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(2) X — Y and X are pr1 -related.
(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 .

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


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