The Lie algebra zA of ZA then consists of all X ∈ g such that a. exp(tX).a’1 =

exp(tX) for all a ∈ A, i.e. zA = {X ∈ g : Ad(a)X = X for all a ∈ A}.

If A is itself a connected Lie subgroup of G, then zA = {X ∈ g : ad(Y )X =

0 for all Y ∈ a}. This set is also called the centralizer of a in g. If A = G then

ZG is called the center of G and zG = {X ∈ g : [X, Y ] = 0 for all Y ∈ g} is then

the center of the Lie algebra g.

5.9. The normalizer of a subset A of a connected Lie group G is the subgroup

NA = {x ∈ G : »x (A) = ρx (A)} = {x ∈ G : conjx (A) = A}. If A is closed then

NA is also closed.

If A is a connected Lie subgroup of G then NA = {x ∈ G : Ad(x)a ‚ a} and

its Lie algebra is nA = {X ∈ g : ad(X)a ‚ a} is then the idealizer of a in g.

5.10. Group actions. A left action of a Lie group G on a manifold M is a

smooth mapping : G — M ’ M such that x —¦ y = xy and e = IdM , where

x (z) = (x, z).

A right action of a Lie group G on a manifold M is a smooth mapping

r : M — G ’ M such that rx —¦ ry = ryx and re = IdM , where rx (z) = r(z, x).

A G-space is a manifold M together with a right or left action of G on M .

We will describe the following notions only for a left action of G on M . They

make sense also for right actions.

The orbit through z ∈ M is the set G.z = (G, z) ‚ M . The action is called

transitive, if M is one orbit, i.e. for all z, w ∈ M there is some g ∈ G with

g.z = w. The action is called free, if g1 .z = g2 .z for some z ∈ M implies already

g1 = g2 . The action is called e¬ective, if x = y implies x = y, i.e. if : G ’

Di¬(M ) is injective, where Di¬(M ) denotes the group of all di¬eomorphisms of

M.

More generally, a continuous transformation group of a topological space M

is a pair (G, M ) where G is a topological group and where to each element x ∈ G

there is given a homeomorphism x of M such that : G—M ’ M is continuous,

and x —¦ y = xy . The continuity is an obvious geometrical requirement, but

in accordance with the general observation that group properties often force

more regularity than explicitly postulated (cf. 5.6), di¬erentiability follows in

many situations. So, if G is locally compact, M is a smooth or real analytic

manifold, all x are smooth or real analytic homeomorphisms and the action is

e¬ective, then G is a Lie group and is smooth or real analytic, respectively,

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

5. Lie subgroups and homogeneous Spaces 45

see [Montgomery, Zippin, 55, p. 212]. The latter result is deeply re¬‚ected in the

theory of bundle functors and will be heavily used in chapter V.

5.11. Homogeneous spaces. Let G be a Lie group and let H ‚ G be a closed

subgroup. By theorem 5.5 H is a Lie subgroup of G. We denote by G/H the

space of all right cosets of G, i.e. G/H = {xH : x ∈ G}. Let p : G ’ G/H

be the projection. We equip G/H with the quotient topology, i.e. U ‚ G/H is

open if and only if p’1 (U ) is open in G. Since H is closed, G/H is a Hausdor¬

space.

G/H is called a homogeneous space of G. We have a left action of G on G/H,

¯

which is induced by the left translation and is given by »x (zH) = xzH.

Theorem. If H is a closed subgroup of G, then there exists a unique structure

of a smooth manifold on G/H such that p : G ’ G/H is a submersion. So

dim G/H = dim G ’ dim H.

Proof. Surjective submersions have the universal property 2.4, thus the manifold

structure on G/H is unique, if it exists. Let h be the Lie algebra of the Lie

subgroup H. We choose a complementary linear subspace k such that g = h • k.

Claim 1. We consider the mapping f : k — H ’ G, given by f (X, h) := exp X.h.

Then there is an open 0-neighborhood W in k and an open e-neighborhood U in

G such that f : W — H ’ U is a di¬eomorphism.

By claim 5 in the proof of theorem 5.5 there are open 0-neighborhoods V in

h, W in k, and an open e-neighborhood U in G such that • : W — V ’ U is a

di¬eomorphism, where •(X, Y ) = exp X. exp Y , and such that U © H = exp V .

Now we choose W in k so small that exp(W )’1 . exp(W ) ‚ U . We will check

that this W satis¬es claim 1.

Claim 2. f |W — H is injective.

f (X1 , h1 ) = f (X2 , h2 ) means exp X1 .h1 = exp X2 .h2 , consequently we have

h2 h’1 = (exp X2 )’1 exp X1 ∈ exp(W )’1 exp(W ) © H ‚ U © H = exp V . So

1

there is a unique Y ∈ V with h2 h’1 = exp Y . But then •(X1 , 0) = exp X1 =

1

’1

exp X2 .h2 .h1 = exp X2 . exp Y = •(X2 , Y ). Since • is injective, X1 = X2 and

Y = 0, so h1 = h2 .

Claim 3. f |W — H is a local di¬eomorphism.

The diagram

w

Id — exp

W —V W — (U © H)

•

u u

f

wU

incl

•(W — V )

commutes, and IdW — exp and • are di¬eomorphisms. So f |W — (U © H) is

a local di¬eomorphism. Since f (X, h) = f (X, e).h we conclude that f |W — H

is everywhere a local di¬eomorphism. So ¬nally claim 1 follows, where U =

f (W — H).

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

46 Chapter I. Manifolds and Lie groups

Now we put g := p —¦ (exp |W ) : k ⊃ W ’ G/H. Then the following diagram

w

commutes: f

W —H U

p

u u

pr1

w G/H.

g

W

¯

Claim 4. g is a homeomorphism onto p(U ) =: U ‚ G/H.

Clearly g is continuous, and g is open, since p is open. If g(X1 ) = g(X2 ) then

exp X1 = exp X2 .h for some h ∈ H, so f (X1 , e) = f (X2 , h). By claim 1 we get

¯

X1 = X2 , so g is injective. Finally g(W ) = U , so claim 4 follows.

¯¯ ¯

¯ ¯

For a ∈ G we consider Ua = »a (U ) = a.U and the mapping ua := g ’1 —¦ »a’1 :

¯

Ua ’ W ‚ k.

¯

¯ ¯

Claim 5. (Ua , ua = g ’1 —¦ »a’1 : Ua ’ W )a∈G is a smooth atlas for G/H.

¯ ¯

Let a, b ∈ G such that Ua © Ub = …. Then

¯ ¯

ua —¦ u’1 = g ’1 —¦ »a’1 —¦ »b —¦ g : ub (Ua © Ub ) ’ ua (Ua © Ub )

¯ ¯ ¯ ¯

b

¯

= g ’1 —¦ »a’1 b —¦ p —¦ (exp |W )

= g ’1 —¦ p —¦ »a’1 b —¦ (exp |W )

= pr1 —¦ f ’1 —¦ »a’1 b —¦ (exp |W ) is smooth.

5.12. Let : G — M ’ M be a left action. Then we have partial mappings

x

: G ’ M , given by a (x) = x (a) = (a, x) = a.x.

a : M ’ M and

M

For any X ∈ g we de¬ne the fundamental vector ¬eld ζX = ζX ∈ X(M ) by

ζX (x) = Te ( x ).X = T(e,x) .(X, 0x ).

Lemma. In this situation the following assertions hold:

(1) ζ : g ’ X(M ) is a linear mapping.

(2) Tx ( a ).ζX (x) = ζAd(a)X (a.x).

(3) RX — 0M ∈ X(G — M ) is -related to ζX ∈ X(M ).

(4) [ζX , ζY ] = ’ζ[X,Y ] .

Proof. (1) is clear.

(2) a x (b) = abx = aba’1 ax = ax conja (b), so we get Tx ( a ).ζX (x) =

Tx ( a ).Te ( x ).X = Te ( a —¦ x ).X = Te ( ax ). Ad(a).X = ζAd(a)X (ax).

(3) —¦ (Id — a ) = —¦ (ρa — Id) : G — M ’ M , so we get ζX ( (a, x)) =

T(e,ax) .(X, 0ax ) = T .(Id —T ( a )).(X, 0x ) = T .(T (ρa )—Id).(X, 0x ) = T .(RX —

0M )(a, x).

(4) [RX — 0M , RY — 0M ] = [RX , RY ] — 0M = ’R[X,Y ] — 0M is -related to

[ζX , ζY ] by (3) and by 3.10. On the other hand ’R[X,Y ] — 0M is -related to

’ζ[X,Y ] by (3) again. Since is surjective we get [ζX , ζY ] = ’ζ[X,Y ] .

5.13. Let r : M — G ’ M be a right action, so r : G ’ Di¬(M ) is a group

ˇ

anti homomorphism. We will use the following notation: ra : M ’ M and

rx : G ’ M , given by rx (a) = ra (x) = r(x, a) = x.a.

M

For any X ∈ g we de¬ne the fundamental vector ¬eld ζX = ζX ∈ X(M ) by

ζX (x) = Te (rx ).X = T(x,e) r.(0x , X).

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

5. Lie subgroups and homogeneous Spaces 47

Lemma. In this situation the following assertions hold:

(1) ζ : g ’ X(M ) is a linear mapping.

(2) Tx (ra ).ζX (x) = ζAd(a’1 )X (x.a).

(3) 0M — LX ∈ X(M — G) is r-related to ζX ∈ X(M ).

(4) [ζX , ζY ] = ζ[X,Y ] .

5.14. Theorem. Let : G — M ’ M be a smooth left action. For x ∈ M let

Gx = {a ∈ G : ax = x} be the isotropy subgroup of x in G, a closed subgroup

of G. Then x : G ’ M factors over p : G ’ G/Gx to an injective immersion

¯

ix : G/Gx ’ M , which is G-equivariant, i.e. a —¦ ix = ix —¦ »a for all a ∈ G. The

image of ix is the orbit through x.

The fundamental vector ¬elds span an integrable distribution on M in the

sense of 3.20. Its leaves are the connected components of the orbits, and each

orbit is an initial submanifold.

Proof. Clearly x factors over p to an injective mapping ix : G/Gx ’ M ; by

the universal property of surjective submersions ix is smooth, and obviously

¯ ¯

it is equivariant. Thus Tp(a) (ix ).Tp(e) (»a ) = Tp(e) (ix —¦ »a ) = Tp(e) ( a —¦ ix ) =

Tx ( a ).Tp(e) (ix ) for all a ∈ G and it su¬ces to show that Tp(e) (ix ) is injective.

Let X ∈ g and consider its fundamental vector ¬eld ζX ∈ X(M ). By 3.14 and

5.12.3 we have

(exp(tX), x) = (FlRX —0M (e, x)) = FlζX ( (e, x)) = FlζX (x).

(1) t t

t

So exp(tX) ∈ Gx , i.e. X ∈ gx , if and only if ζX (x) = 0x . In other words,

0x = ζX (x) = Te ( x ).X = Tp(e) (ix ).Te p.X if and only if Te p.X = 0p(e) . Thus ix

is an immersion.

Since the connected components of the orbits are integral manifolds, the fun-

damental vector ¬elds span an integrable distribution in the sense of 3.20; but

also the condition 3.25.2 is satis¬ed. So by theorem 3.22 each orbit is an initial

submanifold in the sense of 2.14.

¯

5.15. A mapping f : M ’ M between two manifolds with left (or right) actions

and ¯ of a Lie group G is called G-equivariant if f —¦ a = ¯a —¦f ( or f —¦ra = ra —¦f )

¯

for all a ∈ G. Sometimes we say in short that f is a G-mapping. From formula

5.14.(1) we get

Lemma. If G is connected, then f is G-equivariant if and only if the funda-

¯

mental ¬eld mappings are f related, i.e. T f —¦ ζX = ζX —¦ f for all X ∈ g.

Proof. The image of the exponential mapping generates the connected compo-

nent of the unit.

5.16. Semidirect products of Lie groups. Let H and K be two Lie groups

and let : H — K ’ K be a left action of H in K such that each h : K ’ K

is a group homomorphism. So the associated mapping ˇ : H ’ Aut(K) is a

homomorphism into the automorphism group of K. Then we can introduce the

following multiplication on K — H

(1) (k, h)(k , h ) := (k h (k ), hh ).

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

48 Chapter I. Manifolds and Lie groups

It is easy to see that this de¬nes a Lie group G = K H called the semidirect

product of H and K with respect to . If the action is clear from the context we

write G = K H only. The second projection pr2 : K H ’ H is a surjective

smooth homomorphism with kernel K —{e}, and the insertion inse : H ’ K H,

inse (h) = (e, h) is a smooth group homomorphism with pr2 —¦ inse = IdH .

Conversely we consider an exact sequence of Lie groups and homomorphisms

j p

{e} ’ K ’ G ’ H ’ {e}.

’’

(2)

So j is injective, p is surjective, and the kernel of p equals the image of j.

We suppose furthermore that the sequence splits, so that there is a smooth

homomorphism i : H ’ G with p —¦ i = IdH . Then the rule h (k) = i(h)ki(h’1 )

(where we suppress j) de¬nes a left action of H on K by automorphisms. It

H ’ G given by (k, h) ’ ki(h) is an

is easily seen that the mapping K

isomorphism of Lie groups. So we see that semidirect products of Lie groups

correspond exactly to splitting short exact sequences.

Semidirect products will appear naturally also in another form, starting from

right actions: Let H and K be two Lie groups and let r : K — H ’ K be a right

action of H in K such that each rh : K ’ K is a group homomorphism. Then

the multiplication on H — K is given by

¯¯ ¯

¯¯

(h, k)(h, k) := (hh, rh (k)k).

(3)

This de¬nes a Lie group G = H r K, also called the semidirect product of H

and K with respect to r. If the action r is clear from the context we write

G = H K only. The ¬rst projection pr1 : H K ’ H is a surjective smooth

homomorphism with kernel {e} — K, and the insertion inse : H ’ H K,

inse (h) = (h, e) is a smooth group homomorphism with pr1 —¦ inse = IdH .

Conversely we consider again a splitting exact sequence of Lie groups and

homomorphisms

j p

{e} ’ K ’ G ’ H ’ {e}.

’’

The splitting is given by a homomorphism i : H ’ G with p —¦ i = IdH . Then

the rule rh (k) = i(h’1 )ki(h) (where we suppress j) de¬nes now a right action

of H on K by automorphisms. It is easily seen that the mapping H r K ’ G

given by (h, k) ’ i(h)k is an isomorphism of Lie groups.

Remarks

The material in this chapter is standard. The concept of initial submani-

folds in 2.14“2.17 is due to Pradines, the treatment given here follows [Albert,

Molino]. The proof of theorem 3.16 is due to [Mauhart, 90]. The main re-

sults on distributions of non constant rank (3.18“3.25) are due to [Sussman, 73]

and [Stefan, 74], the treatment here follows [Lecomte]. The proof of the Baker-

Campbell-Hausdor¬ formula 4.29 is adapted from [Sattinger, Weaver, 86], see

also [Hilgert, Neeb, 91]. ¦

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

49

CHAPTER II.

DIFFERENTIAL FORMS

This chapter is still devoted to the fundamentals of di¬erential geometry,

but here the deviation from the standard presentations is already large. In

the section on vector bundles we treat the Lie derivative for natural vector

bundles, i.e. functors which associate vector bundles to manifolds and vector

bundle homomorphisms to local di¬eomorphisms. We give a formula for the Lie

derivative of the form of a commutator, but it involves the tangent bundle of the

vector bundle in question. So we also give a careful treatment to this situation.

The Lie derivative will be discussed in detail in chapter XI; here it is presented

in a somewhat special situation as an illustration of the categorical methods we

are going to apply later on. It follows a standard presentation of di¬erential

forms and a thorough treatment of the Fr¨licher-Nijenhuis bracket via the study

o

of all graded derivations of the algebra of di¬erential forms. This bracket is a

natural extension of the Lie bracket from vector ¬elds to tangent bundle valued

di¬erential forms. We believe that this bracket is one of the basic structures of

di¬erential geometry (see also section 30), and in chapter III we will base nearly

all treatment of curvature and the Bianchi identity on it.

6. Vector bundles

6.1. Vector bundles. Let p : E ’ M be a smooth mapping between mani-

folds. By a vector bundle chart on (E, p, M ) we mean a pair (U, ψ), where U is

an open subset in M and where ψ is a ¬ber respecting di¬eomorphism as in the

following diagram:

w U —V

ee ψ

E|U := p’1 (U )

eg pr

e£

p

1

U.

Here V is a ¬xed ¬nite dimensional vector space, called the standard ¬ber or the

typical ¬ber, real as a rule, unless otherwise speci¬ed.

Two vector bundle charts (U1 , ψ1 ) and (U2 , ψ2 ) are called compatible, if ψ1 —¦

’1 ’1

ψ2 is a ¬ber linear isomorphism, i.e. (ψ1 —¦ ψ2 )(x, v) = (x, ψ1,2 (x)v) for some

mapping ψ1,2 : U1,2 := U1 © U2 ’ GL(V ). The mapping ψ1,2 is then unique and

smooth, and it is called the transition function between the two vector bundle

charts.

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

50 Chapter II. Di¬erential forms

A vector bundle atlas (U± , ψ± )±∈A for (E, p, M ) is a set of pairwise compatible

vector bundle charts (U± , ψ± ) such that (U± )±∈A is an open cover of M . Two

vector bundle atlases are called equivalent, if their union is again a vector bundle

atlas.

A vector bundle (E, p, M ) consists of manifolds E (the total space), M (the

base), and a smooth mapping p : E ’ M (the projection) together with an

equivalence class of vector bundle atlases; so we must know at least one vector

bundle atlas. The projection p turns out to be a surjective submersion.

The tangent bundle (T M, πM , M ) of a manifold M is the ¬rst example of a

vector bundle.

6.2. Let us ¬x a vector bundle (E, p, M ) for the moment. On each ¬ber Ex :=

p’1 (x) (for x ∈ M ) there is a unique structure of a real vector space, induced

from any vector bundle chart (U± , ψ± ) with x ∈ U± . So 0x ∈ Ex is a special

element and 0 : M ’ E, 0(x) = 0x , is a smooth mapping, the zero section.

A section u of (E, p, M ) is a smooth mapping u : M ’ E with p —¦ u = IdM .

The support of the section u is the closure of the set {x ∈ M : u(x) = 0x } in

M . The space of all smooth sections of the bundle (E, p, M ) will be denoted by

either C ∞ (E) = C ∞ (E, p, M ) = C ∞ (E ’ M ). Clearly it is a vector space with

¬ber wise addition and scalar multiplication.

If (U± , ψ± )±∈A is a vector bundle atlas for (E, p, M ), then any smooth map-

’1

ping f± : U± ’ V (the standard ¬ber) de¬nes a local section x ’ ψ± (x, f± (x))

on U± . If (g± )±∈A is a partition of unity subordinated to (U± ), then a global

’1

section can be formed by x ’ ± g± (x) · ψ± (x, f± (x)). So a smooth vector

bundle has ˜many™ smooth sections.

6.3. Let (E, p, M ) and (F, q, N ) be vector bundles. A vector bundle homomor-

phism • : E ’ F is a ¬ber respecting, ¬ber linear smooth mapping

wF

•

E

u u

p q

w N.

•

M

So we require that •x : Ex ’ F•(x) is linear. We say that • covers •. If • is

invertible, it is called a vector bundle isomorphism.

The smooth vector bundles together with their homomorphisms form a cate-

gory VB.

6.4. We will now give a formal description of the amount of vector bundles with

¬xed base M and ¬xed standard ¬ber V , up to isomorphisms which cover the

identity on M .

Let us ¬rst ¬x an open cover (U± )±∈A of M . If (E, p, M ) is a vector bundle

which admits a vector bundle atlas (U± , ψ± ) with the given open cover, then

’1

we have ψ± —¦ ψβ (x, v) = (x, ψ±β (x)v) for transition functions ψ±β : U±β =

U± © Uβ ’ GL(V ), which are smooth. This family of transition functions

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

6. Vector bundles 51

satis¬es

ψ±β (x) · ψβγ (x) = ψ±γ (x) for each x ∈ U±βγ = U± © Uβ © Uγ ,

(1)

for all x ∈ U± .

ψ±± (x) = e