Ricci tensor

Ricci tensor

Definition.

The

Ricci curvature tensor

is a rank

2

,

symmetric tensor

that
arises naturally in

pseudo-Riemannian geometry

. Let

(

M

,

g

i

⁢

j

)

be a
smooth,

n

-dimensional

pseudo-Riemannian manifold

, and let

R

i

j

⁢

k

⁢

l

denote the corresponding

Riemann curvature tensor

. The

Ricci tensor

R

i

⁢

j

is commonly defined as the following

contraction

of the full curvature tensor:

R

i

⁢

j

=

R

k

.

i

⁢

k

⁢

j

The index

symmetry

of

R

i

⁢

j

, so defined, follows from the symmetry
properties of the Riemann curvature. To wit,

R

i

⁢

j

=

R

k

=

i

⁢

k

⁢

j

R

k

⁢

i

=

k

j

R

k

=

j

⁢

k

⁢

i

R

j

⁢

i

.

It is also convenient to regard the Ricci tensor as a

symmetric bilinear
form

. To that end for vector-fields

X

,

Y

we will write

Ric

⁡

(

X

,

Y

)

=

X

i

⁢

Y

j

⁢

R

i

⁢

j

.

Related objects.

Contracting the Ricci tensor, we obtain an important scalar

invariant

R

=

R

i

,

i

called the

scalar curvature

, and sometimes also called
the Ricci scalar. Closely related to the Ricci tensor is the tensor

G

i

⁢

j

=

R

i

⁢

j

-

1

2

⁢

R

⁢

g

i

⁢

j

,

called the

Einstein
tensor

. The Einstein tensor is also known as the trace-reversed Ricci
tensor owing to the fact that

G

i

=

i

-

R

.

Another related tensor is

S

i

⁢

j

=

R

i

⁢

j

-

1

n

⁢

R

⁢

g

i

⁢

j

.

This is called
the
trace-free Ricci tensor, owing to the fact that the above definition
implies that

S

i

=

i

0

.

Geometric interpretation.

In Riemannian geometry, the Ricci tensor represents the

average value

of the

sectional curvature

along a particular

direction

.
Let

K

x

⁢

(

u

,

v

)

=

R

x

⁢

(

u

,

v

,

v

,

u

)

g

x

⁢

(

u

,

u

)

⁢

g

x

⁢

(

v

,

v

)

-

g

x

⁢

(

u

,

v

)

2

denote the sectional curvature of

M

along the plane spanned by
vectors

u

,

v

∈

T

x

⁢

M

. Fix a point

x

∈

M

and a

tangent vector

v

∈

T

x

⁢

M

, and let

S

x

⁢

(

v

)

=

{

u

∈

T

x

⁢

M

:

g

x

⁢

(

u

,

u

)

=

1

,

g

x

⁢

(

u

,

v

)

=

0

}

denote the

n

-

2

dimensional
sphere of those

unit vectors

at

x

that are

perpendicular

to

v

.
Let

μ

x

denote the natural

(

n

-

2

)

-dimensional

volume measure

on

T

x

⁢

M

, normalized so that

∫

S

x

⁢

(

v

)

μ

x

=

1

.

In this way, the quantity

∫

S

x

⁢

(

v

)

K

x

⁢

(

⋅

,

v

)

⁢

μ

x

,

describes the average value of the sectional curvature for all planes
in

T

x

⁢

M

that contain

v

. It is possible to show that

Ric

x

⁡

(

v

,

v

)

=

(

1

-

n

)

⁢

∫

S

x

⁢

(

v

)

K

x

⁢

(

⋅

,

v

)

⁢

μ

x

,

thereby giving us the desired geometric

interpretation

.

Decomposition of the curvature tensor.

For

n

≥

3

, the Ricci tensor can be characterized in terms of the

decomposition

of the full curvature tensor into three covariantly
defined summands, namely

F

i

⁢

j

⁢

k

⁢

l

=

1

n

-

2

⁢

(

S

j

⁢

l

⁢

g

i

⁢

k

+

S

i

⁢

k

⁢

g

j

⁢

l

-

S

i

⁢

l

⁢

g

j

⁢

k

-

S

j

⁢

k

⁢

g

i

⁢

l

)

,

E

i

⁢

j

⁢

k

⁢

l

=

1

n

⁢

(

n

-

1

)

⁢

R

⁢

(

g

j

⁢

l

⁢

g

i

⁢

k

-

g

i

⁢

l

⁢

g

j

⁢

k

)

,

W

i

⁢

j

⁢

k

⁢

l

=

R

i

⁢

j

⁢

k

⁢

l

-

F

i

⁢

j

⁢

k

⁢

l

-

E

i

⁢

j

⁢

k

⁢

l

.

The

W

i

⁢

j

⁢

k

⁢

l

is called the

Weyl curvature tensor

. It is
the conformally invariant, trace-free part of the curvature tensor.
Indeed, with the above definitions, we have

W

k

=

i

⁢

k

⁢

j

0

.

The

E

i

⁢

j

⁢

k

⁢

l

and

F

i

⁢

j

⁢

k

⁢

l

correspond to the
trace-free part of the Ricci curvature tensor, and to the Ricci
scalar. Indeed, we can recover

S

i

⁢

j

and

R

from

E

i

⁢

j

⁢

k

⁢

l

and

F

i

⁢

j

⁢

k

⁢

l

as follows:

S

i

⁢

j

=

F

k

,

i

⁢

k

⁢

j

E

i

⁢

j

i

⁢

j

=

R

.

Relativity.

The Ricci tensor also plays an important role
in the theory of general relativity. In this keystone application,

M

is a 4-dimensional pseudo-Riemannian manifold with

signature

(

3

,

1

)

. The Einstein field equations assert that the energy-momentum
tensor is proportional to the Einstein tensor. In particular, the
equation

R

i

⁢

j

=

0

is the field equation for a vacuum space-time. In geometry, a
pseudo-Riemannian manifold that

satisfies

this equation is called
Ricci-flat. It is possible to prove that a

manifold

is Ricci flat if
and only if locally, the manifold, is conformally equivalent to flat space.

Title

Ricci tensor

Canonical name

RicciTensor

Date of creation

2013-03-22 15:02:38

Last modified on

2013-03-22 15:02:38

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

9

Author

rmilson (146)

Entry type

Definition

Classification

msc 83C05

Defines

scalar curvature

Defines

Einstein tensor

Defines

ricci scalar

Defines

Weyl tensor

Defines

Weyl curvature tensor

Generated on Sat Feb 10 13:22:22 2018 by

LaTeXML