937: Lie Bracket (Commutator) of $C^\infty$ Vectors Fields on $C^\infty$ Manifold with Boundary|T.B.P.

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definition of Lie bracket (commutator) of

C^{\infty}

vectors fields on

C^{\infty}

manifold with boundary

Topics

About: $C^{\infty}$ manifold

Starting Context

The reader knows a definition of $C^{\infty}$ vectors field on $C^{\infty}$ manifold with boundary.

Target Context

The reader will have a definition of Lie bracket (commutator) of $C^{\infty}$ vectors fields on $C^{\infty}$ manifold with boundary.

Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

Main Body

1: Structured Description

Here is the rules of Structured Description.

Entities:

M

\in {the C^{\infty} manifolds with boundary}

(T M, M, π)

= the tangent vectors bundle over M

V_{1}

: M \to T M

\in {the C^{\infty} vectors fields on M}

V_{2}

: M \to T M

\in {the C^{\infty} vectors fields on M}

* [V_{1}, V_{2}]

= V_{1} V_{2} - V_{2} V_{1}

\in {the C^{\infty} vectors fields on M}

//

Conditions:
//

2: Note

Let us see that

[V_{1}, V_{2}]

is indeed a

C^{\infty}

vectors field on

M

.

Let us see that at each

m \in M

[V_{1}, V_{2}]_{m}

is indeed a tangent vector, which is about that it is a derivation of

C^{\infty}

functions.

Let

f \in C^{\infty} (M)

be any.

1st, let us understand the meaning of

V_{1} V_{2} - V_{2} V_{1}

. By the proposition that any vectors field is

C^{\infty}

if and only if its operation result on any

C^{\infty}

function is

C^{\infty}

V_{2} (f), V_{1} (f) \in C^{\infty} (M)

, and so,

V_{1} (V_{2} (f)) - V_{2} (V_{1} (f))

is possible. For

V_{1} (V_{2} (f))

m

, we need

V_{2}

as a vectors field (not just a tangent vector at

m

), because we need

V_{2} (f)

as a function (not just a number at

m

), but we need

V_{1}

only as a tangent vector at

m

, and likewise for

V_{2} (V_{1} (f))

, so, we can write

[V_{1}, V_{2}]_{m}

V_{1, m} V_{2} - V_{2, m} V_{1}

[V_{1}, V_{2}]_{m} (f_{1} f_{2}) = (V_{1, m} V_{2} - V_{2, m} V_{1}) (f_{1} f_{2}) = V_{1, m} V_{2} (f_{1} f_{2}) - V_{2, m} V_{1} (f_{1} f_{2}) = V_{1, m} (V_{2} (f_{1}) f_{2} + f_{1} V_{2} (f_{2})) - V_{2, m} (V_{1} (f_{1}) f_{2} + f_{1} V_{1} (f_{2})) = V_{1, m} (V_{2} (f_{1})) f_{2} + V_{2} (f_{1}) V_{1, m} (f_{2}) + V_{1, m} (f_{1}) V_{2} (f_{2}) + f_{1} V_{1, m} (V_{2} (f_{2})) - (V_{2, m} (V_{1} (f_{1})) f_{2} + V_{1} (f_{1}) V_{2, m} (f_{2}) + V_{2, m} (f_{1}) V_{1} (f_{2}) + f_{1} V_{2, m} (V_{1} (f_{2}))) = V_{1, m} (V_{2} (f_{1})) f_{2} - V_{2, m} (V_{1} (f_{1})) f_{2} - (f_{1} V_{2, m} (V_{1} (f_{2})) - f_{1} V_{1, m} (V_{2} (f_{2}))) = (V_{1, m} (V_{2} (f_{1})) - V_{2, m} (V_{1} (f_{1}))) f_{2} - (f_{1} (V_{2, m} (V_{1} (f_{2})) - V_{1, m} (V_{2} (f_{2})))) = (V_{1, m} V_{2} - V_{2, m} V_{1}) (f_{1}) f_{2} - f_{1} (V_{2, m} V_{1} - V_{1, m} V_{2}) (f_{2}) = (V_{1, m} V_{2} - V_{2, m} V_{1}) (f_{1}) f_{2} + f_{1} (V_{1, m} V_{2} - V_{2, m} V_{1}) (f_{2}) = [V_{1}, V_{2}] (f_{1}) f_{2} + f_{1} [V_{1}, V_{2}] (f_{2})

[V_{1}, V_{2}]

C^{\infty}

, by the proposition that any vectors field is

C^{\infty}

if and only if its operation result on any

C^{\infty}

function is

C^{\infty}

.

With the Lie bracket, the set of the

C^{\infty}

vectors fields on

M

Γ (T M)

, becomes a

R

Lie algebra:

Γ (T M)

is an

R

vectors space; for each

V_{1}, V_{2}, V_{3} \in Γ (T M)

and each

r_{1}, r_{2} \in R

, 1)

[r_{1} V_{1} + r_{2} V_{2}, V_{3}] = (r_{1} V_{1} + r_{2} V_{2}) V_{3} - V_{3} (r_{1} V_{1} + r_{2} V_{2}) = r_{1} V_{1} V_{3} + r_{2} V_{2} V_{3} - r_{1} V_{3} V_{1} - r_{2} V_{3} V_{2} = r_{1} (V_{1} V_{3} - V_{3} V_{1}) + r_{2} (V_{2} V_{3} - V_{3} V_{2}) = r_{1} [V_{1}, V_{3}] + r_{2} [V_{2}, V_{3}]

\land

[V_{3}, r_{1} V_{1} + r_{2} V_{2}] = V_{3} (r_{1} V_{1} + r_{2} V_{2}) - (r_{1} V_{1} + r_{2} V_{2}) V_{3} = r_{1} V_{3} V_{1} - r_{1} V_{1} V_{3} + r_{2} V_{3} V_{2} - r_{2} V_{2} V_{3} = r_{1} (V_{3} V_{1} - V_{1} V_{3}) + r_{2} (V_{3} V_{2} - V_{2} V_{3}) = r_{1} [V_{3}, V_{1}] + r_{2} [V_{3}, V_{2}]

; 2)

[V_{2}, V_{1}] = V_{2} V_{1} - V_{1} V_{2} = - (V_{1} V_{2} - V_{2} V_{1}) = - [V_{1}, V_{2}]

; 3)

\sum_{c y c l i c} [V_{1}, [V_{2}, V_{3}]] = [V_{1}, [V_{2}, V_{3}]] + [V_{3}, [V_{1}, V_{2}]] + [V_{2}, [V_{3}, V_{1}]] = [V_{1}, V_{2} V_{3} - V_{3} V_{2}] + [V_{3}, V_{1} V_{2} - V_{2} V_{1}] + [V_{2}, V_{3} V_{1} - V_{1} V_{3}] = V_{1} (V_{2} V_{3} - V_{3} V_{2}) - (V_{2} V_{3} - V_{3} V_{2}) V_{1} + V_{3} (V_{1} V_{2} - V_{2} V_{1}) - (V_{1} V_{2} - V_{2} V_{1}) V_{3} + V_{2} (V_{3} V_{1} - V_{1} V_{3}) - (V_{3} V_{1} - V_{1} V_{3}) V_{2} = 0

.

A major reason for our taking

[V_{1}, V_{2}]

instead of just

V_{1} V_{2}

is that

V_{1} V_{2}

is not any

C^{\infty}

vectors field: it is not any derivation: just taking a

C^{\infty}

function and producing a

C^{\infty}

function does not make it a

C^{\infty}

vectors field.

References

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T.B.P.

2025-01-07