

a

b

c

d

)

]

)

⋅

Ψ

⁢

(

[

(

e

f

g

h

)

]

)

=

Ψ

⁢

(

[

(

a

b

c

d

)

]

⋅

[

(

e

f

g

h

)

]

)

.

A

monomorphism

:

If

Ψ

⁢

(

[

(

a

b

c

d

)

]

)

=

Ψ

⁢

(

[

(

a

′

b

′

c

′

d

′

)

]

)

, then it follows that

(

a

′

,

b

′

,

c

′

,

d

′

)

=

t

⁢

(

a

,

b

,

c

,

d

)

,
so that

[

(

a

b

c

d

)

]

=

[

(

a

′

b

′

c

′

d

′

)

]

.

An

epimorphism

:

Any Möbius transformation

z

↦

a

⁢

z

+

b

c

⁢

z

+

d

is the image

Ψ

⁢

(

[

(

a

b

c

d

)

]

)

.

Title

isomorphism of the group PSL_2(C) with the group of Möbius transformations

Canonical name

IsomorphismOfTheGroupPSL2CWithTheGroupOfMobiusTransformations

Date of creation

2013-03-22 12:43:30

Last modified on

2013-03-22 12:43:30

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Result

Classification

msc 57S25

Generated on Sat Feb 10 11:49:22 2018 by

LaTeXML
Yetter-Drinfel’d module

Yetter-Drinfel’d module

Definition 0.1

.

Let

H

be a

quasi-bialgebra

(

http://planetmath.org/Bialgebra

) with reassociator

Φ

. A left

H

-module

M

together with a left

H

-coaction

λ

M

:

M

→

H

⊗

M

,

λ

M

⁢

(

m

)

=

∑

m

(

Ã

⁢

¢

⁢

Ë

⁢



⁢

â

⁢



⁢



⁢

1

)

⊗

m

0

is called a left

Yetter-Drinfeld

module if the following equalities hold, for all

h

∈

H

and

m

∈

M

:

∑

X

1

m

(

Ã

⁢

¢

⁢

Ë

⁢



⁢

â

⁢



⁢



⁢

1

)

⊗

(

X

2

.

m

(

0

)

)

(

Ã

⁢

¢

⁢

Ë

⁢



⁢

â

⁢



⁢



⁢

1

)

X

3

⊗

(

X

2

.

m

(

0

)

)

0

=

∑

X

1

(

Y

1

×

m

)

(

Ã

⁢

¢

⁢

Ë

⁢



⁢

â

⁢



⁢



⁢

1

)

⁢

1

Y

2

⊗

X

2

×

(

Y

1

x

m

)

(

Ã

⁢

¢

⁢

Ë

⁢



⁢

â

⁢



⁢



⁢

1

)

⁢

2

×

Y

3

⊗

X

3

x

(

Y

1

x

m

)

(

0

)

,

and

∑

ϵ

⁢

(

m

(

Ã

⁢

¢

⁢

Ë

⁢



⁢

â

⁢



⁢



⁢

1

)

)

⁢

m

0

=

m

,

and

∑

h

1

m

(

Ã

⁢

¢

⁢

Ë

⁢



⁢

â

⁢



⁢



⁢

1

)

⊗

h

2

×

m

0

=

∑

(

h

1

.

m

)

(

Ã

⁢

¢

⁢

Ë

⁢



⁢

â

⁢



⁢



⁢

1

)

h

2

⊗

(

h

1

.

m

)

0

.

Remark:

This module (ref.

[

1

]

) is essential for solving the quasi-Yang-Baxter equation which is
an important relation in

Mathematical Physics

.

Drinfel’d modules:

Let us consider a module that operates over a ring of functions on a curve over a

finite field

, which is called an

elliptic module

. Such modules were first studied by Vladimir Drinfel’d in 1973 and called accordingly
Drinfel’d modules.

References

1

Bulacu, D, Caenepeel, S, Torrecillas, B, Doi-Hopf modules and Yetter-Drinfeld modules for quasi-Hopf algebras.

Communications in

Algebra

, 34 (9), pp. 3413-3449, 2006.

2

D. Bulacu, S. Caenepeel, A and F. Panaite. 2003.

http://arxiv.org/PS_cache/math/pdf/0311/0311381v1.pdf

More

Properties

of Yetter-Drinfeld modules over Quasi-Hopf Algebras., Preprint.

Title

Yetter-Drinfel’d module

Canonical

name

YetterDrinfeldModule

Date of creation

2013-03-22 18:24:15

Last modified on

2013-03-22 18:24:15

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

19

Author

bci1 (20947)

Entry type

Definition

Classification

msc 57T05

Classification

msc 13-00

Classification

msc 81R50

Classification

msc 81R15

Classification

msc 46L05

Classification

msc 16W30

Synonym

Drinfel’d module

Synonym

quasi-bialgebra

Related topic

QuantumGroups

Related topic

Module

Related topic

GrassmannHopfAlgebroidCategoriesAndGrassmannCategories

Related topic

GrassmanHopfAlgebrasAndTheirDualCoAlgebras

Related topic

C_cG

Related topic

LocallyCompact

Related topic

LocallyCompactGroupoids

Related topic

WeakHopfCAlgebra2

Related topic

Bialgebra

Related topic

ExampleOfModule2

Defines

H-module

Defines

bialgebras

Generated on Sat Feb 10 11:49:24 2018 by

LaTeXML
formulas for differential forms of small valence

formulas for differential forms of small valence

Coboundary formulas.

Given a function

f

(same thing as
a

differential

0

-form), a differential 1-form

α

and a
differential 2-form

β

, and for vector fields

u

,

v

,

w

, we have

d

⁢

f

⁢

(

u

)

=

u

⁢

(

f

)

,

d

⁢

α

⁢

(

u

,

v

)

=

u

⁢

(

α

⁢

(

v

)

)

-

v

⁢

(

α

⁢

(

u

)

)

-

α

⁢

(

[

u

,

v

]

)

;

d

⁢

β

⁢

(

u

,

v

,

w

)

=

u

⁢

(

β

⁢

(

v

,

w

)

)

+

v

⁢

(

β

⁢

(

w

,

u

)

)

+

w

⁢

(

β

⁢

(

u

,

v

)

)

-

β

⁢

(

[

u

,

v

]

,

w

)

-

β

⁢

(

[

v

,

w

]

,

u

)

-

β

⁢

(

[

w

,

u

]

,

v

)

.

Local coordinate formulas.

Let

f

be a function,

v

=

v

i

⁢

∂

i

a vector field, and

α

=

α

i

⁢

d

⁢

x

i

and

β

=

β

i

⁢

d

⁢

x

i

be 1-forms, and

γ

=

1

2

⁢

γ

i

⁢

j

⁢

d

⁢

x

i

∧

d

⁢

x

j

a

2

-form, expressed relative to a system of

local
coordinates

. The corresponding

interior product

expressions are:

ι

v

⁢

(

α

)

=

v

i

⁢

α

i

,

ι

v

⁢

(

γ

)

=

v

i

⁢

γ

i

⁢

j

⁢

d

⁢

x

j

.

The exterior product formulas are:

α

∧

β

=

α

i

⁢

β

j

⁢

d

⁢

x

i

∧

d

⁢

x

j

=

1

2

⁢

(

α

i

⁢

β

j

-

α

j

⁢

β

i

)

⁢

d

⁢

x

i

∧

d

⁢

x

j

=

∑

i

<

j

(

α

i

⁢

β

j

-

α

j

⁢

β

i

)

⁢

d

⁢

x

i

∧

d

⁢

x

j

;

α

∧

γ

=

1

2

⁢

α

i

⁢

γ

j

⁢

k

⁢

d

⁢

x

i

∧

d

⁢

x

j

∧

d

⁢

x

k

=

1

6

⁢

(

α

i

⁢

γ

j

⁢

k

+

α

j

⁢

γ

k

⁢

i

+

α

k

⁢

γ

i

⁢

j

)

⁢

d

⁢

x

i

∧

d

⁢

x

j

∧

d

⁢

x

k

=

∑

i

<

j

<

k

(

α

i

⁢

γ

j

⁢

k

+

α

j

⁢

γ

k

⁢

i

+

α

k

⁢

γ

i

⁢

j

)

⁢

d

⁢

x

i

∧

d

⁢

x

j

∧

d

⁢

x

k

.

The

exterior derivative

formulas are:

d

⁢

f

=

∂

i

⁡

f

⁢

d

⁢

x

i

,

d

⁢

α

=

∂

i

⁡

α

j

⁢

d

⁢

x

i

∧

d

⁢

x

j

=

1

2

⁢

(

∂

i

⁡

α

j

-

∂

j

⁡

α

i

)

⁢

d

⁢

x

i

∧

d

⁢

x

j

=

∑

i

<

j

(

∂

i

⁡

α

j

-

∂

j

⁡

α

i

)

⁢

d

⁢

x

i

∧

d

⁢

x

j

;

d

⁢

γ

=

1

2

⁢

∂

i

⁡

γ

j

⁢

k

⁢

d

⁢

x

i

∧

d

⁢

x

j

∧

d

⁢

x

k

=

1

6

⁢

(

∂

i

⁡

γ

j

⁢

k

+

∂

j

⁡

γ

k

⁢

i

+

∂

k

⁡

γ

i

⁢

j

)

⁢

d

⁢

x

i

∧

d

⁢

x

j

∧

d

⁢

x

k

=

∑

i

<

j

<

k

(

∂

i

⁡

γ

j

⁢

k

+

∂

j

⁡

γ

k

⁢

i

+

∂

k

⁡

γ

i

⁢

j

)

⁢

d

⁢

x

i

∧

d

⁢

x

j

∧

d

⁢

x

k

.

Title

formulas for differential forms of small valence

Canonical name

FormulasForDifferentialFormsOfSmallValence

Date of creation

2013-03-22 15:13:04

Last modified on

2013-03-22 15:13:04

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

10

Author

rmilson (146)

Entry type

Theorem

Classification

msc 58A10

Generated on Sat Feb 10 11:49:59 2018 by

LaTeXML
volume element

volume element

If

M

is an

n

dimensional

manifold

, then a

differential

n

form (

http://

planetmath

.org/DifferentialForms

) that is never zero is called a

volume element

or a

volume form

. Usually one volume form is associated with the manifold. The volume element is sometimes denoted
by

d

⁢

V

,

ω

or

vol

n

.

If the manifold is a

Riemannian manifold

with

g

,

then the natural volume form is defined in local coordinates

x

1

⁢

…

⁢

x

n

by

d

⁢

V

:=

|

g

|

⁢

d

⁢

x

1

∧

…

∧

d

⁢

x

n

.

It is not hard to show that a manifold has a volume form if and only if it is

orientable

.

If the manifold is

ℝ

n

,

then
the usual volume element

d

⁢

V

=

d

⁢

x

1

∧

d

⁢

x

2

∧

…

∧

d

⁢

x

n

is called the

Euclidean volume element

or

Euclidean volume form

.
In this context,

ℂ

n

is usually treated as

ℝ

2

⁢

n

unless stated otherwise.

When

n

=

2

, then the form is frequently called the

area element

or

area form

and frequently denoted
by

d

⁢

A

. Furthermore, when the manifold is a

submanifold

of

ℝ

3

, then many authors will refer to
a

surface area element

or

surface area form

.

When the context is measure theoretic, this form is sometimes called a

volume measure

,

area measure

,
etc…

References

1

Michael Spivak.
,
W.A. Benjamin, Inc., 1965.

2

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

Title

volume element

Canonical

name

VolumeElement

Date of creation

2013-03-22 17:40:58

Last modified on

2013-03-22 17:40:58

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Definition

Classification

msc 58A10

Classification

msc 53-00

Synonym

volume form

Synonym

volume measure

Defines

area element

Defines

area form

Defines

area measure

Defines

Euclidean volume element

Defines

Euclidean volume form

Defines

euclidean volume measure

Defines

surface area measure

Defines

surface area element

Defines

surface area form

Generated on Sat Feb 10 11:50:17 2018 by

LaTeXML
rectifiable current

rectifiable current

An

m

-dimensional

rectifiable current

is a

current

T

whose action against a

m

-form

ω

can be written as

T

⁢

(

ω

)

=

∫

S

θ

⁢

(

x

)

⁢

⟨

ξ

⁢

(

x

)

,

ω

⁢

(

x

)

⟩

⁢

𝑑

ℋ

m

⁢

(

x

)

where

S

is an

m

-dimensional bounded

rectifiable set

,

ξ

is an orientation of

S

i.e.

ξ

⁢

(

x

)

is a unit

m

-vector representing the approximate tangent plane of

S

at

x

for

ℋ

m

-a.e.

x

∈

S

and, finally,

θ

⁢

(

x

)

is an integer valued

measurable function

defined a.e. on

S

(called

multiplicity

).
The space of

m

-dimensional rectifiable currents is denoted by

ℛ

m

.

An

m

-dimensional rectifiable current

T

such that the boundary

∂

⁡

T

is itself an

(

m

-

1

)

-dimensional rectifiable current, is called

integral current

. The space of integral currents is denoted by

𝐈

m

.
We point out that the word “

integral

” refers to the fact that the multiplicity

θ

is integer valued.

Also notice that rectifiable and integral currents are not vector subspaces of the space of currents. In fact while the sum of two rectifiable currents is again a rectifiable current, the multiplication by a real number gives a rectifiable current only if the number is an integer.

The compactness theorem makes the space of integral currents a good space where geometric problems can be ambiented.

On rectifiable currents one can define an

integral flat norm

ℱ

⁢

(

T

)

:=

inf

⁡

{

𝐌

⁢

(

A

)

+

𝐌

⁢

(

B

)

:

T

=

A

+

∂

⁡

B

,

A

∈

ℛ

m

,

B

∈

ℛ

m

+

1

}

.

The closure of the space

ℛ

m

under the integral flat norm is called the space of

integral flat chains

and is denoted by

ℱ

m

.

As a consequence of the closure theorem, one finds that

ℛ

m

=

{

T

∈

ℱ

m

:

𝐌

⁢

(

T

)

<

∞

}

where

𝐌

is the mass norm of a current.

Title

rectifiable current

Canonical name

RectifiableCurrent

Date of creation

2013-03-22 14:28:34

Last modified on

2013-03-22 14:28:34

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

4

Author

paolini (1187)

Entry type

Definition

Classification

msc 58A25

Defines

integral current

Defines

integral flat norm

Defines

integral flat chains

Generated on Sat Feb 10 11:50:31 2018 by

LaTeXML
conormal bundle

conormal bundle

Let

X

be an immersed

submanifold

of

M

, with

immersion

i

:

X

→

M

. Then as with the

normal bundle

, we can pull the

cotangent bundle

back to

X

, forming a bundle

i

*

⁢

T

*

⁢

M

. This has a

canonical

pairing with

i

*

⁢

T

⁢

M

, essentially by

definition

. Since

T

⁢

X

is a natural

subbundle

of

i

*

⁢

T

⁢

M

, we can consider its

annihilator

: the subbundle of

i

*

⁢

T

*

⁢

M

given by

{

(

x

,

λ

)

|

x

∈

X

,

λ

∈

T

i

⁢

(

x

)

*

⁢

M

,

λ

⁢

(

v

)

=

0

⁢

∀

v

∈

T

x

⁢

X

}

.

This subbundle is denoted

N

*

⁢

X

, and called the

conormal bundle

of

X

.

The conormal bundle to any submanifold is a natural

Lagrangian submanifold

of

T

*

⁢

M

.

Title

conormal bundle

Canonical name

ConormalBundle

Date of creation

2013-03-22 13:59:09

Last modified on

2013-03-22 13:59:09

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Definition

Classification

msc 58A32

Generated on Sat Feb 10 11:50:33 2018 by

LaTeXML
cotangent bundle

cotangent bundle

Overview

Let

M

be a

differentiable manifold

. Analogously to the construction of the

tangent bundle

, we can make the set of covectors on a given

manifold

into a

vector bundle

over

M

, denoted

T

*

⁢

M

and called the

cotangent

bundle of

M

.

Rigorous

Definition

To make this definition precise it is convenient to use the classical definition of a manifold (

http://planetmath.org/NotesOnTheClassicalDefinitionOfAManifold

). Let

M

be an

n

-dimensional differentiable manifold, let

{

V

α

∣

α

∈

𝒜

}

(each

V

α

is an open subset of

ℝ

n

) be an atlas of

M

with

transition functions

σ

α

⁢

β

.

As an atlas for

T

*

⁢

(

M

)

, we may take

{

V

α

×

ℝ

n

∣

α

∈

𝒜

}

. We may construct transition functions

σ

′

α

⁢

β

as follows:

(

σ

′

α

⁢

β

⁢

(

x

1

,

…

,

x

2

⁢

n

)

)

i

=

(

σ

α

⁢

β

⁢

(

x

1

,

…

,

x

n

)

)

i

1

≤

i

≤

n

(

σ

′

α

⁢

β

⁢

(

x

1

,

…

,

x

2

⁢

n

)

)

i

+

n

=

∑

j

=

1

n

∂

(

σ

α

⁢

β

(

x

1

,

…

,

x

n

)

)

i

∂

⁡

x

j

⁢

x

j

+

n

1

≤

i

≤

n

For these to be valid transition functions, they must

satisfy

the three criteria. For a verification that these criteria are satisfied, please see the attachment.

Bundle

Structure

The

cotangent bundle

is a

G

⁢

L

⁢

(

n

)

vector bundle over the manifold

M

. To substantiate this claim, we must specify a

projection map

onto the manifold

M

and

local trivializations

and transition functions and verify that they satisfies the defining properties of a bundle. In terms of the local coordinates used above, it is easy to describe the projection map

π

:

π

⁢

(

x

1

,

…

,

x

2

⁢

n

)

i

=

x

i

The local trivializations are also somewhat

trivial

:

ϕ

α

⁢

(

x

1

,

…

,

x

2

⁢

n

)

=

x

i

+

n

Finally, the transition functions are given as follows:

g

α

⁢

β

⁢

(

x

1

,

…

,

x

2

⁢

n

)

j

i

=

∂

(

σ

α

⁢

β

(

x

1

,

…

x

n

)

)

i

∂

⁡

x

j

For a verification that

(

T

*

⁢

M

,

π

,

ϕ

α

,

g

α

⁢

β

)

satisfies the three criteria for a bundle, please see the attachment.

Properties

The cotangent bundle

T

*

⁢

M

is the vector bundle dual to the tangent bundle

T

⁢

M

. On any differentiable manifold,

T

*

⁢

M

≅

T

⁢

M

(for example, by the

existence

of a

Riemannian metric

), but this identification is by no means canonical, and thus it is useful to distinguish between these two objects.

The cotangent bundle to any manifold has a natural

symplectic structure

given in terms of the Poincaré 1-form, which is in some sense unique. This is not true of the tangent bundle. The existence of a symplectic structure implies that the cotangent bundle is always

orientable

, even if the original manifold is not.

Title

cotangent bundle

Canonical name

CotangentBundle

Date of creation

2013-03-22 13:59:02

Last modified on

2013-03-22 13:59:02

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

17

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 58A32

Generated on Sat Feb 10 11:50:35 2018 by

LaTeXML
cotangent bundle is a bundle

cotangent bundle is a bundle

Verifying the first criterion is simply a matter of writing it out:

(

x

1

,

…

,

x

2

⁢

n

)

↦

(

π

⁢

(

x

1

,

…

,

x

2

⁢

n

)

,

ϕ

α

⁢

(

x

1

,

…

,

x

n

)

)

=

(

(

x

1

,

…

,

x

n

)

,

(

x

n

+

1

,

…

,

x

2

⁢

n

)

)

This is obviously a

homeomorphism

.

As for the second criterion,

∑

j

=

1

n

g

α

⁢

β

j

i

⁢

(

x

1

,

…

,

x

2

⁢

n

)

⁢

ϕ

β

j

⁢

(

x

1

,

…

,

x

2

⁢

n

)

=

∑

j

=

n

2

⁢

n

∂

(

σ

α

⁢

β

(

x

1

,

…

x

n

)

)

i

∂

⁡

x

j

⁢

x

j

+

n

=

σ

′

α

⁢

β

i

+

n

⁢

(

x

1

,

…

,

x

2

⁢

n

)

=

ϕ

α

i

⁢

(

x

1

,

…

,

x

2

⁢

n

)

The third criterion follows from the chain rule:

g

α

⁢

β

j

i

⁢

g

β

⁢

γ

k

j

=

∂

(

σ

α

⁢

β

(

x

1

,

…

x

n

)

)

i

∂

⁡

x

j

⁢

∂

(

σ

β

⁢

γ

(

x

1

,

…

x

n

)

)

j

∂

⁡

x

k

Title

cotangent bundle

is a bundle

Canonical name

CotangentBundleIsABundle

Date of creation

2013-03-22 14:54:31

Last modified on

2013-03-22 14:54:31

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 58A32

Generated on Sat Feb 10 11:50:38 2018 by

LaTeXML
normal bundle

normal bundle

Let

X

be an immersed

submanifold

of

M

, with

immersion

i

:

X

→

M

. Then we can restrict the

tangent bundle

of

M

to

N

or more properly, take the pullback

i

*

⁢

T

⁢

M

. This, as an

vector bundle

over

X

should contain a lot of information about the

embedding

of

X

into

M

. But there is a natural injection

T

⁢

X

→

i

*

⁢

T

⁢

M

, and the

subbundle

which is the image of this only has information on the intrinsic

properties

of

X

, and thus is useless in obtaining information about the embedding of

X

into

M

. Instead, to get information on this, we take the

quotient

i

*

⁢

T

⁢

M

/

T

⁢

X

=

N

⁢

X

, the

normal bundle

of

X

. The normal bundle is very strongly dependent on the immersion

i

. If

E

is any vector bundle on

X

, then

E

is the normal bundle for the embedding of

X

into

E

as the

zero section

.

The normal bundle determines the local

geometry

of the embedding of

X

into

M

in the following sense: In

M

, there exists an open neighborhood

U

⊃

X

which is

diffeomorphic

to

N

⁢

X

by a

diffeomorphism

taking

X

to the zero section.

Title

normal bundle

Canonical

name

NormalBundle

Date of creation

2013-03-22 13:59:06

Last modified on

2013-03-22 13:59:06

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Definition

Classification

msc 58A32

Generated on Sat Feb 10 11:50:40 2018 by

LaTeXML
Poincaré 1-form

Poincaré

1

-form

Definition 1

.

Suppose

M

is a

manifold

, and

T

∗

⁢

M

is its

cotangent bundle

.
Then the ,

α

∈

Ω

1

⁢

(

T

∗

⁢

M

)

, is locally
defined as

α

=

∑

i

=

1

n

y

i

⁢

d

⁢

x

i

where

x

i

,

y

i

are

canonical

local coordinates for

T

∗

⁢

M

.

Let us show that the Poincaré

1

-form is globally defined. That is,

α

has the same

expression

in all local coordinates. Suppose

x

i

,

x

~

i

are overlapping coordinates for

M

. Then we have
overlapping local coordinates

(

x

i

,

y

i

)

,

(

x

~

i

,

y

~

i

)

for

T

∗

⁢

M

with the transformation rule

y

~

i

=

∂

⁡

x

~

j

∂

⁡

x

i

⁢

y

j

.

Hence

∑

i

=

1

n

y

~

i

⁢

d

⁢

x

~

i

=

∑

i

=

1

n

y

~

i

⁢

∂

⁡

x

~

i

∂

⁡

x

k

⁢

d

⁢

x

k

=

∑

i

=

1

n

∂

⁡

x

~

j

∂

⁡

x

i

⁢

y

j

⁢

∂

⁡

x

~

i

∂

⁡

x

k

⁢

d

⁢

x

k

=

∑

k

=

1

n

y

k

⁢

d

⁢

x

k

.

Properties

1.

The Poincaré

1

-form play a crucial role in

symplectic geometry

.
The form

d

⁢

α

is the canonical

symplectic form

for

T

∗

⁢

M

.

2.

Suppose

π

:

T

∗

⁢

M

→

M

is the canonical projection.
Then

α

⁢

(

w

)

=

ξ

⁢

(

(

D

⁢

π

)

⁢

(

w

)

)

,

w

∈

T

ξ

⁢

(

T

∗

⁢

M

)

,

which is an alternative

definition

of

α

without local coordinates.

3.

The restriction of this form to the unit cotangent bundle, is a

contact form

.

Title

Poincaré

1

-form

Canonical name

Poincare1form

Date of creation

2013-03-22 14:45:44

Last modified on

2013-03-22 14:45:44

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

7

Author

matte (1858)

Entry type

Definition

Classification

msc 58A32

Synonym

Liouville one-form

Generated on Sat Feb 10 11:50:42 2018 by

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