Cartan structural equations

Cartan structural equations

To deduce the

Cartan structural equations

in a coordinated frame we are going to use the

definition

of the

Christoffel symbols

(connection coefficients) and where we always are going to use the Einstein sum convention:

∇

∂

i

⁡

∂

j

=

Γ

s

i

⁢

j

⁢

∂

s

and the curvature tensor

R

⁢

(

X

,

Y

)

⁢

Z

=

∇

X

⁡

∇

Y

⁡

Z

-

∇

Y

⁡

∇

X

⁡

Z

-

∇

[

X

,

Y

]

⁡

Z

where

X

,

Y

,

Z

are any three vector fields in a

riemannian manifold

ℳ

with the

Levi-Civita connection

∇

.

First, we define through the relation

∇

X

⁡

∂

i

=

ω

s

i

⁢

(

X

)

⁢

∂

s

a set of scalar function

ω

s

i

which are easily to see that they actually are 1-forms. We observe that

ω

s

i

⁢

(

∂

j

)

=

Γ

s

i

⁢

j

.

They satisfy skew-symmetry rule:

ω

s

⁢

i

=

-

ω

i

⁢

s

,
which arises from the covariant constancy of the metric tensor

g

k

⁢

l

i.e.

0

=

∇

X

⁡

g

k

⁢

l

=

∇

X

⁡

⟨

∂

k

,

∂

l

⟩

=

⟨

∇

X

⁡

∂

k

,

∂

l

⟩

+

⟨

∂

k

,

∇

X

⁡

∂

l

⟩

=

⟨

ω

s

k

⁢

(

X

)

⁢

∂

s

,

∂

l

⟩

+

⟨

∂

k

,

ω

s

l

⁢

(

X

)

⁢

∂

s

⟩

=

ω

s

k

⁢

(

X

)

⁢

g

s

⁢

l

+

ω

s

l

⁢

(

X

)

⁢

g

k

⁢

s

0

=

ω

l

⁢

k

⁢

(

X

)

+

ω

k

⁢

l

⁢

(

X

)

that last equation is valid for each vector field

X

, then

ω

l

⁢

k

=

-

ω

k

⁢

l

.

Next we define through the relation

R

⁢

(

X

,

Y

)

⁢

∂

i

=

Ω

s

i

⁢

(

X

,

Y

)

⁢

∂

s

the scalars

Ω

s

i

⁢

(

X

,

Y

)

which are the so called

connection

2-forms.
That they are really 2-forms is an easy caligraphic exercise.

Now by the use of the

Riemann curvature tensor

above we see

R

⁢

(

X

,

Y

)

⁢

∂

i

=

∇

X

⁡

∇

Y

⁡

∂

i

-

∇

Y

⁡

∇

X

⁡

∂

i

-

∇

[

X

,

Y

]

⁡

∂

i

=

∇

X

⁡

(

ω

s

i

⁢

(

Y

)

⁢

∂

s

)

-

∇

Y

⁡

(

ω

s

i

⁢

(

X

)

⁢

∂

s

)

-

ω

s

i

⁢

[

X

,

Y

]

⁢

∂

s

=

X

(

ω

s

i

(

Y

)

)

∂

s

+

ω

s

i

(

Y

)

∇

X

∂

s

-

Y

(

ω

s

i

(

X

)

∂

s

-

ω

s

i

(

X

)

∇

Y

∂

s

-

ω

s

i

[

X

,

Y

]

∂

s

=

X

(

ω

s

i

(

Y

)

)

∂

s

+

ω

s

i

(

Y

)

ω

t

s

(

X

)

∂

t

-

Y

(

ω

s

i

(

X

)

∂

s

-

ω

s

i

(

X

)

ω

t

s

(

Y

)

∂

t

-

ω

s

i

[

X

,

Y

]

∂

s

=

[

X

⁢

(

ω

s

i

⁢

(

Y

)

)

+

ω

t

i

⁢

(

Y

)

⁢

ω

s

t

⁢

(

X

)

-

Y

⁢

(

ω

s

i

⁢

(

X

)

)

-

ω

t

i

⁢

(

X

)

⁢

ω

s

t

⁢

(

Y

)

-

ω

s

i

⁢

[

X

,

Y

]

]

⁢

∂

s

Ω

s

i

⁢

(

X

,

Y

)

⁢

∂

s

=

[

X

⁢

(

ω

s

i

⁢

(

Y

)

)

-

Y

⁢

(

ω

s

i

⁢

(

X

)

)

-

ω

s

i

⁢

[

X

,

Y

]

+

ω

s

t

⁢

(

X

)

⁢

ω

t

i

⁢

(

Y

)

-

ω

s

t

⁢

(

Y

)

⁢

ω

t

i

⁢

(

X

)

]

⁢

∂

s

In this last relation we recognize -in the first three terms- the

exterior derivative

of

ω

s

i

evaluated at

(

X

,

Y

)

i.e.

d

⁢

ω

s

i

⁢

(

X

,

Y

)

=

X

⁢

(

ω

s

i

⁢

(

Y

)

)

-

Y

⁢

(

ω

s

i

⁢

(

X

)

)

-

ω

s

i

⁢

[

X

,

Y

]

and in the last two terms its wedge product

ω

s

t

∧

ω

t

i

⁢

(

X

,

Y

)

=

ω

s

t

⁢

(

X

)

⁢

ω

t

i

⁢

(

Y

)

-

ω

s

t

⁢

(

Y

)

⁢

ω

t

i

⁢

(

X

)

all these for any two fields

X

,

Y

. Hence

Ω

s

i

=

d

⁢

ω

s

i

+

ω

s

t

∧

ω

t

i

which is called the second Cartan structural equation for the coordinated frame field

∂

i

.

More interesting things happen in an an-holonomic basis.

Title

Cartan structural equations

Canonical

name

CartanStructuralEquations

Date of creation

2013-03-22 17:35:46

Last modified on

2013-03-22 17:35:46

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

12

Author

juanman (12619)

Entry type

Result

Classification

msc 53A45

Classification

msc 58A12

Classification

msc 58A10

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