algebra of vector fields

8/8/2019 Algebra of Vector Fields

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INTRODUCTION TO MANIFOLDS — III

Algebra of vector fields. Lie derivative (s ).

1. Notations. The space of all C ∞ -smooth vector elds on a manifold M isdenoted by X (M ). If v ∈ X (M ) is a vector eld, then v(x ) ∈ T x M R n is itsvalue at a point x ∈M .

The flow of a vector eld v is denoted by vt :

∀t ∈ R vt : M → M

is a smooth map (automorphism) of M taking a point x ∈ M into the pointv t (x ) ∈M which is the t-endpoint of an integral trajectory for the eld v, startingat the point x .♣ Problem 1. Prove that the ow maps for a eld v on a compact manifold M form a one-parameter group:

∀t, s ∈ R vt + s = vt ◦ vs = vs ◦ vt ,

and all vt are diffeomorphisms of M .

♣ Problem 2. What means the formula

dds s =0

vs = v

and is it true?2. Star conventions. The space of all C ∞ -smooth functions is denoted byC ∞ (M ). If F : M → M is a smooth map (not necessary a diffeomorphism), thenthere appears a contravariant map

F ∗ : C ∞ (M ) → C ∞ (M ), F ∗ : f → F ∗ f, F ∗ (x ) = f (F (x )) .

If F : M → N is a smooth map between two different manifolds, then

F ∗ : C ∞ (N ) → C ∞ (M ).

Note that the direction of the arrows is reversed!

♣ Problem 3. Prove that C ∞ (M ) is a commutative associative algebra overR with respect to pointwise addition, subtraction and multiplication of functions.Prove that F ∗ is a homomorphism of this algebra (preserves all the operations). If F : M → N , then F ∗ : C ∞ (N ) → C ∞ (M ) is a homomorphism also.

Typeset by A M S -TEX

1



2 ALGEBRA OF VECTOR FIELDS. LIE DERIVATIVE(S).

Another star is associated with differentials: if F : M 1 → M 2 is a diffeomor-phism , then

F ∗ : X (M 1) → X (M 2), v → F ∗ v, (F ∗ v)(x ) =∂F ∂x

(x ) · v(x ),

is a covariant (acts in the same direction) map which is:(1) additive: F ∗ (v + w) = F ∗ v + F ∗ w;(2) homogeneous: ∀f ∈ C ∞ (M ) F ∗ (fv ) = ( F ∗ )− 1 f · F ∗ v. (explain this for-

mula!),Why F ∗ is in general not dened, if F is just a smooth map and not a diffeomor-phism?

3. Vector elds as differential operators.

♥ Denition. If v ∈X (M ), then the Lie derivative L v is

L v : C ∞ (M ) → C ∞ (M ), L v f = limt → 0

1t

(v t )∗ f − f .

In coordinates:

L v f (a ) = limt → 0

f (a + tv + o(t)) − f (a )t

=n

j =1

∂f ∂x j

(a )vj .

Properties of the Lie derivative:(1) L v : C ∞ (M ) → C ∞ (M ) is a linear operator :

L v (f + g) = L v f + L v g, L v (λf ) = λL v f ;

(2) the Leibnitz identity holds:L v (fg ) = L v f · g + f · L v g.

(3) The Lie derivative linearly depends on v:

∀f ∈C ∞ (M ), v, w ∈X (M ) L f v = fL v , L v + w = L v + L w .

♣ Problem 4. Prove that the Lie derivative is local: for any function f ∈C ∞ (M )and any vector eld v the value L v f (a ) depends only on v(a ), so that for any othereld w such that w(a ) = v(a ), L v f (a ) = L w f (a ).

Theorem. Any differential operator, that is, a map D : C ∞

(M ) → C ∞

(M ) satis-fying

D (f + g) = Df + Dg, D (λf ) = λDf, D (fg ) = f Dg + Df · g, (DiffOper )



INTRODUCTION TO MANIFOLDS — III 3

is a Lie derivative along a certain vector eld v ∈X .

Idea of the proof. In local coordinates any function can be written as

f (x ) = f (a ) +n

k =1

(x k − a k )f k (x ), f k (a ) =∂f

∂x k(a ).

Applying the Leibnitz identity, we conclude thatD

=L

v , wherev

is the vectoreld with components vk = D (x k − a k ).

Thus sometimes the notation

v =n

k =1

vk (x )∂

∂x k

is used: such a notation understood as a differential oper-ator, is a vector eld from the geometric point of view.

4. Commutator. If v, w ∈ X (M ), then D = L v L w − L w L v is a differentialoperator. Indeed, the Leibnitz formula is trivially satised, therefore D = L u ,where u ∈X (M ).♣ Problem 5. Check it!♥ Denition. If L u = L v L w − L w L v , then u is a commutator of v and w:

u = [ v, w ].

In coordinates:

L u f = L v

k

∂f ∂x k

wk − L w (· · · ) =

k,j

∂ 2f ∂x k ∂x j

wk vj +∂f ∂x k

∂w k

∂x jvj − (· · · ) =

j k

∂w j

∂x kvk −

k

∂v j

∂x kwk ∂f

∂x j,

therefore

[v, w ] =j k

∂w j

∂x kvk −

k

∂v j

∂x kwk

∂ ∂x j

.

♣ Problem 6.∂ 2

∂s∂t s =0 ,t =0(f ◦ v t ◦ w s − f ◦ ws ◦ v t ) = L [v,w ]f.

♣ Problem 7.[v, w ] = − [w, v ].

♣ Problem 8. Prove the Jacobi identity

[[u, v ], w ] + [[v, w ], u ] + [[w, u ], v ] = 0 .



4 ALGEBRA OF VECTOR FIELDS. LIE DERIVATIVE(S).

5. Lie derivation of vector elds.

♥ Denition. The Lie derivative of a vector eld w along another eld v is

L v w = limt → 0

1t

(v t∗w − w ◦ v t ).

♣ Problem 9. Check that the above denition makes sense.

Properties of the Lie derivative: if v, w ∈ X (M ), f ∈C ∞ (M ), then:

(1) L v v = 0 .(2) L v is linear map from X (M ) to itself.(3) L v (fw ) = ( L v f )w + f L v w (the Leibnitz property ).

Theorem.L v w = [v, w ] (or [w, v ]?)

Proof. Let

a =∂ 2

∂s∂t s =0 ,t =0(f ◦ vt ◦ ws − f ◦ ws ◦ vt ).

Thena = lim

t → 0

1t

∂ ∂s s =0

(· · · ) ,

but∂ ∂s s =0

vt ◦ ws = vt∗w,

therefore∂ ∂s s =0

f ◦ v t ◦ ws = L v t∗

w f,

while∂

∂s s =0f ◦ ws ◦ vt = L w ◦ v t f,

and nallya = L L v w f.

♣ Problem 10. Is the Lie derivative of a vector eld local in the following sense:if two elds v1 , v2 ∈ X (M ) are coinciding on an open neighborhood of a certainpoint a ∈M , then for any other eld v ∈X (M )

(L v 1 w)(a ) = ( L v 2 w)(a ).

Is it true that the above value is determined by the (common) value vi (a )?

/black/users2/yakov/pub, lename 4.pptE-mail address : [email protected], [email protected]

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