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3.17. Theorem. Let X1 , . . . , Xm be vector ¬elds on M de¬ned in a neighbor-
hood of a point x ∈ M such that X1 (x), . . . , Xm (x) are a basis for Tx M and
[Xi , Xj ] = 0 for all i, j.

Then there is a chart (U, u) of M centered at x such that Xi |U = ‚ui .
Proof. For small t = (t1 , . . . , tm ) ∈ Rm we put

f (t1 , . . . , tm ) = (FlX1 —¦ · · · —¦ FlXm )(x).
tm
t1

By 3.15 we may interchange the order of the ¬‚ows arbitrarily. Therefore
Xi
—¦ FlX1 —¦ · · · )(x) = Xi ((Flx11 —¦ · · · )(x)).
1
, tm ) =
‚ ‚
‚ti f (t , . . . ‚ti (Flti t1 t

So T0 f is invertible, f is a local di¬eomorphism, and its inverse gives a chart
with the desired properties.
3.18. Distributions. Let M be a manifold. Suppose that for each x ∈ M
we are given a sub vector space Ex of Tx M . The disjoint union E = x∈M Ex
is called a distribution on M . We do not suppose, that the dimension of Ex is
locally constant in x.
Let Xloc (M ) denote the set of all locally de¬ned smooth vector ¬elds on M ,
i.e. Xloc (M ) = X(U ), where U runs through all open sets in M . Furthermore
let XE denote the set of all local vector ¬elds X ∈ Xloc (M ) with X(x) ∈ Ex
whenever de¬ned. We say that a subset V ‚ XE spans E, if for each x ∈ M the
vector space Ex is the linear span of the set {X(x) : X ∈ V}. We say that E is a
smooth distribution if XE spans E. Note that every subset W ‚ Xloc (M ) spans
a distribution denoted by E(W), which is obviously smooth (the linear span of
the empty set is the vector space 0). From now on we will consider only smooth
distributions.
An integral manifold of a smooth distribution E is a connected immersed
submanifold (N, i) (see 2.8) such that Tx i(Tx N ) = Ei(x) for all x ∈ N . We
will see in theorem 3.22 below that any integral manifold is in fact an initial
submanifold of M (see 2.14), so that we need not specify the injective immersion
i. An integral manifold of E is called maximal if it is not contained in any strictly
larger integral manifold of E.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
3. Vector ¬elds and ¬‚ows 25


3.19. Lemma. Let E be a smooth distribution on M . Then we have:
1. If (N, i) is an integral manifold of E and X ∈ XE , then i— X makes sense
and is an element of Xloc (N ), which is i|i’1 (UX )-related to X, where UX ‚ M
is the open domain of X.
2. If (Nj , ij ) are integral manifolds of E for j = 1, 2, then i’1 (i1 (N1 ) ©
1
’1
i2 (N2 )) and i2 (i1 (N1 ) © i2 (N2 )) are open subsets in N1 and N2 , respectively;
furthermore i’1 —¦ i1 is a di¬eomorphism between them.
2
3. If x ∈ M is contained in some integral submanifold of E, then it is contained
in a unique maximal one.
Proof. 1. Let UX be the open domain of X ∈ XE . If i(x) ∈ UX for x ∈ N ,
we have X(i(x)) ∈ Ei(x) = Tx i(Tx N ), so i— X(x) := ((Tx i)’1 —¦ X —¦ i)(x) makes
sense. It is clearly de¬ned on an open subset of N and is smooth in x.
2. Let X ∈ XE . Then i— X ∈ Xloc (Nj ) and is ij -related to X. So by lemma
j
3.14 for j = 1, 2 we have
i— X X
ij —¦ Fltj = F lt —¦ ij .
Now choose xj ∈ Nj such that i1 (x1 ) = i2 (x2 ) = x0 ∈ M and choose vector
¬elds X1 , . . . , Xn ∈ XE such that (X1 (x0 ), . . . , Xn (x0 )) is a basis of Ex0 . Then
i— X1 i— Xn
fj (t1 , . . . , tn ) := (Fltj —¦ · · · —¦ Fltj )(xj )
n
1



is a smooth mapping de¬ned near zero Rn ’ Nj . Since obviously ‚tk |0 fj =
i— Xk (xj ) for j = 1, 2, we see that fj is a di¬eomorphism near 0. Finally we have
j

i— X1 i— Xn
(i’1 —¦ i1 —¦ f1 )(t1 , . . . , tn ) = (i’1 —¦ i1 —¦ Flt1 —¦ · · · —¦ Flt1 )(x1 )
n
2 2 1


= (i’1 —¦ FlX1 —¦ · · · —¦ FlXn —¦i1 )(x1 )
tn
2 t1
i— X1 i— Xn
—¦i’1 —¦ i1 )(x1 )
—¦ · · · —¦ Flt2
= (Flt2 n 2
1


= f2 (t1 , . . . , tn ).

So i’1 —¦ i1 is a di¬eomorphism, as required.
2
3. Let N be the union of all integral manifolds containing x. Choose the union
of all the atlases of these integral manifolds as atlas for N , which is a smooth
atlas for N by 2. Note that a connected immersed submanifold of a separable
manifold is automatically separable (since it carries a Riemannian metric).
3.20. Integrable distributions and foliations.
A smooth distribution E on a manifold M is called integrable, if each point
of M is contained in some integral manifold of E. By 3.19.3 each point is
then contained in a unique maximal integral manifold, so the maximal integral
manifolds form a partition of M . This partition is called the foliation of M
induced by the integrable distribution E, and each maximal integral manifold
is called a leaf of this foliation. If X ∈ XE then by 3.19.1 the integral curve
t ’ FlX (t, x) of X through x ∈ M stays in the leaf through x.
Note, however, that usually a foliation is supposed to have constant dimen-
sions of the leafs, so our notion here is sometimes called a singular foliation.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
26 Chapter I. Manifolds and Lie groups


Let us now consider an arbitrary subset V ‚ Xloc (M ). We say that V is
stable if for all X, Y ∈ V and for all t for which it is de¬ned the local vector ¬eld
(FlX )— Y is again an element of V.
t
If W ‚ Xloc (M ) is an arbitrary subset, we call S(W) the set of all local vector
¬elds of the form (FlX1 —¦ · · · —¦ FlXk )— Y for Xi , Y ∈ W. By lemma 3.14 the ¬‚ow
tk
t1
of this vector ¬eld is

Fl((FlX1 —¦ · · · —¦ FlXk )— Y, t) = FlXkk —¦ · · · —¦ FlX11 —¦ FlY —¦ FlX1 —¦ · · · —¦ FlXk ,
’t
’t
tk tk
t1 t t1

so S(W) is the minimal stable set of local vector ¬elds which contains W.
Now let F be an arbitrary distribution. A local vector ¬eld X ∈ Xloc (M ) is
called an in¬nitesimal automorphism of F , if Tx (FlX )(Fx ) ‚ FFlX (t,x) whenever
t
de¬ned. We denote by aut(F ) the set of all in¬nitesimal automorphisms of F .
By arguments given just above, aut(F ) is stable.
3.21. Lemma. Let E be a smooth distribution on a manifold M . Then the
following conditions are equivalent:
(1) E is integrable.
(2) XE is stable.
(3) There exists a subset W ‚ Xloc (M ) such that S(W) spans E.
(4) aut(E) © XE spans E.

Proof. (1) =’ (2). Let X ∈ XE and let L be the leaf through x ∈ M , with

i : L ’ M the inclusion. Then FlX —¦i = i —¦ Fli X by lemma 3.14, so we have
’t ’t

Tx (FlX )(Ex ) = T (FlX ).Tx i.Tx L = T (FlX —¦i).Tx L
’t ’t ’t

= T i.Tx (Fli X ).Tx L
’t
= T i.TF li— X (’t,x) L = EF lX (’t,x) .

This implies that (FlX )— Y ∈ XE for any Y ∈ XE .
t
(2) =’ (4). In fact (2) says that XE ‚ aut(E).
(4) =’ (3). We can choose W = aut(E) © XE : for X, Y ∈ W we have
(FlX )— Y ∈ XE ; so W ‚ S(W) ‚ XE and E is spanned by W.
t
(3) =’ (1). We have to show that each point x ∈ M is contained in some
integral submanifold for the distribution E. Since S(W) spans E and is stable
we have

T (FlX ).Ex = EFlX (t,x)
(5) t

for each X ∈ S(W). Let dim Ex = n. There are X1 , . . . , Xn ∈ S(W) such that
X1 (x), . . . , Xn (x) is a basis of Ex , since E is smooth. As in the proof of 3.19.2
we consider the mapping

f (t1 , . . . , tn ) := (FlX1 —¦ · · · —¦ FlXn )(x),
tn
t1

de¬ned and smooth near 0 in Rn . Since the rank of f at 0 is n, the image
under f of a small open neighborhood of 0 is a submanifold N of M . We claim

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
3. Vector ¬elds and ¬‚ows 27


that N is an integral manifold of E. The tangent space Tf (t1 ,... ,tn ) N is linearly
generated by
X
—¦ · · · —¦ FlXn )(x) = T (FlX1 —¦ · · · —¦ Fltk’1 )Xk ((FlXk —¦ · · · —¦ FlXn )(x))
(FlX1
‚ k’1
tn tn
t1 t1
‚tk tk
X
= ((FlX11 )— · · · (Fl’tk’1 )— Xk )(f (t1 , . . . , tn )).
k’1
’t

Since S(W) is stable, these vectors lie in Ef (t) . From the form of f and from (5)
we see that dim Ef (t) = dim Ex , so these vectors even span Ef (t) and we have
Tf (t) N = Ef (t) as required.
3.22. Theorem (local structure of foliations). Let E be an integrable
distribution of a manifold M . Then for each x ∈ M there exists a chart (U, u)
with u(U ) = {y ∈ Rm : |y i | < µ for all i} for some µ > 0, and an at most
countable subset A ‚ Rm’n , such that for the leaf L through x we have

u(U © L) = {y ∈ u(U ) : (y n+1 , . . . , y m ) ∈ A}.

Each leaf is an initial submanifold.
If furthermore the distribution E has locally constant rank, this property
holds for each leaf meeting U with the same n.
This chart (U, u) is called a distinguished chart for the distribution or the
foliation. A connected component of U © L is called a plaque.
Proof. Let L be the leaf through x, dim L = n. Let X1 , . . . , Xn ∈ XE be local
vector ¬elds such that X1 (x), . . . , Xn (x) is a basis of Ex . We choose a chart
(V, v) centered at x on M such that the vectors

X1 (x), . . . , Xn (x), ‚v‚ |x , . . . , ‚vm |x

n+1


form a basis of Tx M . Then

f (t1 , . . . , tm ) = (FlX1 —¦ · · · —¦ FlXn )(v ’1 (0, . . . , 0, tn+1 , . . . , tm ))
tn
t1

is a di¬eomorphism from a neighborhood of 0 in Rm onto a neighborhood of x
in M . Let (U, u) be the chart given by f ’1 , suitably restricted. We have

y ∈ L ⇐’ (FlX1 —¦ · · · —¦ FlXn )(y) ∈ L
tn
t1

for all y and all t1 , . . . , tn for which both expressions make sense. So we have

f (t1 , . . . , tm ) ∈ L ⇐’ f (0, . . . , 0, tn+1 , . . . , tm ) ∈ L,

and consequently L © U is the disjoint union of connected sets of the form
{y ∈ U : (un+1 (y), . . . , um (y)) = constant}. Since L is a connected immersed
submanifold of M , it is second countable and only a countable set of constants
can appear in the description of u(L©U ) given above. From this description it is
clear that L is an initial submanifold (2.14) since u(Cx (L©U )) = u(U )©(Rn —0).
The argument given above is valid for any leaf of dimension n meeting U , so
also the assertion for an integrable distribution of constant rank follows.


Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
28 Chapter I. Manifolds and Lie groups


3.23. Involutive distributions. A subset V ‚ Xloc (M ) is called involutive if
[X, Y ] ∈ V for all X, Y ∈ V. Here [X, Y ] is de¬ned on the intersection of the
domains of X and Y .
A smooth distribution E on M is called involutive if there exists an involutive
subset V ‚ Xloc (M ) spanning E.
For an arbitrary subset W ‚ Xloc (M ) let L(W) be the set consisting of
all local vector ¬elds on M which can be written as ¬nite expressions using
Lie brackets and starting from elements of W. Clearly L(W) is the smallest
involutive subset of Xloc (M ) which contains W.

3.24. Lemma. For each subset W ‚ Xloc (M ) we have

E(W) ‚ E(L(W)) ‚ E(S(W)).

In particular we have E(S(W)) = E(L(S(W))).

Proof. We will show that for X, Y ∈ W we have [X, Y ] ∈ XE(S(W)) , for then by
induction we get L(W) ‚ XE(S(W)) and E(L(W)) ‚ E(S(W)).
Let x ∈ M ; since by 3.21 E(S(W)) is integrable, we can choose the leaf L
through x, with the inclusion i. Then i— X is i-related to X, i— Y is i-related to
Y , thus by 3.10 the local vector ¬eld [i— X, i— Y ] ∈ Xloc (L) is i-related to [X, Y ],
and [X, Y ](x) ∈ E(S(W))x , as required.

3.25. Theorem. Let V ‚ Xloc (M ) be an involutive subset. Then the distribu-
tion E(V) spanned by V is integrable under each of the following conditions.
(1) M is real analytic and V consists of real analytic vector ¬elds.
(2) The dimension of E(V) is constant along all ¬‚ow lines of vector ¬elds in
V.

Proof. (1) For X, Y ∈ V we have dt (FlX )— Y = (FlX )— LX Y , consequently
d
t t
dk X— X— k
(Flt ) Y = (Flt ) (LX ) Y , and since everything is real analytic we get for
dtk
x ∈ M and small t

tk d k tk
(FlX )— Y | (FlX )— Y (x) = (LX )k Y (x).
(x) = k0
t t
k! dt k!
k≥0 k≥0



Since V is involutive, all (LX )k Y ∈ V. Therefore we get (FlX )— Y (x) ∈ E(V)x
t
for small t. By the ¬‚ow property of FlX the set of all t satisfying (FlX )— Y (x) ∈
t
E(V)x is open and closed, so it follows that 3.21.2 is satis¬ed and thus E(V) is
integrable.
(2) We choose X1 , . . . , Xn ∈ V such that X1 (x), . . . , Xn (x) is a basis of
E(V)x . For X ∈ V, by hypothesis, E(V)FlX (t,x) has also dimension n and ad-
mits X1 (FlX (t, x)), . . . , Xn (FlX (t, x)) as basis for small t. So there are smooth

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
3. Vector ¬elds and ¬‚ows 29


functions fij (t) such that
n
X
fij (t)Xj (FlX (t, x)).
[X, Xi ](Fl (t, x)) =
j=1
X X
= T (FlX ).[X, Xi ](FlX (t, x)) =
d
dt T (Fl’t ).Xi (Fl (t, x)) ’t
n
fij (t)T (FlX ).Xj (FlX (t, x)).
= ’t
j=1


So the Tx M -valued functions gi (t) = T (FlX ).Xi (FlX (t, x)) satisfy the linear
’t
n
d
ordinary di¬erential equation dt gi (t) = j=1 fij (t)gj (t) and have initial values
in the linear subspace E(V)x , so they have values in it for all small t. There-
fore T (FlX )E(V)FlX (t,x) ‚ E(V)x for small t. Using compact time intervals
’t
and the ¬‚ow property one sees that condition 3.21.2 is satis¬ed and E(V) is
integrable.
Example. The distribution spanned by W ‚ Xloc (R2 ) is involutive, but not
integrable, where W consists of all global vector ¬elds with support in R2 \ {0}

and the ¬eld ‚x1 ; the leaf through 0 should have dimension 1 at 0 and dimension
2 elsewhere.
3.26. By a time dependent vector ¬eld on a manifold M we mean a smooth
mapping X : J — M ’ T M with πM —¦ X = pr2 , where J is an open interval.
An integral curve of X is a smooth curve c : I ’ M with c(t) = X(t, c(t)) for

all t ∈ I, where I is a subinterval of J.
¯ ¯
There is an associated vector ¬eld X ∈ X(J — M ), given by X(t, x) =
(1t , X(t, x)) ∈ Tt R — Tx M .
By the evolution operator of X we mean the mapping ¦X : J — J — M ’ M ,
de¬ned in a maximal open neighborhood of the diagonal in M —M and satisfying
the di¬erential equation

dX
= X(t, ¦X (t, s, x))
dt ¦ (t, s, x)
X
¦ (s, s, x) = x.

¯
It is easily seen that (t, ¦X (t, s, x)) = FlX (t ’ s, (s, x)), so the maximally de¬ned
evolution operator exists and is unique, and it satis¬es

¦X = ¦ X —¦ ¦ X
t,s t,r r,s


whenever one side makes sense (with the restrictions of 3.7), where ¦X (x) =
t,s
¦(t, s, x).




Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
30 Chapter I. Manifolds and Lie groups


4. Lie groups

4.1. De¬nition. A Lie group G is a smooth manifold and a group such that
the multiplication µ : G — G ’ G is smooth. We shall see in a moment, that
then also the inversion ν : G ’ G turns out to be smooth.
We shall use the following notation:
µ : G — G ’ G, multiplication, µ(x, y) = x.y.
»a : G ’ G, left translation, »a (x) = a.x.
ρa : G ’ G, right translation, ρa (x) = x.a.
ν : G ’ G, inversion, ν(x) = x’1 .
e ∈ G, the unit element.
Then we have »a —¦ »b = »a.b , ρa —¦ ρb = ρb.a , »’1 = »a’1 , ρ’1 = ρa’1 , ρa —¦ »b =
a a
»b —¦ ρa . If • : G ’ H is a smooth homomorphism between Lie groups, then we
also have • —¦ »a = »•(a) —¦ •, • —¦ ρa = ρ•(a) —¦ •, thus also T •.T »a = T »•(a) .T •,
etc. So Te • is injective (surjective) if and only if Ta • is injective (surjective) for
all a ∈ G.
4.2. Lemma. T(a,b) µ : Ta G — Tb G ’ Tab G is given by

T(a,b) µ.(Xa , Yb ) = Ta (ρb ).Xa + Tb (»a ).Yb .

Proof. Let ria : G ’ G — G, ria (x) = (a, x) be the right insertion and let
lib : G ’ G — G, lib (x) = (x, b) be the left insertion. Then we have

T(a,b) µ.(Xa , Yb ) = T(a,b) µ.(Ta (lib ).Xa + Tb (ria ).Yb ) =
= Ta (µ —¦ lib ).Xa + Tb (µ —¦ ria ).Yb = Ta (ρb ).Xa + Tb (»a ).Yb .

4.3. Corollary. The inversion ν : G ’ G is smooth and

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