Riemannian manifold

Riemannian manifold

A

Riemannian metric

tensor

is a covariant, type

(

0

,

2

)

tensor
field

g

∈

Γ

⁢

(

T

*

⁡

M

⊗

T

*

⁡

M

)

such that at each point

p

∈

M

,
the

bilinear form

g

p

:

T

p

⁡

M

×

T

p

⁡

M

→

ℝ

is

symmetric

and

positive definite

. Here

T

*

⁢

M

is the

cotangent bundle

of

M

(defined as a sheaf),

Γ

is the set of

global sections

of

T

*

⁢

M

⊗

T

*

⁢

M

, and

g

p

is the value of the function

g

at the point

p

∈

M

.

Let

(

x

1

,

…

,

x

n

)

be a system of

local coordinates

on an open
subset

U

⊂

M

, let

d

⁢

x

i

,

i

=

1

,

…

,

n

be the corresponding

coframe

of 1-forms, and let

∂

∂

⁡

x

i

,

i

=

1

,

…

,

n

be the corresponding dual frame of

vector fields

. Using the local
coordinates, the

metric tensor

has the unique

expression

g

=

∑

i

,

j

=

1

n

g

i

⁢

j

⁢

d

⁢

x

i

⊗

d

⁢

x

j

,

where the metric
tensor

components

g

i

⁢

j

=

g

⁢

(

∂

∂

⁡

x

i

,

∂

∂

⁡

x

j

)

are

smooth
functions

on

U

.

Once we fix the local coordinates, the functions

g

i

⁢

j

completely
determine the Riemannian metric. Thus, at each point

p

∈

U

, the
matrix

(

g

i

⁢

j

⁢

(

p

)

)

is symmetric, and positive definite. Indeed, it
is possible to define a

Riemannian structure

on a

manifold

M

by
specifying an atlas over

M

together with a matrix of functions

g

i

⁢

j

on each

coordinate chart

which are symmetric and positive
definite, with the proviso that the

g

i

⁢

j

’s must be

compatible

with
each other on overlaps.

A manifold

M

together with a Riemannian metric tensor

g

is called
a

Riemannian manifold

.

Note:

A Riemannian metric tensor on

M

is not a distance metric
on

M

. However, on a connected manifold every Riemannian metric
tensor on

M

induces a distance metric on

M

, given by

d

⁢

(

x

,

y

)

:=

inf

⁡

{

∫

0

1

[

g

⁢

(

d

⁢

c

d

⁢

t

,

d

⁢

c

d

⁢

t

)

c

⁢

(

t

)

]

1

/

2

⁢

𝑑

t

}

,

x

,

y

∈

M

,

where the infimum is taken over all

rectifiable curves

c

:

[

0

,

1

]

→

M

with

c

⁢

(

0

)

=

x

and

c

⁢

(

1

)

=

y

.

Often, it is the

g

i

⁢

j

that are referred to as the “Riemannian
metric”. This, however, is a misnomer. Properly speaking, the

g

i

⁢

j

should be called local coordinate components of a metric
tensor, where as “Riemannian metric” should refer to the distance
function defined above. However, the practice of calling the
collection of

g

i

⁢

j

’s by the misnomer “Riemannian metric” appears
to have stuck.

Remarks:

•

Both the Riemannian manifold and Riemannian metric tensor are fundamental concepts
in Einstein’s General Relativity (GR) theory where the “Riemannian metric” and

curvature

of the
physical Riemannian space-time are changed by the presence of massive bodies and energy
according to Einstein’s fundamental GR field equations (

http://planetmath.org/EinsteinFieldEquations

).

•

The category of Riemannian manifolds (or ‘spaces’) provides an alternative framework for GR theories
as well as algebraic quantum field theories (AQFTs);

•

The category of ‘pseudo-Riemannian’ manifolds, deals in fact with extensions of

Minkowski spaces

, does not
possess the Riemannian metric defined in this entry on Riemannian manifolds, and is claimed as a useful
approach to defining 4D-spacetimes in relativity theories.

Title

Riemannian manifold

Canonical

name

RiemannianManifold

Date of creation

2013-03-22 13:02:54

Last modified on

2013-03-22 13:02:54

Owner

djao (24)

Last modified by

djao (24)

Numerical id

31

Author

djao (24)

Entry type

Definition

Classification

msc 53B20

Classification

msc 53B21

Synonym

Riemann space and metric

Related topic

QuantumGeometry2

Related topic

Gradient

Related topic

CategoryOfRiemannianManifolds

Related topic

HomotopyCategory

Related topic

CWComplexDefinitionRelatedToSpinNetworksAndSpinFoams

Related topic

QuantumAlgebraicTopologyOfCWComplexRepresentationsNewQATResultsForQuantumStateSpacesOfSpinNetworks

Defines

Riemannian metric

Defines

Riemannian structure

Defines

metric tensor

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