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.

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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