aarjaneiro/planetmath huggingface dataset:
Brouwer degree

Brouwer degree

Suppose that

M

and

N

are two

oriented

differentiable manifolds

of

dimension

n

(without boundary) with

M

compact and

N

connected and suppose that

f

:

M

→

N

is a

differentiable mapping

. Let

D

⁢

f

⁢

(

x

)

denote the

differential

mapping at the point

x

∈

M

,
that is the linear mapping

D

⁢

f

⁢

(

x

)

:

T

x

⁢

(

M

)

→

T

f

⁢

(

x

)

⁢

(

N

)

. Let

sign

⁡

D

⁢

f

⁢

(

x

)

denote the sign
of the determinant of

D

⁢

f

⁢

(

x

)

. That is the sign is

positive

if

f

preserves

orientation and negative if

f

reverses orientation.

Definition

.

Let

y

∈

N

be a

regular value

, then we define the

Brower degree

(or just
degree) of

f

by

deg

⁡

f

:=

∑

x

∈

f

-

1

⁢

(

y

)

sign

⁡

D

⁢

f

⁢

(

x

)

.

It can be shown that the degree does not depend on the regular value

y

that we pick so that

deg

⁡

f

is well defined.

Note that this degree coincides with the degree (

http://planetmath.org/Degree5

) as defined for maps of spheres.

References

1

John W. Milnor.
.
The University Press of Virginia, Charlottesville, Virginia, 1969.

Title

Brouwer degree

Canonical

name

BrouwerDegree

Date of creation

2013-03-22 14:52:37

Last modified on

2013-03-22 14:52:37

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 57R35

Synonym

degree

Related topic

DegreeMod2OfAMapping

Generated on Sat Feb 10 11:48:42 2018 by

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degree mod 2 of a mapping

degree mod 2 of a mapping

Suppose that

M

and

N

are two

differentiable manifolds

of dimension

n

(without boundary) with

M

compact and

N

connected and
suppose that

f

:

M

→

N

is a

differentiable mapping

. If

y

∈

N

is a

regular
value

of

f

, then we denote by

#

⁢

f

-

1

⁢

(

y

)

the number of points in

M

that map to

y

.

Definition

.

Let

y

∈

N

be a regular value, then we define the

degree mod 2

of

f

by

deg

2

⁡

f

:=

#

⁢

f

-

1

⁢

(

y

)

(

mod

2

)

.

It can be shown that the degree mod 2 does not depend on the regular value

y

that we pick so that

deg

2

⁡

f

is well defined.

This is similar to the

Brouwer degree

but does not require

oriented manifolds

. In fact

deg

2

⁡

f

=

deg

⁡

f

(

mod

2

)

.

References

1

John W. Milnor.
.
The University Press of Virginia, Charlottesville, Virginia, 1969.

Title

degree mod 2 of a mapping

Canonical name

DegreeMod2OfAMapping

Date of creation

2013-03-22 14:52:39

Last modified on

2013-03-22 14:52:39

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Definition

Classification

msc 57R35

Synonym

degree mod 2

Synonym

degree modulo 2

Related topic

BrouwerDegree

Generated on Sat Feb 10 11:48:44 2018 by

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differentiable functions are continuous

differentiable functions are continuous

Proposition 1

.

Suppose

I

is an

open interval

on

R

,
and

f

:

I

→

C

is

differentiable

at

x

∈

I

. Then

f

is

continuous at

x

. Further, if

f

is differentiable on

I

,
then

f

is

continuous

on

I

.

Proof.

Suppose

x

∈

I

. Let us show that

f

⁢

(

y

)

→

f

⁢

(

x

)

, when

y

→

x

. First, if

y

∈

I

is distinct to

x

,
then

f

⁢

(

x

)

-

f

⁢

(

y

)

=

f

⁢

(

x

)

-

f

⁢

(

y

)

x

-

y

⁢

(

x

-

y

)

.

Thus, if

f

′

⁢

(

x

)

is the

derivative

of

f

at

x

, we have

lim

y

→

x

⁡

f

⁢

(

x

)

-

f

⁢

(

y

)

=

lim

y

→

x

⁡

f

⁢

(

x

)

-

f

⁢

(

y

)

x

-

y

⁢

(

x

-

y

)

=

lim

y

→

x

⁡

f

⁢

(

x

)

-

f

⁢

(

y

)

x

-

y

⁢

lim

y

→

x

⁡

(

x

-

y

)

=

f

′

⁢

(

x

)

⁢

0

=

0

,

where the second equality is justified since both limits on the second line
exist. The second claim follows since

f

is continuous on

I

if and only
if

f

is continuous at

x

for all

x

∈

I

.
∎

Title

differentiable functions are continuous

Canonical name

DifferentiableFunctionsAreContinuous

Date of creation

2013-03-22 14:35:27

Last modified on

2013-03-22 14:35:27

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

8

Author

matte (1858)

Entry type

Theorem

Classification

msc 57R35

Classification

msc 26A24

Related topic

DifferentiableFunctionsAreContinuous2

Related topic

LimitsOfNaturalLogarithm

Generated on Sat Feb 10 11:48:46 2018 by

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example of differentiable function which is not continuously differentiable

example of differentiable function which is not continuously differentiable

Let

f

be defined in the following way:

f

⁢

(

x

)

=

{

x

2

⁢

sin

⁡

(

1

x

)

if

⁢

x

≠

0

0

if

⁢

x

=

0

.

Then if

x

≠

0

,

f

′

⁢

(

x

)

=

2

⁢

x

⁢

sin

⁡

(

1

x

)

-

cos

⁡

(

1

x

)

using the usual rules for calculating

derivatives

. If

x

=

0

, we must compute the derivative by evaluating the limit

lim

ϵ

→

0

⁡

f

⁢

(

ϵ

)

-

f

⁢

(

0

)

ϵ

which we can simplify to

lim

ϵ

→

0

⁡

ϵ

⁢

sin

⁡

(

1

ϵ

)

.

We know

|

sin

⁡

(

x

)

|

≤

1

for every

x

, so this limit is just

0

. Combining this with our previous calculation, we see that

f

′

⁢

(

x

)

=

{

2

⁢

x

⁢

sin

⁡

(

1

x

)

-

cos

⁡

(

1

x

)

if

⁢

x

≠

0

0

if

⁢

x

=

0

.

This is just a slightly modified version of the topologist’s sine curve; in particular,

lim

x

→

0

⁡

f

′

⁢

(

x

)

diverges

, so that

f

′

⁢

(

x

)

is not

continuous

, even though it is defined for every

real number

. Put another way,

f

is

differentiable

but not

C

1

.

Title

example of differentiable function which is not continuously differentiable

Canonical name

ExampleOfDifferentiableFunctionWhichIsNotContinuouslyDifferentiable

Date of creation

2013-03-22 14:10:18

Last modified on

2013-03-22 14:10:18

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

8

Author

Koro (127)

Entry type

Example

Classification

msc 57R35

Classification

msc 26A24

Generated on Sat Feb 10 11:48:48 2018 by

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function differentiable at only one point

function differentiable at only one point

Let

f

:

ℝ

→

ℝ

be the function

f

⁢

(

x

)

=

{

x

,

when

x

is rational

,

-

x

,

when

x

is irrational

.

See this entry (

http://

planetmath

.org/FunctionContinuousAtOnlyOnePoint

).
Let

g

:

ℝ

→

ℝ

be the function

g

⁢

(

x

)

=

f

⁢

(

x

)

⁢

x

.

Then

g

differentiable

at

0

,
but everywhere else non-differentiable.

Indeed, since

g

′

⁢

(

0

)

=

lim

h

→

0

⁡

f

⁢

(

h

)

⁢

h

-

f

⁢

(

0

)

⁢

0

h

=

lim

h

→

0

⁡

f

⁢

(

h

)

=

0

g

is differentiable at

0

.
If

g

would be

continuous at

x

≠

0

, then

f

⁢

(

x

)

=

g

⁢

(

x

)

/

x

would be continuous at

x

.
This result (

http://planetmath.org/DifferentiableFunctionsAreContinuous

)
implies that

g

is non-differentiable away from the origin.

Title

function differentiable at only one point

Canonical name

FunctionDifferentiableAtOnlyOnePoint

Date of creation

2013-03-22 15:48:16

Last modified on

2013-03-22 15:48:16

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

6

Author

matte (1858)

Entry type

Example

Classification

msc 57R35

Classification

msc 26A24

Related topic

FunctionContinuousAtOnlyOnePoint

Generated on Sat Feb 10 11:48:50 2018 by

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Hopf theorem

Hopf theorem

In the following we will assume that the term “smooth” implies just

C

1

(once

continuously differentiable

). By smooth homotopy we will that the homotopy mapping is itself continuously differentiable

Theorem

.

Suppose that

M

is a connected,

oriented

(

http://planetmath.org/Orientation2

)

smooth manifold

without boundary of dimension

m

and suppose

f

,

g

:

M

→

S

m

are smooth mappings to the

m

-sphere. Then

f

and

g

are smoothly homotopic if and only if

f

and

g

have the same

Brouwer degree

.

When

M

is not orientable, then we can always “flip” the orientation by following a closed loop on the

manifold

and one can then prove the following
result.

Theorem

.

Suppose that

M

is not orientable, connected smooth manifold without boundary
of dimension

m

, and suppose

f

,

g

:

M

→

S

m

are smooth mappings to
the

m

-sphere.
Then

f

and

g

are smoothly homotopic if and only if

f

and

g

have the
same

degree mod 2

.

References

1

John W. Milnor.
.
The University Press of Virginia, Charlottesville, Virginia, 1969.

Title

Hopf theorem

Canonical name

HopfTheorem

Date of creation

2013-03-22 14:52:34

Last modified on

2013-03-22 14:52:34

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

8

Author

jirka (4157)

Entry type

Theorem

Classification

msc 57R35

Generated on Sat Feb 10 11:48:52 2018 by

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Sard’s theorem

Sard’s theorem

Theorem

.

Let

U

be an open set in

R

m

,

V

be an open set in

R

n

and

f

:

U

→

V

be a

C

k

function

. If

k

≥

m

/

n

, then the set of
critical values of

f

has

n

-dimensional

measure zero

.

References

1

S.G. Krantz, H. R. Parks,

The

Implicit Function Theorem

:
History, Theory, and Applications

, Birkhäuser, Boston, c. 2002

2

H. Federer,

Geometric Measure Theory

, Springer-Verlag,
New York, 1969

Title

Sard’s theorem

Canonical name

SardsTheorem

Date of creation

2013-03-22 16:12:33

Last modified on

2013-03-22 16:12:33

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 57R35

Generated on Sat Feb 10 11:48:53 2018 by

LaTeXML
Sard’s theorem

Sard’s theorem

Let

ϕ

:

X

n

→

Y

m

be a

smooth map

on

smooth manifolds

. A

critical point

of

ϕ

is a point

p

∈

X

such that the

differential

ϕ

*

:

T

p

⁢

X

→

T

ϕ

⁢

(

p

)

⁢

Y

considered as a

linear transformation

of

real vector spaces

has rank (

http://

planetmath

.org/RankLinearMapping

)

<

m

. A

critical value

of

ϕ

is the image of a critical point. A

regular value

of

ϕ

is a point

q

∈

Y

which is not the image of any critical point. In particular,

q

is a regular value of

ϕ

if

q

∈

Y

∖

ϕ

⁢

(

X

)

.

Following Spivak

[

Spivak

]

, we say a subset

V

of

Y

m

has measure zero

if there is a sequence of

coordinate charts

(

x

i

,

U

i

)

whose union contains

V

and such that

x

i

⁢

(

U

i

∩

V

)

has measure 0 (in the usual sense) in

ℝ

m

for all

i

. With that definition, we can now state:

Sard’s Theorem.

Let

ϕ

:

X

→

Y

be a smooth map on smooth manifolds. Then the set of critical values of

ϕ

has measure zero.

References

Spivak

Spivak, Michael.

A Comprehensive Introduction to

Differential Geometry

.

Volume I, Third Edition. Publish of Perish, Inc. Houston, Texas. 1999.

Title

Sard’s theorem

Canonical name

SardsTheorem

Date of creation

2013-03-22 13:04:09

Last modified on

2013-03-22 13:04:09

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 57R35

Related topic

Residual

Related topic

BaireCategoryTheorem

Defines

critical point

Defines

critical value

Defines

regular value

Generated on Sat Feb 10 11:48:55 2018 by

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abstract embedding

abstract embedding

From the most general -categorical- point of view, an

embedding

between two objects

A

,

B

in a

category

𝒞

is a

𝒞

-

morphism

f

:

A

→

B

which is injective. So, for example, the adjectives topological, algebraic or geometrical should be used, respectively, when we talk of embeddings between

topological spaces

,

algebraic structures

or geometries.

Title

abstract embedding

Canonical name

AbstractEmbedding

Date of creation

2013-03-22 16:30:15

Last modified on

2013-03-22 16:30:15

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

8

Author

juanman (12619)

Entry type

Definition

Classification

msc 57R40

Synonym

embedding

Generated on Sat Feb 10 11:48:57 2018 by

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embedding

embedding

Let

M

and

N

be

manifolds

and

f

:

M

→

N

a

smooth map

. Then

f

is an

embedding

if

1.

f

⁢

(

M

)

is a

submanifold

of

N

, and

2.

f

:

M

→

f

⁢

(

M

)

is a

diffeomorphism

. (There’s an abuse of notation here. This should really be restated as the map

g

:

M

→

f

⁢

(

M

)

defined by

g

⁢

(

p

)

=

f

⁢

(

p

)

is a diffeomorphism.)

The above

characterization

can be equivalently stated:

f

:

M

→

N

is an embedding if

1.

f

is an

immersion

, and

2.

by abuse of notation,

f

:

M

→

f

⁢

(

M

)

is a

homeomorphism

.

Remark

. A celebrated

theorem

of Whitney states that every

n

dimensional manifold admits an embedding into

ℝ

2

⁢

n

+

1

.

Title

embedding

Canonical

name

Embedding

Date of creation

2013-03-22 14:52:46

Last modified on

2013-03-22 14:52:46

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 57R40

Synonym

differential embedding

Defines

Whitney’s theorem

Generated on Sat Feb 10 11:48:59 2018 by

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immersion

immersion

Let

X

and

Y

be

manifolds

, and let

f

be a mapping

f

:

X

→

Y

. Choose

x

∈

X

, and let

y

=

f

⁢

(

x

)

. Recall that

d

⁢

f

x

:

T

x

⁢

(

X

)

→

T

y

⁢

(

Y

)

is the

derivative

of

f

at

x

, and

T

z

⁢

(

Z

)

is the

tangent space

of manifold

Z

at point

z

.

If

d

⁢

f

x

is injective, then

f

is said to be an

immersion

at x

. If

f

is an immersion at every point, it is called an

immersion

.

If the image of

f

is also closed, then

f

is called a

closed immersion

.

The notion of closed immersion (

http://planetmath.org/ClosedImmersion

) for schemes is the analog of this notion in

algebraic geometry

.

Title

immersion

Canonical

name

Immersion

Date of creation

2013-03-22 12:35:04

Last modified on

2013-03-22 12:35:04

Owner

bshanks (153)

Last modified by

bshanks (153)

Numerical id

8

Author

bshanks (153)

Entry type

Definition

Classification

msc 57R42

Related topic

Submersion

Defines

closed immersion

Generated on Sat Feb 10 11:49:00 2018 by

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diffeotopy

diffeotopy

Let

M

be a

manifold

and

I

=

[

0

,

1

]

the closed

unit interval

. A

smooth map

h

:

M

×

I

→

M

is called a

diffeotopy

(on

M

) if for every

t

∈

I

:

h

t

:=

h

⁢

(

-

,

t

)

:

M

→

M

is a

diffeomorphism

.

Two diffeomorphisms

f

,

g

:

M

→

M

are said to be

diffeotopic

if there is a diffeotopy

h

:

M

×

I

→

M

such that

1.

h

0

=

f

, and

2.

h

1

=

g

.

Remark

. Diffeotopy is an

equivalence relation

among diffeomorphisms. In particular, those diffeomorphisms that are diffeotopic to the

identity map

form a group.

Two points

a

,

b

∈

M

are said to be

isotopic

if there is a diffeotopy

h

on

M

such that

1.

h

0

=

i

⁢

d

M

, the identity map on

M

, and

2.

h

1

⁢

(

a

)

=

b

.

Remark

. If

M

is a connected manifold, then every pair of points on

M

are isotopic.

Pairs of isotopic points in a manifold can be generazlied to pairs of isotopic sets. Two arbitrary sets

A

,

B

⊆

M

are said to be

isotopic

if there is a diffeotopy

h

on

M

such that

1.

h

0

=

i

⁢

d

M

, and

2.

h

1

⁢

(

A

)

=

B

.

Remark

. One special example of isotopic sets is the

isotopy

of curves. In

ℝ

3

, curves that are isotopic to the

unit circle

are the

trivial knots

.

Title

diffeotopy

Canonical

name

Diffeotopy

Date of creation

2013-03-22 14:52:43

Last modified on

2013-03-22 14:52:43

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 57R50

Defines

isotopic

Defines

diffeotopic

Generated on Sat Feb 10 11:49:02 2018 by

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examples of mapping class group

examples of mapping class group

An example of this concept is to take the 2-sphere

S

2

, then one can calculate that

ℳ

⁢

(

S

2

)

=

1

,

but

ℳ

*

⁢

(

S

2

)

=

ℤ

2

.

For the genus one

orientable surface

, i.e. the torus

T

=

S

1

×

S

1

, it is known that its (extended)

mapping class group

ℳ

*

⁢

(

T

)

=

G

⁢

L

2

⁢

(

ℤ

)

,

but usually by the (non-extended) mapping class group, that is, the group of

isotopy

classes of

homeomorphisms

that preserve

orientations

(the Dehn’s twists) is just

ℳ

⁢

(

T

)

=

S

⁢

L

2

⁢

(

ℤ

)

.

In these two examples we see that

ℳ

*

is an

extension

of

ℳ

by

ℤ

2

, trivial for the 2-sphere and non trivial for the torus.

For the

projective plane

ℝ

⁢

P

2

we have

ℳ

⁢

(

ℝ

⁢

P

2

)

=

ℳ

*

⁢

(

ℝ

⁢

P

2

)

=

1

And what about the

Klein bottle

?

ℳ

⁢

(

K

)

=

ℤ

2

ℳ

*

⁢

(

K

)

=

ℤ

2

⊕

ℤ

2

Title

examples of mapping class group

Canonical name

ExamplesOfMappingClassGroup

Date of creation

2013-03-22 15:41:19

Last modified on

2013-03-22 15:41:19

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

8

Author

juanman (12619)

Entry type

Example

Classification

msc 57R50

Synonym

first homeotopy group

Related topic

isotopy

Related topic

group

Related topic

Group

Related topic

Isotopy

Generated on Sat Feb 10 11:49:04 2018 by

LaTeXML
mapping class group

mapping class group

Since the set of

diffeomorphisms

of a

manifold

which are

diffeotopic

(

http://planetmath.org/Diffeotopy

) to the

identity

is a

normal subgroup

of the group of all diffeomorphisms, we can form the

quotient

of the diffeomorphism group by this

subgroup

. This quotient is known as the

mapping class group

.

Title

mapping class group

Canonical name

MappingClassGroup

Date of creation

2013-03-22 14:52:51

Last modified on

2013-03-22 14:52:51

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 57R50

Generated on Sat Feb 10 11:49:06 2018 by

LaTeXML
isotopy

isotopy

Let

M

and

N

be manifolds and

I

=

[

0

,

1

]

the closed unit interval. A smooth map

h

:

M

×

I

→

N

is called an

isotopy

if the

restriction

map

h

t

:=

h

⁢

(

-

,

t

)

:

M

→

N

is an

embedding

for all

t

∈

I

.

In particular, a diffeotopy is an isotopy.

Remark

. Given an isotopy

h

:

M

×

I

→

N

, there exists a diffeotopy

g

:

N

×

I

→

N

such that

h

t

=

g

t

∘

h

0

.

Title

isotopy

Canonical

name

Isotopy

Date of creation

2013-03-22 14:52:49

Last modified on

2013-03-22 14:52:49

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 57R52

Related topic

ExampleOfMappingClassGroup

Related topic

Homeotopy

Generated on Sat Feb 10 11:49:08 2018 by

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homology sphere

homology sphere

A

compact

n

-manifold

M

is called a

homology sphere

if its

homology

is that of the

n

-sphere

S

n

, i.e.

H

0

⁢

(

M

;

ℤ

)

≅

H

n

⁢

(

M

;

ℤ

)

≅

ℤ

and is zero otherwise.

An application of the Hurewicz theorem and homological

Whitehead theorem

shows that any

simply connected

homology sphere is in fact

homotopy equivalent

to

S

n

, and hence

homeomorphic

to

S

n

for

n

≠

3

, by the higher dimensional

equivalent

of the Poincaré

conjecture

.

The original version of the Poincaré conjecture stated that every 3 dimensional homology sphere was homeomorphic to

S

3

, but Poincaré himself found a counter-example. There are, in fact, a number of interesting 3-dimensional homology spheres.

Title

homology sphere

Canonical

name

HomologySphere

Date of creation

2013-03-22 13:56:10

Last modified on

2013-03-22 13:56:10

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

4

Author

bwebste (988)

Entry type

Definition

Classification

msc 57R60

Generated on Sat Feb 10 11:49:09 2018 by

LaTeXML
Poincare conjecture

Poincare conjecture

Until its proof in 2003,
the Poincaré Conjecture was the central problem of low-dimensional topology.
It is one of the

http://www.claymath.org

Clay Mathematics Institute

http://www.claymath.org/millennium/

Millennium Prize Problems,
and so far the only one to be solved.

Theorem

Every

3-manifold

without

boundary

that is

homotopy equivalent

to the

3

-sphere
is in fact

homeomorphic

to it.

Or, in a more elementary form:

every simply-connected

compact

3

-

manifold

without boundary
is homeomorphic to

S

3

.

The first statement is also true
when

3

is replaced by any other positive integer,
but the 3-dimensional case turned out to be much harder than the other cases.

The Poincaré Conjecture was eventually proved by Grigoriy Perelman,
who gave his proof in a series of preprints posted on arXiv.org in 2002 and 2003.
For this work Perelman was offered a Fields Medal in 2006, though he declined it.
On 18 March 2010, he was awarded the $1,000,000 Millennium Prize
for resolution of the Poincaré Conjecture, but has also declined this prize.
(The long delay in the awarding of the Millennium Prize
was largely due to the way that Perelman chose to publish his results.
Full details of his proof did not appear
in a peer reviewed mathematical publication until 2006,
and the Millennium Prize rules require a waiting period of two years
after such publication before the prize can be awarded.)

See also

http://www.claymath.org/poincare/millenniumPrizeFull.pdf

Clay Mathematics Institute press release on the Millennium Prize for the Poincaré Conjecture (PDF file),
and the

http://www.claymath.org/millennium/Poincare_Conjecture/perelman+expositions.php

Clay Mathematics Institute’s page on Perelman’s work.

Title

Poincare conjecture

Canonical name

PoincareConjecture

Date of creation

2013-03-22 13:56:16

Last modified on

2013-03-22 13:56:16

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

20

Author

yark (2760)

Entry type

Conjecture

Classification

msc 57R60

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

LaTeXML
round function

round function

Let

M

be a manifold. By a

round function

we a

function

M

→

ℝ

whose

critical points

form

connected components

, each of which is

homeomorphic

to the circle

S

1

.

For example, let

M

be the torus. Let

K

=

]

0

,

2

π

[

×

]

0

,

2

π

[

. Then we know that a map

X

:

K

→

ℝ

3

given by

X

⁢

(

θ

,

ϕ

)

=

(

(

2

+

cos

⁡

θ

)

⁢

cos

⁡

ϕ

,

(

2

+

cos

⁡

θ

)

⁢

sin

⁡

ϕ

,

sin

⁡

θ

)

is a parametrization for almost all of

M

. Now, via the projection

π

3

:

ℝ

3

→

ℝ

we get the

restriction

G

=

π

3

|

M

:

M

→

ℝ

whose critical sets are determined by

∇

⁡

G

⁢

(

θ

,

ϕ

)

=

(

∂

⁡

G

∂

⁡

θ

,

∂

⁡

G

∂

⁡

ϕ

)

⁢

(

θ

,

ϕ

)

=

(

0

,

0

)

if and only if

θ

=

π

2

,

3

⁢

π

2

.

These two values for

θ

give the critical set

X

⁢

(

π

2

,

ϕ

)

=

(

2

⁢

cos

⁡

ϕ

,

2

⁢

sin

⁡

ϕ

,

1

)

X

⁢

(

3

⁢

π

2

,

ϕ

)

=

(

2

⁢

cos

⁡

ϕ

,

2

⁢

sin

⁡

ϕ

,

-

1

)

which represent two extremal circles over the torus

M

.

Observe that the

Hessian

for this function is

d

2

⁢

(

G

)

=

(

-

sin

⁡

θ

0

0

0

)

which clearly it reveals itself as of

rank

⁢

(

d

2

⁢

(

G

)

)

=

1

at the tagged circles,
making the critical point degenerate, that is, showing that the critical points are not

isolated

.

Title

round function

Canonical name

RoundFunction

Date of creation

2013-03-22 15:44:12

Last modified on

2013-03-22 15:44:12

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

11

Author

juanman (12619)

Entry type

Definition

Classification

msc 57R70

Synonym

functions with critical loops

Related topic

DifferntiableFunction

Generated on Sat Feb 10 11:49:17 2018 by

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