Christoffel symbols

Christoffel symbols

A

vector field

in

ℝ

n

can be seen as a

differentiable

(

C

∞

) map

V

:

ℝ

n

→

ℝ

n

.

Or as a

section

ℝ

n

→

V

T

⁢

(

ℝ

n

)

where

T

⁢

ℝ

n

≡

ℝ

n

×

ℝ

n

is the

ℝ

n

’s

trivial

tangent bundle obeying

p

↦

(

p

,

V

(

p

)

∈

T

p

(

ℝ

n

)

)

with

T

p

⁢

(

ℝ

n

)

≡

ℝ

n

being the

tangent space

at

p

.

Another viewpoint about

tangent vectors

is that they are also

linear operators

called

derivations

and they act
over scalars

f

:

ℝ

n

→

ℝ

via

p

↦

V

⁢

f

|

p

=

V

⁢

(

p

)

⋅

∇

⁡

f

|

p

.

Let

X

be one of them and

d

⁢

X

|

p

its

Jacobian matrix

evaluated at the point

p

∈

ℝ

n

.
Then, for any other vector field

Y

:

ℝ

n

→

ℝ

n

,

d

⁢

X

|

p

⁢

(

Y

⁢

(

p

)

)

measures how

X

varies in the

direction

Y

at

p

.

We have

d

⁢

X

|

p

⁢

(

Y

⁢

(

p

)

)

=

(

Y

⁢

(

p

)

⋅

∇

⁡

X

1

|

p

,

…

,

Y

⁢

(

p

)

⋅

∇

⁡

X

n

|

p

)

, where

X

=

∑

s

X

s

⁢

e

s

in

components

.
Also, it is obvious that

p

↦

d

⁢

X

|

p

⁢

(

Y

⁢

(

p

)

)

defines a new vector field in

ℝ

n

which is symbolized as

D

Y

⁢

X

We can be consider it as a

bilinear map

D

:

T

⁢

(

ℝ

n

)

×

T

⁢

(

ℝ

n

)

→

T

⁢

(

ℝ

n

)

.

(

X

,

Y

)

↦

D

X

⁢

Y

Further, it is easy to see that for any scalar

f

:

ℝ

n

→

ℝ

1.

D

f

⁢

Y

⁢

X

=

f

⁢

D

Y

⁢

X

2.

D

Y

⁢

(

f

⁢

X

)

=

(

Y

⁢

f

)

⁢

X

+

f

⁢

D

Y

⁢

X

3.

D

X

⁢

Y

-

D

Y

⁢

X

=

[

X

,

Y

]

4.

X

⁢

(

Y

⋅

Z

)

=

D

X

⁢

Y

⋅

Z

+

X

⋅

D

X

⁢

Z

Here we have abbreviated (as usual)

Y

⁢

f

=

Y

⋅

∇

⁡

F

and the

operation

[

X

,

Y

]

is the

Lie bracket

.

This

D

is called the

standard

connection

of

ℝ

n

.

Now, let

M

be a n-dimensional

differentiable manifold

and let

T

⁢

M

be its tangent bundle.
The set of differentiable sections

Γ

⁢

(

M

)

=

{

X

:

M

→

T

⁢

M

}

is a differentiable Lie algebra which is endowed with a differentiable

inner product

g

:

Γ

⁢

(

M

)

×

Γ

⁢

(

M

)

→

ℝ

via

g

⁢

(

X

,

Y

)

|

p

=

X

⁢

(

p

)

⋅

Y

⁢

(

p

)

in each

T

p

⁢

(

M

)

≡

ℝ

n

.

It is possible construct a bilinear operator

∇

∇

:

Γ

⁢

(

M

)

×

Γ

⁢

(

M

)

→

Γ

⁢

(

M

)

compatible

with

g

and which

satisfies

the following properties

1.

∇

f

⁢

Y

⁡

X

=

f

⁢

∇

Y

⁡

X

2.

∇

Y

⁡

(

f

⁢

X

)

=

(

Y

⁢

f

)

⁢

X

+

f

⁢

∇

Y

⁡

X

3.

∇

X

⁡

Y

-

∇

Y

⁡

X

=

[

X

,

Y

]

4.

X

⁢

g

⁢

(

Y

,

Z

)

=

g

⁢

(

∇

X

⁡

Y

,

Z

)

+

g

⁢

(

X

,

∇

X

⁡

Z

)

The

Fundamental Theorem of Riemannian Geometry

establishes that this

∇

exists and it is unique,
and it is called the

Levi-Civita connection

for the metric

g

on

M

.

Now, if one uses a coordinated patch in

M

one has a set of n-coordinated vector fields

∂

1

,

.

.

,

∂

n

meaning

∂

i

=

∂

∂

⁡

u

i

being

u

i

the

coordinate

functions

.
These are also dubbed holonomic derivations.

So it makes sense to speak about the

derivatives

∇

∂

i

⁡

∂

j

and since the

∂

i

are

tangent

which generate at a point

T

p

⁢

(

M

)

, then

∇

∂

i

⁡

∂

j

is also tangent, so there are

n

×

n

numbers (functions if one varies position)

Γ

i

⁢

j

s

which enters
in the

relation

∇

∂

i

⁡

∂

j

=

∑

s

Γ

i

⁢

j

s

⁢

∂

s

.

These

coefficients

Γ

i

⁢

j

s

are called

Christoffel symbols

and an easy calculation shows that

Γ

i

⁢

j

k

=

1

2

⁢

∑

s

g

k

⁢

s

⁢

[

g

s

⁢

j

,

i

+

g

i

⁢

s

,

j

-

g

i

⁢

j

,

s

]

where

g

i

⁢

j

=

g

⁢

(

∂

i

,

∂

j

)

,

g

i

⁢

j

are the entries of the matrix

[

g

i

⁢

j

]

-

1

and

g

i

⁢

j

,

k

=

∂

k

⁡

(

g

i

⁢

j

)

.

Routinely one can check that under a

change of coordinates

u

i

→

w

j

these functions transform as

Γ

¯

k

⁢

l

i

=

∂

⁡

w

i

∂

⁡

u

m

⁢

∂

⁡

u

n

∂

⁡

w

k

⁢

∂

⁡

u

p

∂

⁡

w

l

⁢

Γ

n

⁢

p

m

+

∂

2

⁡

u

p

∂

⁡

w

k

⁢

∂

⁡

w

l

⁢

∂

⁡

w

i

∂

⁡

u

p

here we have used Einstein’s sum convention (

m

,

n

,

p

-sums) and the term

∂

2

⁡

u

p

∂

⁡

w

k

⁢

∂

⁡

w

l

⁢

∂

⁡

w

i

∂

⁡

u

p

shows that the

Γ

k

⁢

l

i

are not tensors.

For a proof please see the last part in:

http://planetmath.org/?op=getobj&from=collab&id=64

http://planetmath.org/?op=getobj&from=collab&id=64

Connection with base vectors.

Let us assume that coordinates

u

i

are referred to a

right-handed

orthogonal

Cartesian system with attached

constant

base vectors

𝐞

i

≡

𝐞

i

and coordinates

w

j

referred to a general curvilinear system attached to a local covariant base vectors

𝐠

j

and local contravariant base vectors

𝐠

k

, both systems embedded in the

Euclidean space

ℝ

n

. We shall also suppose diffeomorphic the transfomation

u

i

↦

w

j

. Then, by definition

𝐠

j

:=

∂

⁡

u

i

∂

⁡

w

j

⁢

𝐞

i

,

𝐠

j

:=

∂

⁡

w

j

∂

⁡

u

i

⁢

𝐞

i

,

(1)

and its

inverses

𝐞

i

=

𝐞

i

=

∂

⁡

u

i

∂

⁡

w

j

⁢

𝐠

j

=

∂

⁡

w

j

∂

⁡

u

i

⁢

𝐠

j

.

(2)

Let us consider

differentiation

of base vectors

𝐠

j

, which may be written from (1),(2)

∂

⁡

𝐠

j

∂

⁡

w

k

=

∂

2

⁡

u

i

∂

⁡

w

j

⁢

∂

⁡

w

k

⁢

𝐞

i

=

∂

2

⁡

u

i

∂

⁡

w

j

⁢

∂

⁡

w

k

⁢

∂

⁡

u

i

∂

⁡

w

s

⁢

𝐠

s

=

∂

2

⁡

u

i

∂

⁡

w

j

⁢

∂

⁡

w

k

⁢

∂

⁡

w

s

∂

⁡

u

i

⁢

𝐠

s

≡

∂

⁡

𝐠

k

∂

⁡

w

j

,

and using the Christoffel symbols this becomes

∂

⁡

𝐠

j

∂

⁡

w

k

=

Γ

j

⁢

k

⁢

s

⁢

𝐠

s

=

Γ

j

⁢

k

r

⁢

𝐠

r

,

(3)

where

Γ

j

⁢

k

⁢

s

=

∂

2

⁡

u

i

∂

⁡

w

j

⁢

∂

⁡

w

k

⁢

∂

⁡

u

i

∂

⁡

w

s

,

Γ

j

⁢

k

r

=

g

r

⁢

s

⁢

Γ

j

⁢

k

⁢

s

.

(4)

Since the

transformation

of covariant and contravariant

metric tensors

are given by

g

j

⁢

k

=

∂

⁡

u

i

∂

⁡

w

j

⁢

∂

⁡

u

l

∂

⁡

w

k

⁢

δ

i

⁢

l

,

g

j

⁢

k

=

∂

⁡

w

j

∂

⁡

u

i

⁢

∂

⁡

w

k

∂

⁡

u

l

⁢

δ

i

⁢

l

,

is easy to see from here that Christoffel symbol

Γ

j

⁢

k

⁢

s

enjoy the property

Γ

j

⁢

k

⁢

s

=

1

2

(

∂

⁡

g

j

⁢

s

∂

⁡

w

k

+

∂

⁡

g

k

⁢

s

∂

⁡

w

j

-

∂

⁡

g

j

⁢

k

∂

⁡

w

s

)

⋅

(5)

In a

similar

way we find for the derivative of the contravariant base vectors

∂

⁡

𝐠

j

∂

⁡

w

k

=

-

Γ

k

⁢

s

j

⁢

𝐠

s

.

(6)

Is easy to show the following results:

Γ

j

⁢

k

⁢

s

=

Γ

k

⁢

j

⁢

s

=

𝐠

s

⋅

∂

⁡

𝐠

k

∂

⁡

w

j

=

𝐠

s

⋅

∂

⁡

𝐠

j

∂

⁡

w

k

,

Γ

j

⁢

k

r

=

Γ

k

⁢

j

r

=

𝐠

r

⋅

∂

⁡

𝐠

j

∂

⁡

w

k

=

𝐠

r

⋅

∂

⁡

𝐠

k

∂

⁡

w

j

=

-

𝐠

j

⋅

∂

⁡

𝐠

r

∂

⁡

w

k

,

Γ

i

⁢

r

i

=

1

2

⁢

g

i

⁢

s

⁢

(

g

i

⁢

s

,

r

+

g

r

⁢

s

,

i

-

g

i

⁢

r

,

s

)

=

1

2

⁢

g

i

⁢

s

⁢

g

i

⁢

s

,

r

=

1

2

⁢

g

⁢

∂

⁡

g

∂

⁡

g

i

⁢

s

⁢

∂

⁡

g

i

⁢

s

∂

⁡

w

r

=

1

g

⁢

∂

⁡

g

∂

⁡

w

r

,

Γ

j

⁢

s

⁢

k

+

Γ

k

⁢

s

⁢

j

=

g

j

⁢

k

,

s

,

comma denoting differentiation with respect to the

curvilinear coordinates

w

j

and

g

=

|

g

j

⁢

k

|

. When the coordinate curves are orthogonal we have the following formulae for the Christoffel symbols:

(repeated indices are not to be summed)

Γ

j

⁢

k

⁢

s

=

0

,

Γ

j

⁢

k

s

=

0

,

(

j

≠

k

≠

s

≠

j

)

,

Γ

i

⁢

i

⁢

r

=

-

1

2

∂

⁡

g

i

⁢

i

∂

⁡

w

r

,

Γ

i

⁢

i

r

=

-

1

2

⁢

g

r

⁢

r

∂

⁡

g

i

⁢

i

∂

⁡

w

r

,

(

r

≠

i

)

,

Γ

i

⁢

r

⁢

i

=

Γ

r

⁢

i

⁢

i

=

1

2

∂

⁡

g

i

⁢

i

∂

⁡

w

r

,

Γ

r

⁢

i

r

=

Γ

i

⁢

r

r

=

1

2

⁢

g

r

⁢

r

∂

⁡

g

r

⁢

r

∂

⁡

w

i

=

1

2

∂

⁡

log

⁡

g

r

⁢

r

∂

⁡

w

i

⋅

Title

Christoffel symbols

Canonical name

ChristoffelSymbols

Date of creation

2013-03-22 15:43:52

Last modified on

2013-03-22 15:43:52

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

24

Author

juanman (12619)

Entry type

Definition

Classification

msc 53B20

Classification

msc 53-01

Synonym

connection coefficients

Related topic

Connection

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