aarjaneiro/planetmath huggingface dataset:
Lagrangian submanifold

Lagrangian submanifold

If

(

M

,

ω

)

is a symplectic

2

⁢

n

-

manifold

(

http://planetmath.org/SymplecticManifold

), then a

submanifold

L

is called
if it is isotropic, and of

dimension

n

. This is the maximal dimension an

isotropic submanifold

can have, by the non-degeneracy of

ω

.

Title

Lagrangian submanifold

Canonical

name

LagrangianSubmanifold

Date of creation

2013-03-22 13:12:29

Last modified on

2013-03-22 13:12:29

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Definition

Classification

msc 53D05

Related topic

IsotropicSubmanifold

Generated on Fri Feb 9 22:28:49 2018 by

LaTeXML
Moser’s theorem

Moser’s theorem

Let

ω

0

and

ω

1

be

symplectic structures

on a

compact manifold

M

. If there is a path
in the space of symplectic structures of a fixed DeRham

cohomology class

connecting

ω

0

and

ω

1

(in particular

ω

0

and

ω

1

must have the same class),
then

(

M

,

ω

0

)

and

(

M

,

ω

1

)

are symplectomorphic, by a

symplectomorphism

isotopic

to the identity.

Title

Moser’s theorem

Canonical name

MosersTheorem

Date of creation

2013-03-22 13:18:08

Last modified on

2013-03-22 13:18:08

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

6

Author

bwebste (988)

Entry type

Theorem

Classification

msc 53D05

Generated on Fri Feb 9 22:28:51 2018 by

LaTeXML
Poisson bracket

Poisson bracket

Let

M

be a

symplectic manifold

with

symplectic form

Ω

. The

Poisson bracket

is a bilinear operation on the set of

differentiable functions

on

M

. In terms of local

Darboux coordinates

p

1

,

…

,

p

n

,

q

1

,

…

,

q

n

, the Poisson bracket of two functions is defined as follows:

[

f

,

g

]

=

∑

i

=

1

n

∂

⁡

f

∂

⁡

q

i

⁢

∂

⁡

g

∂

⁡

p

i

-

∂

⁡

f

∂

⁡

p

i

⁢

∂

⁡

g

∂

⁡

q

i

It can be shown that the value of

[

f

,

g

]

does not depend on the choice of Darboux coordinates. Therefore, the Poisson bracket is a well-defined operation on the symplectic manifold. Also, some authors use a different sign convention — what they call

[

f

,

g

]

is what would be referred to as

-

[

f

,

g

]

here.

The Poisson bracket can be defined without reference to a special

coordinate system

as follows:

[

f

,

g

]

=

Ω

-

1

⁢

(

d

⁢

f

,

d

⁢

g

)

=

∑

i

=

1

2

⁢

n

Ω

i

⁢

j

⁢

∂

⁡

f

∂

⁡

x

i

⁢

∂

⁡

g

∂

⁡

x

j

Here

Ω

-

1

is the

inverse

of the symplectic form, and its

components

in an arbitrary coordinate system are denoted

Ω

i

⁢

j

.

The Poisson bracket sastisfies several important algebraic identities. It is

antisymmetric

:

[

f

,

g

]

=

-

[

g

,

f

]

It is a derivation:

[

f

⁢

g

,

h

]

=

f

⁢

[

g

,

h

]

+

g

⁢

[

f

,

h

]

It satisfies Jacobi’s identitity:

[

f

,

[

g

,

h

]

]

+

[

g

,

[

h

,

f

]

]

+

[

h

,

[

f

,

g

]

]

=

0

The

Hamilton equations

can be expressed elegantly in terms of the Poisson bracket. If

X

is a smooth function on

M

, we can describe the time-evolution of

X

by the equation

d

⁢

X

d

⁢

t

=

[

X

,

H

]

If

X

is a smooth function on

ℝ

×

M

, we can describe the time-evolution of

X

by the more general equation

d

⁢

X

d

⁢

t

=

∂

⁡

X

∂

⁡

t

-

[

X

,

H

]

Title

Poisson bracket

Canonical name

PoissonBracket

Date of creation

2013-03-22 14:46:04

Last modified on

2013-03-22 14:46:04

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 53D05

Related topic

Quantization

Related topic

CanonicalQuantization

Generated on Fri Feb 9 22:28:54 2018 by

LaTeXML
Poisson ring

Poisson ring

A

Poisson ring

A

is a commutative ring on which a

binary operation

[

,

]

, known as the

Poisson bracket

is defined. This operation must satisfy the following

identities

:

1.

[

f

,

g

]

=

-

[

g

,

f

]

2.

[

f

+

g

,

h

]

=

[

f

,

h

]

+

[

g

,

h

]

3.

[

f

⁢

g

,

h

]

=

f

⁢

[

g

,

h

]

+

g

⁢

[

f

,

h

]

4.

[

f

,

[

g

,

h

]

]

+

[

g

,

[

h

,

f

]

]

+

[

h

,

[

f

,

g

]

]

=

0

If, in addition,

A

is an algebra over a field, then we call

A

a

Poisson algebra

. In this case, we may wish to add the extra requirement

[

s

⁢

f

,

g

]

=

s

⁢

[

f

,

g

]

for all scalars

s

.

Because of properties 2 and 3, for each

g

∈

A

, the operation

a

⁢

d

g

defined as

a

⁢

d

g

⁢

(

f

)

=

[

f

,

g

]

is a derivation. If the set

{

a

⁢

d

g

|

g

∈

A

}

generates the set of derivations of

A

, we say that

A

is

non-degenerate

.

It can be shown that, if

A

is non-degenerate and is isomorphic as a commutative ring to the

algebra

of

smooth functions

on a

manifold

M

, then

M

must be a

symplectic manifold

and

[

,

]

is the Poisson bracket defined by the

symplectic form

.

Many important operations and results of

symplectic geometry

and Hamiltonian mechanics may be formulated in terms of the Poisson bracket and, hence, apply to Poisson algebras as well. This observation is important in studying the classical limit of quantum mechanics — the non-commutative algebra of

operators

on a Hilbert space has the Poisson algebra of functions on a symplectic manifold as a singular limit and properties of the non-commutative algebra pass over to corresponding properties of the Poisson algebra.

In addition to their use in mechanics, Poisson algebras are also used in the study of Lie groups.

Title

Poisson ring

Canonical name

PoissonRing

Date of creation

2013-03-22 14:46:12

Last modified on

2013-03-22 14:46:12

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 53D05

Defines

Poisson algebra

Generated on Fri Feb 9 22:28:57 2018 by

LaTeXML
symplectic manifold

symplectic manifold

Symplectic manifolds

constitute
the mathematical structure for modern Hamiltonian mechanics.
Symplectic manifolds can also be seen as even dimensional
analogues to

contact manifolds

.

Definition 1

.

A

symplectic manifold

is a pair

(

M

,

ω

)

consisting
of a

smooth manifold

M

and a
closed 2-form (

http://planetmath.org/DifferentialForms

)

ω

∈

Ω

2

⁢

(

M

)

, that is non-degenerate
at each point.
Then

ω

is called a

symplectic
form

for

M

.

Properties

1.

Every symplectic manifold is even dimensional. This is
easy to understand in view of the physics. In

Hamilton
equations

, location and momentum vectors always appear in pairs.

2.

A form

ω

∈

Ω

2

⁢

(

M

)

on a

2

⁢

n

-dimensional

manifold

M

is non-degenerate if and only if the

n

-fold product

ω

n

=

ω

∧

⋯

∧

ω

is non-zero.

3.

As a consequence of the last , every symplectic manifold
is

orientable

.

Let

(

M

,

ω

)

and

(

N

,

η

)

be symplectic manifolds. Then a

diffeomorphism

f

:

M

→

N

is
called a

symplectomorphism

if

f

*

⁢

η

=

ω

, that is, if the symplectic form on

N

pulls back to the form on

M

.

Notes

A symplectomorphism is also known as a

canonical transformation

.
This is mostly used in the mechanics literature.

Title

symplectic manifold

Canonical name

SymplecticManifold

Date of creation

2013-03-22 13:12:18

Last modified on

2013-03-22 13:12:18

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

11

Author

matte (1858)

Entry type

Definition

Classification

msc 53D05

Related topic

ContactManifold

Related topic

KahlerManifold

Related topic

HyperkahlerManifold

Related topic

MathbbCIsAKahlerManifold

Defines

symplectic form

Defines

symplectomorphism

Defines

canonical transformation

Generated on Fri Feb 9 22:29:01 2018 by

LaTeXML
symplectic matrix

symplectic matrix

A real

2

⁢

n

×

2

⁢

n

matrix

A

∈

M

2

⁢

n

⁢

(

ℝ

)

is a

symplectic matrix

if

A

⁢

J

⁢

A

T

=

J

, where

A

T

is the

transpose

of

A

, and

J

∈

O

⁢

(

2

⁢

n

)

is the

orthogonal matrix

J

=

(

𝟎

𝐈

n

-

𝐈

n

𝟎

)

.

Here

𝐈

n

∈

M

n

⁢

(

ℝ

)

is the identity

n

×

n

matrix and

𝟎

∈

M

n

⁢

(

ℝ

)

is the zero

n

×

n

matrix.

Symplectic matrices satisfy the following
properties:

1.

The

determinant

of a symplectic matrix equals one.

2.

With

standard matrix multiplication

, symplectic

2

⁢

n

×

2

⁢

n

matrices form a group denoted by

Sp

⁢

(

2

⁢

n

)

.

3.

Suppose

Ψ

=

(

A

B

C

D

)

,
where

A

,

B

,

C

,

D

are

n

×

n

matrices. Then

Ψ

is symplectic if and only if

A

⁢

D

T

-

B

⁢

C

T

=

I

,

A

⁢

B

T

=

B

⁢

A

T

,

C

⁢

D

T

=

D

⁢

C

T

.

4.

If

X

and

Y

are real

n

×

n

matrices, then

U

=

X

+

i

⁢

Y

is

unitary

if and only if

(

X

-

Y

Y

X

)

is symplectic.

Title

symplectic matrix

Canonical name

SymplecticMatrix

Date of creation

2013-03-22 13:32:28

Last modified on

2013-03-22 13:32:28

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

11

Author

matte (1858)

Entry type

Definition

Classification

msc 53D05

Generated on Fri Feb 9 22:29:03 2018 by

LaTeXML
symplectic vector field

symplectic vector field

If

(

M

,

ω

)

is a

symplectic manifold

, then a vector field

X

∈

𝔛

⁢

(

M

)

is

symplectic

if
its flow

preserves

the

symplectic structure

. That is, if the

Lie derivative

ℒ

X

⁢

ω

=

0

.

Title

symplectic vector field

Canonical

name

SymplecticVectorField

Date of creation

2013-03-22 13:14:04

Last modified on

2013-03-22 13:14:04

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

4

Author

bwebste (988)

Entry type

Definition

Classification

msc 53D05

Generated on Fri Feb 9 22:29:05 2018 by

LaTeXML
symplectic vector space

symplectic vector space

A

symplectic vector space

(

V

,

ω

)

is a finite dimensional

real vector space

V

equipped with
an

alternating

non-degenerate

2-tensor, i.e.,
a bilinear map

ω

:

V

×

V

→

ℝ

that

satisfies

the following

properties

:

1.

Alternating: For all

v

,

w

∈

V

,

ω

⁢

(

v

,

w

)

=

-

ω

⁢

(

w

,

v

)

.

2.

Non-degenerate: If

ω

⁢

(

v

,

w

)

=

0

for all

w

∈

V

, then

v

=

0

.

The tensor

ω

is
called a for

V

.

A linear

automorphism

T

∈

Aut

⁢

(

V

)

is called

linear symplectomorphism

when

T

*

⁢

ω

=

ω

, i.e.

ω

⁢

(

T

⁢

v

,

T

⁢

w

)

=

ω

⁢

(

v

,

w

)

∀

v

,

w

∈

W

.

Linear symplectomorphisms of

(

V

,

ω

)

form a group (under

composition

of

linear map

) that is denoted by

Sp

⁢

(

V

,

ω

)

.

One can show that a symplectic vector space is always even dimensional

[

1

]

.

References

1

D. McDuff, D. Salamon,

Introduction

to Symplectic Topology

,
Clarendon Press, 1997.

Title

symplectic vector space

Canonical

name

SymplecticVectorSpace

Date of creation

2013-03-22 13:32:22

Last modified on

2013-03-22 13:32:22

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

11

Author

matte (1858)

Entry type

Definition

Classification

msc 53D05

Defines

symplectic vector space

Defines

linear symplectomorphism

Generated on Fri Feb 9 22:29:07 2018 by

LaTeXML
contact manifold

contact manifold

Let

M

be a

smooth manifold

and

α

a one form on

M

. Then

α

is a

contact form

on

M

if

1.

for each point

m

∈

M

,

α

m

≠

0

and

2.

the restriction

d

⁢

α

m

|

ker

⁡

α

m

of the

differential

of

α

is

nondegenerate

.

Condition 1 ensures that

ξ

=

ker

⁡

α

is a subbundle of the vector bundle

T

⁢

M

. Condition 2 equivalently says

d

⁢

α

is a

symplectic structure

on the vector bundle

ξ

→

M

. A

contact structure

ξ

on a

manifold

M

is a subbundle of

T

⁢

M

so that for each

m

∈

M

, there is a contact form

α

defined on some neighborhood of

m

so that

ξ

=

ker

⁡

α

. A co-oriented contact structure is a subbundle of

T

⁢

M

of the form

ξ

=

ker

⁡

α

for some globally defined contact form

α

.

A (co-oriented)

contact manifold

is a pair

(

M

,

ξ

)

where

M

is a manifold and

ξ

is a (co-oriented) contact structure. Note, symplectic linear algebra implies that

dim

⁡

M

is odd. If

dim

⁡

M

=

2

⁢

n

+

1

for some positive integer

n

, then a one form

α

is a contact form if and only if

α

∧

(

d

⁢

α

)

n

is everywhere nonzero.

Suppose now that

(

M

1

,

ξ

1

=

ker

⁡

α

1

)

and

(

M

2

,

ξ

2

=

ker

⁡

α

2

)

are co-oriented contact manifolds. A

diffeomorphism

ϕ

:

M

1

→

M

2

is called a

contactomorphism

if the pullback along

ϕ

of

α

2

differs from

α

1

by some

positive

smooth function

f

:

M

1

→

ℝ

, that is,

ϕ

*

⁢

α

2

=

f

⁢

α

1

.

Examples

:

1.

ℝ

3

is a contact manifold with the contact structure induced by the one form

α

=

d

⁢

z

+

x

⁢

d

⁢

y

.

2.

Denote by

𝕋

2

the two-torus

𝕋

2

=

S

1

×

S

1

. Then,

ℝ

×

𝕋

2

(with coordinates

t

,

θ

1

,

θ

2

) is a contact manifold with the contact structure induced by

α

=

cos

⁡

t

⁢

θ

1

+

sin

⁡

t

⁢

θ

2

.

Title

contact manifold

Canonical name

ContactManifold

Date of creation

2013-03-22 13:43:27

Last modified on

2013-03-22 13:43:27

Owner

RevBobo (4)

Last modified by

RevBobo (4)

Numerical id

4

Author

RevBobo (4)

Entry type

Definition

Classification

msc 53D10

Related topic

SymplecticManifold

Defines

contact structure

Defines

contact form

Defines

contactomorphism

Generated on Fri Feb 9 22:29:09 2018 by

LaTeXML
momentum map

momentum map

Let

(

M

,

ω

)

be a

symplectic manifold

,

G

a Lie group acting on that

manifold

,

𝔤

its

Lie algebra

, and

𝔤

*

the dual of the Lie algebra. This action induces a map

α

:

𝔤

→

𝔛

⁢

(

M

)

where

𝔛

⁢

(

M

)

is the Lie algebra of vector fields on

M

, such that

exp

⁡

(

t

⁢

X

)

⁢

(

m

)

=

ρ

t

⁢

(

m

)

where

ρ

is the flow of

α

⁢

(

X

)

. Then a

moment map

μ

:

M

→

𝔤

*

for the action of

G

is a map such that

H

μ

⁢

(

X

)

=

α

⁢

(

X

)

.

Here

μ

⁢

(

X

)

⁢

(

m

)

=

μ

⁢

(

m

)

⁢

(

X

)

, that is,

μ

⁢

(

m

)

is a covector, so we apply
it to the vector

X

and get a scalar function

μ

⁢

(

X

)

, and

H

μ

⁢

(

X

)

is its

Hamiltonian vector field

.

Generally, the moment maps we are interested in are equivariant with respect to the coadjoint
action, that is, they satisfy

Ad

g

*

∘

μ

=

μ

∘

g

.

Title

momentum map

Canonical name

MomentumMap

Date of creation

2013-03-22 13:14:36

Last modified on

2013-03-22 13:14:36

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

4

Author

bwebste (988)

Entry type

Definition

Classification

msc 53D20

Synonym

moment map

Generated on Fri Feb 9 22:29:12 2018 by

LaTeXML
Kähler manifold

Kähler manifold

Let

M

be a

complex manifold

with integrable

complex structure

(

http://planetmath.org/AlmostComplexStructure

)

J

.

Suppose

M

is also a

Riemannian manifold

with metric tensor

g

such that

∀

X

,

Y

g

⁢

(

X

,

Y

)

=

g

⁢

(

J

⁢

X

,

J

⁢

Y

)

. We say that

g

is an

Hermitian metric tensor

.

A

differentiable manifold

M

is said to be a

Kähler

manifold

iff all the following conditions are verified:

•

M

is a complex manifold with complex structure

J

•

M

is a Riemannian manifold with an

Hermitian metric

g

•

J

is covariantly constant with regard to the

Levi-Civita connection

(

∇

⁡

J

=

0

)

Kähler manifolds are symplectic in a natural way with

symplectic form

defined by

ω

⁢

(

X

,

Y

)

=

g

⁢

(

J

⁢

X

,

Y

)

Title

Kähler manifold

Canonical name

KahlerManifold

Date of creation

2013-03-22 15:43:26

Last modified on

2013-03-22 15:43:26

Owner

cvalente (11260)

Last modified by

cvalente (11260)

Numerical id

13

Author

cvalente (11260)

Entry type

Definition

Classification

msc 53D99

Synonym

kählerian manifold

Synonym

kähler structure

Related topic

almostcomplexstructure

Related topic

RiemannianMetric

Related topic

HyperkahlerManifold

Related topic

MathbbCIsAKahlerManifold

Related topic

SymplecticManifold

Related topic

aKahlerManifoldIsSymplectic

Related topic

AKahlerManifoldIsSymplectic

Related topic

AlmostComplexStructure

Defines

Hermitian metric tensor

Generated on Fri Feb 9 22:29:18 2018 by

LaTeXML
Kähler potential

Kähler potential

A Kähler potential is a real-valued function

f

defined on some coordinate patch of a
Hermitean manifold such that the metric of the manifold is given by the

expression

g

i

⁢

j

⁣

*

=

∂

2

⁡

f

d

⁢

z

i

⁢

d

⁢

z

¯

j

.

It turns out that, for every Káhler manifold, there will exist a coordinate

neighborhood

of any
given point in which the metric can be expresses in terms of a potential this way.

As an elementary example of a Kähler potential, we may consider

f

⁢

(

z

,

z

¯

)

=

z

⁢

z

¯

. This potential gives rise to the flat metric

d

⁢

s

2

=

d

⁢

z

⁢

d

⁢

z

¯

.

Kähler potentials have applications in physics. For example, this function

f

⁢

(

x

)

=

log

⁡

(

x

)

+

g

⁢

(

x

)

relates to the motion of certain subatomic particles called gauginos.

References

1

T. Barreiro, B. de Carlos & E. J. Copeland, “On non-perturbative corrections to the Kähler potential”

Physical Review

D57 (1998): 7354 - 7360

Title

Kähler potential

Canonical

name

KahlerPotential

Date of creation

2013-03-22 16:33:17

Last modified on

2013-03-22 16:33:17

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 53D99

Synonym

Kahler potential

Generated on Fri Feb 9 22:29:19 2018 by

LaTeXML
ℂ as a Kähler manifold

ℂ

as a Kähler manifold

ℂ

can be interpreted as

ℝ

2

with a

complex structure

(

http://planetmath.org/AlmostComplexStructure

)

J

.

Parametrize

ℝ

2

via the usual coordinates

(

x

,

y

)

.

A point

z

in the complex plane can thus be written

z

=

x

+

i

⁢

y

.

The

tangent space

at each point is generated by the

span

ℝ

⁢

{

∂

∂

⁡

x

,

∂

∂

⁡

y

}

and the complex structure (

http://planetmath.org/AlmostComplexStructure

)

J

is defined by

1

1

notice

J

acts as a counterclockwise rotation by

π

2

, just as expected

J

⁢

(

∂

∂

⁡

x

)

=

∂

∂

⁡

y

(1)

J

⁢

(

∂

∂

⁡

y

)

=

-

∂

∂

⁡

x

(2)

The metric can be the usual metric

g

=

d

⁢

x

⊗

d

⁢

x

+

d

⁢

y

⊗

d

⁢

y

.
This is a flat metric and therefore all the

covariant derivatives

are plain

partial derivatives

in the

(

x

,

y

)

coordinates

2

2

the

Christoffel symbols

on these coordinates vanish

.

So lets verify all the points in the definition.

•

ℂ

is a

Riemannian Manifold

•

g

is Hermitian.

g

⁢

(

J

⁢

∂

∂

⁡

x

,

J

⁢

∂

∂

⁡

y

)

=

g

⁢

(

∂

∂

⁡

y

,

-

∂

∂

⁡

x

)

=

0

=

g

⁢

(

∂

∂

⁡

x

,

∂

∂

⁡

y

)

g

⁢

(

J

⁢

∂

∂

⁡

x

,

J

⁢

∂

∂

⁡

x

)

=

g

⁢

(

∂

∂

⁡

y

,

∂

∂

⁡

y

)

=

1

=

g

⁢

(

∂

∂

⁡

x

,

∂

∂

⁡

x

)

g

⁢

(

J

⁢

∂

∂

⁡

y

,

J

⁢

∂

∂

⁡

y

)

=

g

⁢

(

-

∂

∂

⁡

x

,

-

∂

∂

⁡

x

)

=

1

=

g

⁢

(

∂

∂

⁡

y

,

∂

∂

⁡

y

)

•

J

is covariantly constant because its

components

in the

(

x

,

y

)

coordinates are constant and as previously stated, the covariant derivatives are just partial derivatives in this example.

ℂ

is therefore a Kähler

manifold

.

The

symplectic form

for this example is

ω

=

d

⁢

x

∧

d

⁢

y

This is the simplest example of a Kähler manifold and can be seen as a template for other less trivial examples. Those are generalizations of this example just as Riemannian manifolds are generalizations of

ℝ

n

seen as a metric space.

Title

ℂ

as a Kähler manifold

Canonical name

mathbbCAsAKahlerManifold

Date of creation

2013-03-22 15:46:32

Last modified on

2013-03-22 15:46:32

Owner

cvalente (11260)

Last modified by

cvalente (11260)

Numerical id

16

Author

cvalente (11260)

Entry type

Example

Classification

msc 53D99

Related topic

KahlerManifold

Related topic

AlmostComplexStructure

Related topic

SymplecticManifold

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

LaTeXML
Minkowski space

Minkowski space

Minkowski space

is a 4 dimensional

real vector space

with a

non-degenerate

pseudo-metric of

signature

(

-

+

+

+

)

.

More precisely,

M

with a metric

g

is a Minkowski space iff:

•

M

is a 4 dimensional real vector space

•

g

is a

symmetric

2-covariant tensor (defines a

quadratic form

)

•

g

is non-degenerate (i.e.

∀

x

∈

M

,

g

⁢

(

x

,

y

)

=

0

⟹

y

=

0

)

•

the

diagonalization

(

http://planetmath.org/SylvestersLaw

) of

g

contains one

negative

and three

positive

diagonal

entries

1

1

this convention is sometimes reversed depending on notation

.

Title

Minkowski space

Canonical name

MinkowskiSpace

Date of creation

2013-03-22 16:16:51

Last modified on

2013-03-22 16:16:51

Owner

cvalente (11260)

Last modified by

cvalente (11260)

Numerical id

7

Author

cvalente (11260)

Entry type

Definition

Classification

msc 53Z05

Related topic

PseudoRiemannianManifold

Generated on Fri Feb 9 22:29:24 2018 by

LaTeXML
pseudo-Riemannian manifold

pseudo-Riemannian manifold

A

pseudo-Riemannian

manifold

is a manifold

M

together with a non degenerate (

http://planetmath.org/NonDegenerateBilinearForm

),

symmetric

section

g

of

T

2

0

⁢

(

M

)

(2-covariant tensor bundle over

M

).

Unlike with a

Riemannian manifold

,

g

is not

positive definite

. That is, there exist vectors

v

∈

T

p

⁢

M

such that

g

⁢

(

v

,

v

)

≤

0

.

A well known result from

linear algebra

(

http://planetmath.org/SylvestersLaw

) permits us to make a

change of basis

such that in the new base

g

is represented by a

diagonal matrix

with

-

1

or

1

elements in the

diagonal

. If there are

i

,

-

1

elements in the diagonal and

j

,

1

, the tensor is said to have

signature

(

i

,

j

)

The signature will be

invariant

in every

connected component

of

M

, but usually the

restriction

that it be a global invariant is added to the definition of a

pseudo-Riemannian manifold

.

Unlike a

Riemannian metric

, some manifolds do not admit a

pseudo-Riemannian metric

.

Pseudo-Riemannian manifolds are crucial in Physics and in particular in General Relativity where space-time is modeled as a 4-pseudo Riemannian manifold with signature (1,3)

1

1

also referred to as

(

-

+

+

+

)

.

Intuitively pseudo-Riemannian manifolds are generalizations of Minkowski’s space just as a Riemannian manifold is a generalization of a

vector space

with a positive definite metric.

Title

pseudo-Riemannian manifold

Canonical name

PseudoRiemannianManifold

Date of creation

2013-03-22 15:44:15

Last modified on

2013-03-22 15:44:15

Owner

cvalente (11260)

Last modified by

cvalente (11260)

Numerical id

10

Author

cvalente (11260)

Entry type

Definition

Classification

msc 53Z05

Related topic

EinsteinFieldEquations

Related topic

SylvestersLaw

Related topic

MinkowskiSpace

Related topic

CategoryOfRiemannianManifolds

Defines

pseudo-Riemannian geometry

Defines

pseudo-Riemannian manifold

Generated on Fri Feb 9 22:29:26 2018 by

LaTeXML