(3) X — Y and Y are pr2 -related.

(4) X and X — Y are ins(y)-related if and only if Y (y) = 0, where

ins(y)(x) = (x, y), ins(y) : M ’ M — N .

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20 Chapter I. Manifolds and Lie groups

3.10. Lemma. Consider vector ¬elds Xi ∈ X(M ) and Yi ∈ X(N ) for i = 1, 2,

and a smooth mapping f : M ’ N . If Xi and Yi are f -related for i = 1, 2, then

also »1 X1 + »2 X2 and »1 Y1 + »2 Y2 are f -related, and also [X1 , X2 ] and [Y1 , Y2 ]

are f -related.

Proof. The ¬rst assertion is immediate. To show the second let h ∈ C ∞ (N, R).

Then by assumption we have T f —¦ Xi = Yi —¦ f , thus:

(Xi (h —¦ f ))(x) = Xi (x)(h —¦ f ) = (Tx f.Xi (x))(h) =

= (T f —¦ Xi )(x)(h) = (Yi —¦ f )(x)(h) = Yi (f (x))(h) = (Yi (h))(f (x)),

so Xi (h —¦ f ) = (Yi (h)) —¦ f , and we may continue:

[X1 , X2 ](h —¦ f ) = X1 (X2 (h —¦ f )) ’ X2 (X1 (h —¦ f )) =

= X1 (Y2 (h) —¦ f ) ’ X2 (Y1 (h) —¦ f ) =

= Y1 (Y2 (h)) —¦ f ’ Y2 (Y1 (h)) —¦ f = [Y1 , Y2 ](h) —¦ f.

But this means T f —¦ [X1 , X2 ] = [Y1 , Y2 ] —¦ f .

3.11. Corollary. If f : M ’ N is a local di¬eomorphism (so (Tx f )’1 makes

sense for each x ∈ M ), then for Y ∈ X(N ) a vector ¬eld f — Y ∈ X(M ) is de¬ned

by (f — Y )(x) = (Tx f )’1 .Y (f (x)). The linear mapping f — : X(N ) ’ X(M ) is

then a Lie algebra homomorphism, i.e. f — [Y1 , Y2 ] = [f — Y1 , f — Y2 ].

3.12. The Lie derivative of functions. For a vector ¬eld X ∈ X(M ) and

f ∈ C ∞ (M, R) we de¬ne LX f ∈ C ∞ (M, R) by

X

d

LX f (x) := dt |0 f (Fl (t, x)) or

X—

—¦ FlX ).

d d

LX f := dt |0 (Flt ) f = dt |0 (f t

Since FlX (t, x) is de¬ned for small t, for any x ∈ M , the expressions above make

sense.

Lemma. dt (FlX )— f = (FlX )— X(f ), in particular for t = 0 we have LX f =

d

t t

X(f ) = df (X).

3.13. The Lie derivative for vector ¬elds. For X, Y ∈ X(M ) we de¬ne

LX Y ∈ X(M ) by

X— X

—¦ Y —¦ FlX ),

d d

LX Y := dt |0 (Flt ) Y dt |0 (T (Fl’t )

= t

and call it the Lie derivative of Y along X.

X—

= (FlX )— LX Y = (FlX )— [X, Y ].

d

Lemma. LX Y = [X, Y ] and dt (Flt ) Y t t

Proof. Let f ∈ C ∞ (M, R) be a function and consider the mapping ±(t, s) :=

Y (FlX (t, x))(f —¦ FlX ), which is locally de¬ned near 0. It satis¬es

s

±(t, 0) = Y (FlX (t, x))(f ),

±(0, s) = Y (x)(f —¦ FlX ),

s

X X

‚ ‚ ‚

‚t ±(0, 0) = ‚t 0 Y (Fl (t, x))(f ) = ‚t 0 (Y f )(Fl (t, x)) = X(x)(Y f ),

X X

‚ ‚ ‚

‚s |0 Y (x)(f —¦ Fls ) = Y (x) ‚s |0 (f —¦ Fls ) = Y (x)(Xf ).

‚s ±(0, 0) =

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

3. Vector ¬elds and ¬‚ows 21

But on the other hand we have

(FlX (u, x))(f —¦ FlX ) =

‚ ‚

‚u |0 ±(u, ’u) ‚u |0 Y

= ’u

T (FlX ) —¦ Y —¦ FlX

‚

‚u |0

= (f ) = (LX Y )x (f ),

’u u

x

so the ¬rst assertion follows. For the second claim we compute as follows:

X—

T (FlX ) —¦ T (FlX ) —¦ Y —¦ FlX —¦ FlX

‚ ‚

‚s |0

‚t (Flt ) Y = ’t ’s s t

= T (FlX ) —¦ T (FlX ) —¦ Y —¦ FlX —¦ FlX

‚

‚s |0

’t ’s s t

= T (FlX ) —¦ [X, Y ] —¦ FlX = (FlX )— [X, Y ].

’t t t

3.14. Lemma. Let X ∈ X(M ) and Y ∈ X(N ) be f -related vector ¬elds for

a smooth mapping f : M ’ N . Then we have f —¦ FlX = FlY —¦f , whenever

t t —

both sides are de¬ned. In particular, if f is a di¬eomorphism we have Flf Y =

t

f ’1 —¦ FlY —¦f .

t

Proof. We have dt (f —¦ FlX ) = T f —¦ dt FlX = T f —¦ X —¦ FlX = Y —¦ f —¦ F lt

d d X

t t t

and f (FlX (0, x)) = f (x). So t ’ f (FlX (t, x)) is an integral curve of the vector

¬eld Y on N with initial value f (x), so we have f (FlX (t, x)) = FlY (t, f (x)) or

f —¦ FlX = FlY —¦f .

t t

3.15. Corollary. Let X, Y ∈ X(M ). Then the following assertions are equiva-

lent

(1) LX Y = [X, Y ] = 0.

(2) (FlX )— Y = Y , wherever de¬ned.

t

(3) Flt —¦ FlY = FlY —¦ FlX , wherever de¬ned.

X

s s t

Proof. (1) ” (2) is immediate from lemma 3.13. To see (2) ” (3) we note

X—

that FlX —¦ FlY = FlY —¦ FlX if and only if FlY = FlX —¦ FlY —¦ FlX = Fl(Flt ) Y by

’t

t s s t s s t s

lemma 3.14; and this in turn is equivalent to Y = (FlX )— Y .

t

3.16. Theorem. Let M be a manifold, let •i : R — M ⊃ U•i ’ M be smooth

mappings for i = 1, . . . , k where each U•i is an open neighborhood of {0} — M

in R — M , such that each •i is a di¬eomorphism on its domain, •i = IdM , and

t 0

j j ’1

—¦ (•t ) —¦ •j —¦ •i .

i ’1

‚ i i j i

‚t 0 •t = Xi ∈ X(M ). We put [• , • ]t = [•t , •t ] := (•t ) t t

Then for each formal bracket expression P of lenght k we have

|0 P (•1 , . . . , •k )

‚

for 1 ¤ < k,

0= t t

‚t

1 ‚k 1 k

k! ‚tk |0 P (•t , . . . , •t ) ∈ X(M )

P (X1 , . . . , Xk ) =

in the sense explained in step 2 of the proof. In particular we have for vector

¬elds X, Y ∈ X(M )

Y X Y X

‚

‚t 0 (Fl’t —¦ Fl’t —¦ Flt —¦ Flt ),

0=

1 ‚2 Y X Y X

2 ‚t2 |0 (Fl’t —¦ Fl’t —¦ Flt —¦ Flt ).

[X, Y ] =

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

22 Chapter I. Manifolds and Lie groups

Proof. Step 1. Let c : R ’ M be a smooth curve. If c(0) = x ∈ M , c (0) =

0, . . . , c(k’1) (0) = 0, then c(k) (0) is a well de¬ned tangent vector in Tx M which

is given by the derivation f ’ (f —¦ c)(k) (0) at x.

For we have

k

k

(k) (k)

(f —¦ c)(j) (0)(g —¦ c)(k’j) (0)

((f.g) —¦ c) (0) = ((f —¦ c).(g —¦ c)) (0) = j

j=0

= (f —¦ c)(k) (0)g(x) + f (x)(g —¦ c)(k) (0),

since all other summands vanish: (f —¦ c)(j) (0) = 0 for 1 ¤ j < k.

Step 2. Let • : R — M ⊃ U• ’ M be a smooth mapping where U• is an open

neighborhood of {0} — M in R — M , such that each •t is a di¬eomorphism on

its domain and •0 = IdM . We say that •t is a curve of local di¬eomorphisms

though IdM .

‚j 1 ‚k

From step 1 we see that if ‚tj |0 •t = 0 for all 1 ¤ j < k, then X := k! ‚tk |0 •t

is a well de¬ned vector ¬eld on M . We say that X is the ¬rst non-vanishing

derivative at 0 of the curve •t of local di¬eomorphisms. We may paraphrase this

as (‚t |0 •— )f = k!LX f .

k

t

Claim 3. Let •t , ψt be curves of local di¬eomorphisms through IdM and let

f ∈ C ∞ (M, R). Then we have

k

j k’j

— —

•— )f (‚t |0 ψt )(‚t |0 •— )f.

—

k

k k

‚t |0 (•t —¦ ψt ) f = ‚t |0 (ψt —¦ =

t t

j

j=0

Also the multinomial version of this formula holds:

k! j j

‚t |0 (•1 —¦ . . . —¦ •t )— f = (‚t 1 |0 (•t )— ) . . . (‚t 1 |0 (•1 )— )f.

k

t t

j1 ! . . . j !

j1 +···+j =k

We only show the binomial version. For a function h(t, s) of two variables we

have

k

j k’j

k

k

‚t h(t, t) = ‚t ‚s h(t, s)|s=t ,

j

j=0

since for h(t, s) = f (t)g(s) this is just a consequence of the Leibnitz rule, and

linear combinations of such decomposable tensors are dense in the space of all

functions of two variables in the compact C ∞ -topology, so that by continuity

the formula holds for all functions. In the following form it implies the claim:

k

j k’j

k

k

‚t |0 f (•(t, ψ(t, x))) = ‚t ‚s f (•(t, ψ(s, x)))|t=s=0 .

j

j=0

Claim 4. Let •t be a curve of local di¬eomorphisms through IdM with ¬rst

k

non-vanishing derivative k!X = ‚t |0 •t . Then the inverse curve of local di¬eo-

morphisms •’1 has ¬rst non-vanishing derivative ’k!X = ‚t |0 •’1 .

k

t t

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

3. Vector ¬elds and ¬‚ows 23

For we have •’1 —¦ •t = Id, so by claim 3 we get for 1 ¤ j ¤ k

t

j

j j’i

‚t |0 (•’1 (‚t |0 •— )(‚t (•’1 )— )f =

— j i

—¦ •t ) f =

0= t t

t

i

i=0

j j

= ‚t |0 •— (•’1 )— f + •— ‚t |0 (•’1 )— f,

t

t 0

0

j j

i.e. ‚t |0 •— f = ’‚t |0 (•’1 )— f as required.

t

t

Claim 5. Let •t be a curve of local di¬eomorphisms through IdM with ¬rst

m

non-vanishing derivative m!X = ‚t |0 •t , and let ψt be a curve of local di¬eo-

n

morphisms through IdM with ¬rst non-vanishing derivative n!Y = ‚t |0 ψt .

Then the curve of local di¬eomorphisms [•t , ψt ] = ψt —¦ •’1 —¦ ψt —¦ •t has ¬rst

’1

t

non-vanishing derivative

m+n

|0 [•t , ψt ].

(m + n)![X, Y ] = ‚t

From this claim the theorem follows.

By the multinomial version of claim 3 we have

AN f : = ‚t |0 (ψt —¦ •’1 —¦ ψt —¦ •t )— f

’1

N

t

N! j

(‚t |0 •— )(‚t |0 ψt )(‚t |0 (•’1 )— )(‚t |0 (ψt )— )f.

’1

—

i k

= t

t

i!j!k! !

i+j+k+ =N

Let us suppose that 1 ¤ n ¤ m, the case m ¤ n is similar. If N < n all

summands are 0. If N = n we have by claim 4

AN f = (‚t |0 •— )f + (‚t |0 ψt )f + (‚t |0 (•’1 )— )f + (‚t |0 (ψt )— )f = 0.

’1

—

n n n n

t

t

If n < N ¤ m we have, using again claim 4:

N! j

(‚t |0 ψt )(‚t |0 (ψt )— )f + δN (‚t |0 •— )f + (‚t |0 (•’1 )— )f

’1

— m m m

AN f = t

t

j! !

j+ =N

’1

= (‚t |0 (ψt —¦ ψt )— )f + 0 = 0.

N

Now we come to the di¬cult case m, n < N ¤ m + n.

AN f = ‚t |0 (ψt —¦ •’1 —¦ ψt )— f +

’1

(‚t |0 •— )(‚t ’m |0 (ψt —¦ •’1 —¦ ψt )— )f

’1

N N

N m

t t

t

m

+ (‚t |0 •— )f,

N

(1) t

by claim 3, since all other terms vanish, see (3) below. By claim 3 again we get:

N! j

‚t |0 (ψt —¦ •’1 —¦ ψt )— f =

’1

(‚t |0 ψt )(‚t |0 (•’1 )— )(‚t |0 (ψt )— )f

’1

—

N k

t t

j!k! !

j+k+ =N

j ’1

(‚t ’m |0 ψt )(‚t |0 (•’1 )— )f

(‚t |0 ψt )(‚t |0 (ψt )— )f +

— —

N N N m

(2) = t

j m

j+ =N

(‚t |0 (•’1 )— )(‚t ’m |0 (ψt )— )f + ‚t |0 (•’1 )— f

’1

N N

m N

+ t t

m

(‚t ’m |0 ψt )m!L’X f + m!L’X (‚t ’m |0 (ψt )— )f

’1

—

N N

N N

=0+ m m

+ ‚t |0 (•’1 )— f

N

t

= δm+n (m + n)!(LX LY ’ LY LX )f + ‚t |0 (•’1 )— f

N N

t

= δm+n (m + n)!L[X,Y ] f + ‚t |0 (•’1 )— f

N N

t

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

24 Chapter I. Manifolds and Lie groups

From the second expression in (2) one can also read o¬ that

‚t ’m |0 (ψt —¦ •’1 —¦ ψt )— f = ‚t ’m |0 (•’1 )— f.

’1

N N

(3) t t

If we put (2) and (3) into (1) we get, using claims 3 and 4 again, the ¬nal result

which proves claim 5 and the theorem:

AN f = δm+n (m + n)!L[X,Y ] f + ‚t |0 (•’1 )— f

N N

t

(‚t |0 •— )(‚t ’m |0 (•’1 )— )f + (‚t |0 •— )f

N N

m N

+ t

t t

m

= δm+n (m + n)!L[X,Y ] f + ‚t |0 (•’1 —¦ •t )— f

N N

t

N

= δm+n (m + n)!L[X,Y ] f + 0.