aarjaneiro/planetmath huggingface dataset:
formulas in Riemannian geometry

formulas in Riemannian geometry

The aim of this page is to collect
frequently used formulas in

Riemannian geometry

.

Symbol conventions.

•

g

i

⁢

j

: the

components

of the

metric tensor

;

•

Γ

i

⁢

j

⁢

k

=

Γ

j

⁢

i

⁢

k

: the

Christoffel symbols

;

•

X

i

=

X

j

⁢

g

i

⁢

j

, and

Y

i

: rank 1
tensors;

•

T

i

⁢

j

=

T

i

⁢

g

j

⁢

k

k

: a rank 2 tensor;

•

indices

i

,

j

,

k

,

l

and subscripted versions thereof: components
taken with respect to a

local coordinate

frame;

•

x

i

,

y

j

: systems of local coordinates;

•

∂

i

=

∂

∂

⁡

x

i

: local coordinate frame;

•

boldfaced symbols: the actual geometric quantity, rather
than components; e.g.

𝑿

=

X

i

⁢

∂

i

.

Formulas for the covariant derivative.

∂

k

⁡

g

i

⁢

j

=

Γ

k

⁢

i

⁢

j

+

Γ

k

⁢

j

⁢

i

,

∂

k

⁡

g

i

⁢

j

=

-

(

g

j

⁢

b

Γ

b

⁢

k

+

i

g

i

⁢

a

Γ

a

⁢

k

)

j

,

∇

k

⁡

g

i

⁢

j

=

0

,

Γ

i

⁢

j

⁢

k

=

1

2

⁢

(

∂

i

⁡

g

j

⁢

k

+

∂

j

⁡

g

i

⁢

k

-

∂

k

⁡

g

i

⁢

j

)

,

∇

i

⁡

X

j

=

∂

i

⁡

X

j

+

Γ

i

⁢

k

⁢

X

k

j

,

∇

𝑿

⁡

𝒀

=

X

i

⁢

∇

i

⁡

Y

j

⁢

∂

j

,

∇

i

⁡

X

j

=

∂

i

⁡

X

j

-

Γ

i

⁢

j

⁢

X

k

k

,

∇

i

⁡

T

j

⁢

k

=

∂

i

⁡

T

j

⁢

k

-

Γ

i

⁢

j

⁢

T

l

⁢

k

l

-

Γ

i

⁢

k

⁢

T

j

⁢

l

l

,

∇

i

T

j

k

=

∂

i

T

j

+

k

Γ

i

⁢

l

T

l

j

-

k

Γ

i

⁢

k

T

j

l

.

l

Formulas for geodesics

A

geodesic

is a curve

c

:

I

→

M

satisfying

c

¨

+

i

Γ

j

⁢

k

c

˙

j

i

c

˙

k

=

0

Title

formulas in Riemannian geometry

Canonical name

FormulasInRiemannianGeometry

Date of creation

2013-03-22 15:32:55

Last modified on

2013-03-22 15:32:55

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

8

Author

matte (1858)

Entry type

Definition

Classification

msc 53B21

Classification

msc 53B20

Generated on Fri Feb 9 22:27:25 2018 by

LaTeXML
Hodge star operator

Hodge star operator

Let V be a

n

-dimensional (

n

finite)

vector space

with

inner product

g

. The

Hodge star operator

(denoted by

∗

) is
a

linear operator

mapping

http://planetmath.org/node/3050

p

-forms on

V

to

(

n

-

p

)

-forms, i.e.,

∗

:

Ω

p

(

V

)

→

Ω

n

-

p

(

V

)

.

In terms of a basis

{

e

1

,

…

,

e

n

}

for

V

and the corresponding

dual basis

{

e

1

,

…

,

e

n

}

for

V

*

(the star used to denote the

dual space

is not to be confused with the Hodge star!), with the inner product being expressed in terms of

components

as

g

=

∑

i

,

j

=

1

n

g

i

⁢

j

⁢

e

i

⊗

e

j

, the

∗

-

operator

is defined as the linear operator that maps the basis elements of

Ω

p

⁢

(

V

)

as

∗

(

e

i

1

∧

⋯

∧

e

i

p

)

=

|

g

|

(

n

-

p

)

!

⁢

g

i

1

⁢

l

1

⁢

⋯

⁢

g

i

p

⁢

l

p

⁢

ε

l

1

⁢

⋯

⁢

l

p

⁢

l

p

+

1

⁢

⋯

⁢

l

n

⁢

e

l

p

+

1

∧

⋯

∧

e

l

n

.

Here,

|

g

|

=

det

⁡

g

i

⁢

j

, and

ε

is the

Levi-Civita permutation symbol

This operator may be defined in a coordinate-free manner by the condition

u

∧

*

v

=

g

(

u

,

v

)

𝐕𝐨𝐥

(

g

)

where the notation

g

⁢

(

u

,

v

)

denotes the inner product on

p

-forms (in

coordinates

,

g

⁢

(

u

,

v

)

=

g

i

1

⁢

j

1

⁢

⋯

⁢

g

i

p

⁢

j

p

⁢

u

i

1

⁢

…

⁢

i

p

⁢

v

j

1

⁢

…

⁢

j

p

) and

𝐕𝐨𝐥

(

g

)

is the unit

volume form

associated to the metric. (in coordinates,

𝐕𝐨𝐥

(

g

)

=

det

⁡

(

g

)

⁢

e

1

∧

⋯

∧

e

n

)

Generally

∗

∗

=

(

-

1

)

p

⁢

(

n

-

p

)

id

, where

id

is the

identity operator

in

Ω

p

⁢

(

V

)

. In three

dimensions

,

∗

∗

=

id

for all

p

=

0

,

…

,

3

.
On

ℝ

3

with

Cartesian coordinates

, the

metric tensor

is

g

=

d

⁢

x

⊗

d

⁢

x

+

d

⁢

y

⊗

d

⁢

y

+

d

⁢

z

⊗

d

⁢

z

, and the Hodge
star operator is

∗

d

x

=

d

y

∧

d

z

,

∗

d

y

=

d

z

∧

d

x

,

∗

d

z

=

d

x

∧

d

y

.

The Hodge star

operation

occurs most

frequently in

differential geometry

in the case where

M

n

is a

n

-dimensional orientable manifold with
a Riemannian (or pseudo-Riemannian) tensor

g

and

V

is a cotangent vector space of

M

n

. Also, one can extend this notion to

antisymmetric tensor

fields by computing Hodge star

pointwise

.

Title

Hodge star operator

Canonical

name

HodgeStarOperator

Date of creation

2013-03-22 13:31:41

Last modified on

2013-03-22 13:31:41

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 53B21

Synonym

Hodge operator

Synonym

star operator

Defines

hodge star operator

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

LaTeXML
sectional curvature

sectional curvature

Let

M

be a

Riemannian manifold

. Let

p

be a point in

M

and let

S

be a two-dimensional

subspace

of

T

p

⁢

M

. Then the

sectional curvature

of

S

at

p

is defined as

K

⁢

(

S

)

=

g

⁢

(

R

⁢

(

x

,

y

)

⁢

x

,

y

)

g

⁢

(

x

,

x

)

⁢

g

⁢

(

y

,

y

)

-

g

⁢

(

x

,

y

)

2

where

x

,

y

span

S

,

g

is the

metric tensor

and

R

is the Riemann’s curvature tensor.

This is a natural generalization of the classical

Gaussian curvature

for

surfaces

.

Title

sectional curvature

Canonical name

SectionalCurvature

Date of creation

2013-03-22 15:54:15

Last modified on

2013-03-22 15:54:15

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

5

Author

juanman (12619)

Entry type

Definition

Classification

msc 53B21

Classification

msc 53B20

Related topic

RiemannianMetric

Generated on Fri Feb 9 22:27:29 2018 by

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integral manifold

integral manifold

In the following we will

C

∞

when we say smooth.

Definition

.

Let

M

be a

smooth manifold

of

dimension

m

and let

Δ

be a

distribution

of dimension

n

on

M

. Suppose that

N

is a

connected

submanifold

of

M

such that for every

x

∈

N

we have that

T

x

⁢

(

N

)

(the

tangent space

of

N

at

x

) is contained in

Δ

x

(the distribution at

x

). We can abbreviate this by saying that

T

⁢

(

N

)

⊂

Δ

. We then say that

N

is an

integral manifold

of

Δ

.

Do note that

N

could be of lower dimension then

Δ

and is not required to be a regular submanifold of

M

.

Definition

.

We say that a distribution

Δ

of dimension

n

on

M

is

completely integrable

if for
each point

x

∈

M

there exists an integral manifold

N

of

Δ

passing through

x

such that the dimension of

N

is equal to the dimension
of

Δ

.

An example of an integral manifold is the

integral curve

of a non-vanishing vector field and then of course the span of the
vector field is a completely integrable distribution.

References

1

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

Title

integral manifold

Canonical

name

IntegralManifold

Date of creation

2013-03-22 14:52:00

Last modified on

2013-03-22 14:52:00

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Definition

Classification

msc 53B25

Classification

msc 52-00

Classification

msc 37C10

Related topic

FrobeniussTheorem

Defines

completely integrable

Defines

completely integrable distribution

Generated on Fri Feb 9 22:27:44 2018 by

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example of rewriting a differential equation as a Pfaffian system

example of rewriting a differential equation as a Pfaffian system

To show how one may reformulate a

differential equation

as Pfaff’s problem for a set of

differential forms

, consider the

wave equation

∂

2

⁡

u

∂

⁡

t

2

=

∂

2

⁡

u

∂

⁡

x

2

+

∂

2

⁡

u

∂

⁡

y

2

The first step is to rewrite the equation as a system of first-order equations

∂

⁡

a

∂

⁡

t

-

∂

⁡

b

∂

⁡

x

-

∂

⁡

c

∂

⁡

y

=

0

∂

⁡

u

∂

⁡

t

-

a

=

0

∂

⁡

u

∂

⁡

x

-

b

=

0

∂

⁡

u

∂

⁡

y

-

c

=

0

To

translate

these equations into the language of differential forms, we shall use the fact that

d

⁢

u

=

∂

⁡

u

∂

⁡

t

⁢

d

⁢

t

+

∂

⁡

u

∂

⁡

x

⁢

d

⁢

x

+

∂

⁡

u

∂

⁡

y

⁢

d

⁢

y

from which it follows that

d

⁢

u

∧

d

⁢

x

∧

d

⁢

y

=

∂

⁡

u

∂

⁡

t

⁢

d

⁢

t

∧

d

⁢

x

∧

d

⁢

y

d

⁢

u

∧

d

⁢

y

∧

d

⁢

t

=

∂

⁡

u

∂

⁡

x

⁢

d

⁢

t

∧

d

⁢

x

∧

d

⁢

y

d

⁢

u

∧

d

⁢

t

∧

d

⁢

x

=

∂

⁡

u

∂

⁡

y

⁢

d

⁢

t

∧

d

⁢

x

∧

d

⁢

y

We can do likewise with

a

or

b

or

c

in the place of

u

; there is no point in repeating the formulas for each of these variables.

Multiplying the differential equations through by the form

d

⁢

t

∧

d

⁢

x

∧

d

⁢

y

and using the above identities to eliminate

partial derivatives

, we obtain the following system of differential forms:

d

⁢

a

∧

d

⁢

x

∧

d

⁢

y

-

d

⁢

b

∧

d

⁢

y

∧

d

⁢

t

-

d

⁢

c

∧

d

⁢

t

∧

d

⁢

x

d

⁢

u

∧

d

⁢

x

∧

d

⁢

y

-

a

⁢

d

⁢

t

∧

d

⁢

x

∧

d

⁢

y

d

⁢

u

∧

d

⁢

y

∧

d

⁢

t

-

b

⁢

d

⁢

t

∧

d

⁢

x

∧

d

⁢

y

d

⁢

u

∧

d

⁢

t

∧

d

⁢

x

-

c

⁢

d

⁢

t

∧

d

⁢

x

∧

d

⁢

y

From the way these forms were constructed, it is clear that a three dimensional surface in the seven dimensional space with

coordinates

x

,

y

,

t

,

a

,

b

,

c

,

u

which solves Pfaff’s problem and can be parameterized by

x

,

y

,

t

corresponds to the graph of a solution to the system of differential equations, and hence to a solution of the wave equation.

Note: These considerations are purely local. The global topology of the seven-dimensional space will depend on the domain on which the original wave equation was formulated and on the

boundary conditions

.

Title

example of rewriting a differential equation as a Pfaffian system

Canonical name

ExampleOfRewritingADifferentialEquationAsAPfaffianSystem

Date of creation

2013-03-22 14:38:49

Last modified on

2013-03-22 14:38:49

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Example

Classification

msc 53B99

Generated on Fri Feb 9 22:27:47 2018 by

LaTeXML
germ

germ

GermFernando Sanz GÃÂ¡miz

Definition 1

(Germ).

Let

M

and

N

be

manifolds

and

x

∈

M

. We consider all

smooth
mappings

f

:

U

f

→

N

, where

U

f

is some open neighborhood of

x

in

M

. We define an

equivalence relation

on the set of

mappings

considered, and we put

f

⁢

∼

𝑥

⁢

g

if there is some
open neighborhood

V

of

x

with

f

|

V

=

g

|

V

. The

equivalence
class

of a mapping

f

is called the

germ of f at x

, denoted
by

f

¯

or, sometimes,

g

⁢

e

⁢

r

⁢

m

x

⁢

f

, and we write

f

¯

:

(

M

,

x

)

→

(

N

,

f

⁢

(

x

)

)

Remark 1

.

Germs arise naturally in

differential

topolgy. It is very convenient
when dealing with

derivatives

at the point

x

, as every mapping in
a germ will have the same derivative values and properties in

x

,
and hence can be identified for such purposes: every mapping in a
germ gives rise to the same

tangent vector

of

M

at

x

.

Title

germ

Canonical name

Germ

Date of creation

2013-03-22 17:25:36

Last modified on

2013-03-22 17:25:36

Owner

fernsanz (8869)

Last modified by

fernsanz (8869)

Numerical id

5

Author

fernsanz (8869)

Entry type

Definition

Classification

msc 53B99

Related topic

TangentSpace

Defines

Germ

Defines

function

germ.

Generated on Fri Feb 9 22:27:49 2018 by

LaTeXML
germ of smooth functions

germ of smooth functions

If

x

is a point on a

smooth manifold

M

, then a

germ of smooth functions

near

x

is represented by a pair

(

U

,

f

)

where

U

⊆

M

is an open

neighbourhood

of

x

, and

f

is a

smooth function

U

→

ℝ

. Two such pairs

(

U

,

f

)

and

(

V

,

g

)

are considered

equivalent

if there is a third open neighbourhood

W

of

x

,

contained

in both

U

and

V

, such that

f

|

W

=

g

|

W

. To be precise, a germ of smooth functions near

x

is an

equivalence class

of such pairs.

In more fancy

language

: the set

𝒪

x

of germs at

x

is the stalk at

x

of the sheaf

𝒪

of smooth functions on

M

. It is clearly an

ℝ

-algebra.

Germs are useful for defining the

tangent space

T

x

⁢

M

in a coordinate-free manner: it is simply the space of all

ℝ

-linear maps

X

:

𝒪

x

→

ℝ

satisfying Leibniz’ rule

X

⁢

(

f

⁢

g

)

=

X

⁢

(

f

)

⁢

g

+

f

⁢

X

⁢

(

g

)

. (Such a map is called an

ℝ

-linear

derivation

of

𝒪

x

with values in

ℝ

.)

Title

germ of smooth functions

Canonical name

GermOfSmoothFunctions

Date of creation

2013-03-22 13:05:08

Last modified on

2013-03-22 13:05:08

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

4

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 53B99

Generated on Fri Feb 9 22:27:51 2018 by

LaTeXML
germ space

germ space

Let

X

,

Y

be

topological spaces

and

x

∈

X

. Consider the set of all

continuous functions

C

(

X

,

Y

)

=

{

f

:

X

→

Y

|

f

is continuous

}

.

For any two

functions

f

,

g

:

X

→

Y

we put

f

∼

x

g

if and only if there exists an open

neighbourhood

U

⊆

X

of

x

such that

f

|

U

=

g

|

U

.

The corresponding quotient set is called

the germ space

at

x

∈

X

and we denote it by

G

x

⁢

(

X

,

Y

)

.

More generally, if

X

,

Y

are topological spaces with

x

∈

X

, then consider the following set:

C

x

(

X

,

Y

)

=

{

f

:

U

→

Y

|

f

is continuous and

U

is an open neighbourhood of

x

}

.

Again we define a relation on

C

x

⁢

(

X

,

Y

)

. If

f

:

U

→

Y

and

g

:

U

′

→

Y

, then put

f

∼

x

g

if and only if there exists and open neighbourhood

V

⊆

X

of

x

such that

V

⊆

U

∩

U

′

and

f

|

V

=

g

|

V

.

The corresponding set is called

the generalized germ space

at

x

∈

X

and we denote it by

G

x

*

⁢

(

X

,

Y

)

.

Note that if

Y

=

ℝ

or

Y

=

ℂ

(or

Y

is any

topological ring

), then both

G

x

⁢

(

X

,

Y

)

and

G

x

*

⁢

(

X

,

Y

)

have a

well-defined

ring

structure

via

pointwise addition

and

multiplication

.

Title

germ space

Canonical

name

GermSpace

Date of creation

2013-03-22 19:18:20

Last modified on

2013-03-22 19:18:20

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Definition

Classification

msc 53B99

Generated on Fri Feb 9 22:27:53 2018 by

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Pfaff’s problem

Pfaff’s problem

Given a

manifold

M

, let

S

be a set of differential forms on

M

.

Pfaff’s problem

is to determine all

submanifolds

of

M

such that the

pullback

of any form in

S

vanishes identically.

In the

context

of Pfaff’s problem, a set of differential forms is often called a

Pfaffian system

and a manifold on which the pullback (

http://planetmath.org/PullbackOfAKForm

) of these forms vanishes is called a solution to this system. Sometimes, (especially in older literature) one also sees a notation like

a

⁢

d

⁢

x

+

b

⁢

d

⁢

x

=

0

to indicate that the pullback of the form

a

⁢

d

⁢

x

+

b

⁢

d

⁢

y

is required to vanish. A more accurate notation would be

ι

M

*

⁢

(

a

⁢

d

⁢

x

+

b

⁢

d

⁢

y

)

=

0

which explicitly indicates the pullback to the submanifold.

Given a system of local coordinates on

M

, Pfaff’s problem is

equivalent

to a system of

partial differential equations

.

Conversely

, given a system of differential equations, one can find a set of differential forms such that solutions of the system of equations correspond to solutions of Pfaff’s problem for that set of forms. In this way, Pfaff’s problem

differential geometry

and the theory of

differential equations

. The resulting cross-fertilization has been beneficial for both .

On the one hand, translating a system of differential equations into a Pfaff’s problem makes it easier to discuss the

transformation

properties of that equation since changes of variable for the equation correspond to diffeomorphisms of the manifold on which the Pfaff’s problem is formulated. This is especially true when the transformations mix dependent and independent variables or involve

derivatives

of variables. For this reason, it is often easier to study the transformation properties and

symmetries

of a system of differential equations by examining the transformation properties and symmetries of a system of differential forms corresponding to the system of equations. Also, questions of

consistency

and integrability conditions are usually much more easily discussed for systems of differential forms than for systems of differential equations.

On the other hand, to prove the

existence

or uniqueness of a solution to Pfaff’s problem for a certain class of systems of differential forms, it is typical to first translate the statement to be proven to a statement about the existence or uniqueness of a solution to a system of differential equations and then apply existence and

uniqueness theorems

for differential equations.

Reference: Schouten and v. d. Kulk 1949 in

bibliography for differential geometry

Title

Pfaff’s problem

Canonical name

PfaffsProblem

Date of creation

2013-03-22 14:37:48

Last modified on

2013-03-22 14:37:48

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

13

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 53B99

Generated on Fri Feb 9 22:27:55 2018 by

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relation between germ space and generalized germ space

relation between germ space and generalized germ space

Let

X

,

Y

be a

topological spaces

and

x

∈

X

. Consider the

germ space

(

http://planetmath.org/GermSpace

) and the generalized germ space (

http://planetmath.org/GermSpace

) at

x

:

G

x

⁢

(

X

,

Y

)

;

G

x

*

⁢

(

X

,

Y

)

.

If

f

:

X

→

Y

is a

continuous function

, then we have an induced

element

[

f

]

∈

G

x

*

⁢

(

X

,

Y

)

. It can be easily seen, that if

[

f

]

=

[

g

]

∈

G

x

⁢

(

X

,

Y

)

, then

[

f

]

=

[

g

]

∈

G

x

*

⁢

(

X

,

Y

)

. In particular we have a well-defined

mapping

τ

:

G

x

⁢

(

X

,

Y

)

→

G

x

*

⁢

(

X

,

Y

)

;

τ

⁢

(

[

f

]

)

=

[

f

]

.

Proposition

1.

τ

is

injective

.

Proof.

Indeed, assume that

τ

⁢

(

[

f

]

)

=

τ

⁢

(

[

g

]

)

for some

f

,

g

:

X

→

Y

. Let

f

′

:

U

→

Y

and

g

′

:

U

′

→

Y

be a representatives of

τ

⁢

(

[

f

]

)

and

τ

⁢

(

[

g

]

)

respectively. It follows, that there exists an open

neighbourhood

V

⊆

X

of

x

such that

f

|

V

=

f

|

V

′

=

g

|

V

′

=

g

|

V

.

In particular

[

f

]

=

[

g

]

in

G

x

⁢

(

X

,

Y

)

, which

completes

the proof.

□

Proposition 2.

If

X

is a

normal space

and

Y

is a normal

absolute retract

(for example

Y

=

ℝ

), then

τ

is onto.

Proof.

Assume that

[

f

]

∈

G

x

*

⁢

(

X

,

Y

)

for some

f

:

U

→

Y

. Since

X

is

regular

(because it is normal) then there exists an open neighbourhood

V

⊆

X

such that the

closure

V

¯

⊆

U

. Now since

X

is normal and

Y

is a normal absolute retract, then

f

|

V

¯

can be extended to entire

X

(by the generalized

Tietze extension theorem

). It is easily seen that any such

extension

gives the same element in

G

x

⁢

(

X

,

Y

)

(and it is

independent

on the choice of the representative

f

) and if

F

:

X

→

Y

is an extension of

f

|

V

¯

, then

τ

⁢

(

[

F

]

)

=

[

f

]

because

F

|

V

=

f

|

V

. This completes the proof.

□

Remark.

If in

addition

Y

is a

topological ring

(for example

Y

=

ℝ

), then it can be easily checked that

τ

preserves ring

structures

. In particular if

X

is normal and

Y

=

ℝ

or

Y

=

ℂ

, then

τ

is an

isomorphism

of rings. Also it is a good question whether the

assumptions

in proposition 2 can be weakened.

Title

relation between germ space and generalized germ space

Canonical name

RelationBetweenGermSpaceAndGeneralizedGermSpace

Date of creation

2013-03-22 19:18:23

Last modified on

2013-03-22 19:18:23

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

5

Author

joking (16130)

Entry type

Theorem

Classification

msc 53B99

Generated on Fri Feb 9 22:27:58 2018 by

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Sub-Riemannian manifold

Sub-Riemannian manifold

A

Sub-Riemannian

manifold

is a triple

(

M

,

ℋ

,

g

ℋ

)

where

M

is a manifold,

ℋ

is a

distribution

(that is, a linear

subbundle

of the

tangent bundle

T

⁢

M

of

M

), and

g

ℋ

is a metric on

ℋ

induced by a fiber inner product on

ℋ

. The
distribution

ℋ

is often referred to as the

horizontal

distribution.

Title

Sub-Riemannian manifold

Canonical name

SubRiemannianManifold

Date of creation

2013-03-22 13:58:52

Last modified on

2013-03-22 13:58:52

Owner

RevBobo (4)

Last modified by

RevBobo (4)

Numerical id

4

Author

RevBobo (4)

Entry type

Definition

Classification

msc 53C17

Synonym

Sub-Riemannian geometry

Generated on Fri Feb 9 22:28:00 2018 by

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Nash isometric embedding theorem

Nash isometric embedding theorem

Every

compact

n

-dimensional

Riemannian manifold

M

of class

C

k

(

3

≤

k

≤

∞

) can be

C

k

-isometrically imbedded in any small
portion of a

Euclidean space

ℝ

N

, where

N

=

1

2

⁢

n

⁢

(

3

⁢

n

+

11

)

.

Every non-compact

n

-dimensional Riemannian manifold

M

of class

C

k

(

3

≤

k

≤

∞

) can be

C

k

-isometrically imbedded in any small portion of a Euclidean space

ℝ

N

, where

N

=

(

n

+

1

)

⁢

1

2

⁢

n

⁢

(

3

⁢

n

+

11

)

.

The original proof due to Nash relying on an iteration scheme has been considerably simplified. For an overview, see

[

2

]

.

References

1

Nash, J. F.,

The

imbedding

problem for Riemannian manifold

, Ann. of Math. 63 (1956), 20–63 (MR 17, 782)

2

D. Yang,

Gunther’s proof of Nash’s

isometric embedding

theorem

,

http://www.math.poly.edu/ yang/papers/gunther.pdf

online

Title

Nash isometric embedding theorem

Canonical name

NashIsometricEmbeddingTheorem

Date of creation

2013-03-22 15:38:17

Last modified on

2013-03-22 15:38:17

Owner

Simone (5904)

Last modified by

Simone (5904)

Numerical id

7

Author

Simone (5904)

Entry type

Theorem

Classification

msc 53C20

Classification

msc 53C42

Classification

msc 57R40

Classification

msc 58A05

Generated on Fri Feb 9 22:28:01 2018 by

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sphere theorem from global differential geometry

sphere theorem from global differential geometry

This theorem, as do Carmo refers it, is one of the most beautiful theorems in

Riemannian geometry

:

sphere theorem

.

Let

M

be a n-dimensional compact

simply connected

Riemannian manifold

, whose

sectional curvature

K

satisfies

0

<

K

max

/

4

<

K

≤

K

max

Then

M

is

homeomorphic

to a sphere.

References

1

M. P. do Carmo,

Riemannian Geometry

, Birkhäuser, Boston, 1992.

Title

sphere theorem from global differential geometry

Canonical name

SphereTheoremFromGlobalDifferentialGeometry

Date of creation

2013-03-22 15:54:09

Last modified on

2013-03-22 15:54:09

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

8

Author

juanman (12619)

Entry type

Theorem

Classification

msc 53C21

Classification

msc 53C20

Related topic

Curvature

Related topic

Connection

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