proof of d⁢α⁢(X,Y)=X⁢(α⁢(Y)) - Y⁢(α⁢(X)) - α⁢([X,Y]) (global coordinate-free)

proof of

d

⁢

α

⁢

(

X

,

Y

)

=

X

⁢

(

α

⁢

(

Y

)

)

-

Y

⁢

(

α

⁢

(

X

)

)

-

α

⁢

(

[

X

,

Y

]

)

(global coordinate-free)

In the coordinate-free approach to

differential geometry

, this statement has a rather different ontological status than that which it has in the

coordinate

approach. While, in the latter, it is a theorem, pure and simple, in the former, it is a

combination

of definition and theorem. That is to say, one uses the equation to define the

exterior derivative

of a

one-form

but, in order for this definition to be valid, one must first show that a. the right-hand side of this identity satisfies the requisite properties of antisymmetry and bilinearity and b. the operator

d

implicitly defined is linear.

1

Proof of a.

For

simplicity

, we will break down this demontration into three parts.

1.1

Antisymmetry

It is easy enough to see that both sides of the equation change by a sign upon interchanging

X

with

Y

as a consequence of antisymmetry of two-forms and of the

Lie bracket

:

Y

⁢

(

α

⁢

(

X

)

)

-

X

⁢

(

α

⁢

(

Y

)

)

-

α

⁢

(

[

Y

,

X

]

)

=

-

X

⁢

(

α

⁢

(

Y

)

)

+

Y

⁢

(

α

⁢

(

X

)

)

⁢

X

+

α

⁢

(

[

X

,

Y

]

)

=

-

(

X

⁢

(

α

⁢

(

Y

)

)

-

Y

⁢

(

α

⁢

(

X

)

)

-

α

⁢

(

[

X

,

Y

]

)

)

1.2

Distributivity over addition

For simplicity, we can break down the study of how our expression behaves under linear operations into the study of how it behaves under

addition

and under

scaling

. Because of antisymmetry, it suffices to check linearity in only one of the two

arguments

:

X

⁢

(

α

⁢

(

Y

+

Z

)

)

-

(

Y

+

Z

)

⁢

(

α

⁢

(

X

)

)

-

α

⁢

(

[

X

,

Y

+

Z

]

)

=

X

⁢

(

α

⁢

(

Y

)

)

+

X

⁢

(

α

⁢

(

Z

)

)

-

Y

⁢

(

α

⁢

(

X

)

)

-

Z

⁢

(

α

⁢

(

X

)

)

-

α

⁢

(

[

X

,

Y

]

)

-

α

⁢

(

[

X

,

Z

]

)

=

(

X

⁢

(

α

⁢

(

Y

)

)

-

Y

⁢

(

α

⁢

(

X

)

)

-

α

⁢

(

[

X

,

Y

]

)

)

+

(

X

⁢

(

α

⁢

(

Z

)

)

-

Z

⁢

(

α

⁢

(

X

)

)

-

α

⁢

(

[

X

,

Z

]

)

)

1.3

Scaling

Finally, we verify that our expression transforms properly under rescaling. Again, by antisymmetry, it suffices to look at only one argument.

X

⁢

(

α

⁢

(

s

⁢

Y

)

)

-

(

s

⁢

Y

)

⁢

(

α

⁢

(

X

)

)

-

α

⁢

(

[

X

,

s

⁢

Y

]

)

=

s

⁢

X

⁢

(

α

⁢

(

Y

)

)

+

X

⁢

(

s

)

⁢

α

⁢

(

Y

)

-

s

⁢

Y

⁢

(

α

⁢

(

X

)

)

-

s

⁢

α

⁢

(

[

X

,

Y

]

)

-

X

⁢

(

s

)

⁢

α

⁢

(

Y

)

Cancelling a term and

factoring

out a common ”

s

”,

=

s

⁢

(

X

⁢

(

α

⁢

(

Y

)

)

-

Y

⁢

(

α

⁢

(

X

)

)

-

α

⁢

(

[

X

,

Y

]

)

)

Hence this expression behaves like a two-form should. By the principle ”If it looks like a duck and quacks like a duck, it must be a duck”, we conclude that the expression indeed specifies a two-form.

2

Proof of b.

As before, we will break the proof of linearity into two steps. Note that here the term ”scalar” simply means a

real number

, not a function on the

manifold

.

2.1

Distributivity over addition

d

⁢

(

α

+

β

)

⁢

(

X

,

Y

)

=

X

⁢

(

(

α

+

β

)

⁢

(

Y

)

)

-

Y

⁢

(

(

α

+

β

)

⁢

(

X

)

)

-

(

α

+

β

)

⁢

(

[

X

,

Y

]

)

=

X

⁢

(

α

⁢

(

Y

)

)

+

X

⁢

(

β

⁢

(

Y

)

)

-

Y

⁢

(

α

⁢

(

X

)

)

-

Y

⁢

(

β

⁢

(

X

)

)

-

β

⁢

(

[

X

,

Y

]

)

-

α

⁢

(

[

X

,

Y

]

)

=

d

⁢

α

⁢

(

X

,

Y

)

+

d

⁢

β

⁢

(

X

,

Y

)

2.2

Scaling

d

⁢

(

s

⁢

α

)

⁢

(

X

,

Y

)

=

X

⁢

(

(

s

⁢

α

)

⁢

(

Y

)

)

-

Y

⁢

(

(

s

⁢

α

)

⁢

(

X

)

)

-

(

s

⁢

α

)

⁢

(

[

X

,

Y

]

)

=

s

⁢

X

⁢

(

α

⁢

(

Y

)

)

-

s

⁢

Y

⁢

(

α

⁢

(

X

)

)

-

s

⁢

α

⁢

(

[

X

,

Y

]

)

=

s

⁢

d

⁢

α

⁢

(

X

,

Y

)

Title

proof of

d

⁢

α

⁢

(

X

,

Y

)

=

X

⁢

(

α

⁢

(

Y

)

)

-

Y

⁢

(

α

⁢

(

X

)

)

-

α

⁢

(

[

X

,

Y

]

)

(global coordinate-free)

Canonical name

ProofOfDalphaXYXalphaYYalphaXalphaXYglobalCoordinatefree

Date of creation

2013-03-22 15:34:04

Last modified on

2013-03-22 15:34:04

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 53-00

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