only on the germ of f at a and we have Xa (f · g) = Xa (f ) · g(a) + f (a) · Xa (g)

(the derivation property).

If conversely D : C ∞ (Rn , R) ’ R is linear and satis¬es D(f · g) = D(f ) ·

g(a) + f (a) · D(g) (a derivation at a), then D is given by the action of a tangent

vector with foot point a. This can be seen as follows. For f ∈ C ∞ (Rn , R) we

have

1

d

+ t(x ’ a))dt

f (x) = f (a) + dt f (a

0

n 1

‚f

+ t(x ’ a))dt (xi ’ ai )

= f (a) + ‚xi (a

0

i=1

n

hi (x)(xi ’ ai ).

= f (a) +

i=1

D(1) = D(1 · 1) = 2D(1), so D(constant) = 0. Thus

n

hi (x)(xi ’ ai ))

D(f ) = D(f (a) +

i=1

n n

i i

hi (a)(D(xi ) ’ 0)

D(hi )(a ’ a ) +

=0+

i=1 i=1

n

‚f i

= ‚xi (a)D(x ),

i=1

where xi is the i-th coordinate function on Rn . So we have the expression

n n

i

D(xi ) ‚xi |a .

‚ ‚

) ‚xi |a (f ),

D(f ) = D(x D=

i=1 i=1

n

D(xi )ei ), where (ei ) is the

Thus D is induced by the tangent vector (a, i=1

standard basis of Rn .

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

1. Di¬erentiable manifolds 7

1.6. The tangent space of a manifold. Let M be a manifold and let x ∈

M and dim M = n. Let Tx M be the vector space of all derivations at x of

∞

Cx (M, R), the algebra of germs of smooth functions on M at x. (Using 1.3 it

may easily be seen that a derivation of C ∞ (M, R) at x factors to a derivation of

∞

Cx (M, R).)

So Tx M consists of all linear mappings Xx : C ∞ (M, R) ’ R satisfying Xx (f ·

g) = Xx (f ) · g(x) + f (x) · Xx (g). The space Tx M is called the tangent space of

M at x.

If (U, u) is a chart on M with x ∈ U , then u— : f ’ f —¦ u induces an iso-

morphism of algebras Cu(x) (Rn , R) ∼ Cx (M, R), and thus also an isomorphism

∞

=∞

Tx u : Tx M ’ Tu(x) Rn , given by (Tx u.Xx )(f ) = Xx (f —¦ u). So Tx M is an n-

dimensional vector space. The dot in Tx u.Xx means that we apply the linear

mapping Tx u to the vector Xx ” a dot will frequently denote an application of

a linear or ¬ber linear mapping.

We will use the following notation: u = (u1 , . . . , un ), so ui denotes the i-th

coordinate function on U , and

:= (Tx u)’1 ( ‚xi |u(x) ) = (Tx u)’1 (u(x), ei ).

‚ ‚

‚ui |x

‚

‚ui |x ∈ Tx M is the derivation given by

So

‚(f —¦ u’1 )

‚

‚ui |x (f ) = (u(x)).

‚xi

From 1.5 we have now

n

(Tx u.Xx )(xi ) ‚xi |u(x) =

‚

Tx u.Xx =

i=1

n n

i

Xx (ui ) ‚xi |u(x) .

‚ ‚

Xx (x —¦ u) ‚xi |u(x)

= =

i=1 i=1

1.7. The tangent bundle. For a manifold M of dimension n we put T M :=

x∈M Tx M , the disjoint union of all tangent spaces. This is a family of vec-

tor spaces parameterized by M , with projection πM : T M ’ M given by

πM (Tx M ) = x.

’1

For any chart (U± , u± ) of M consider the chart (πM (U± ), T u± ) on T M ,

’1

where T u± : πM (U± ) ’ u± (U± ) — Rn is given by the formula T u± .X =

(u± (πM (X)), TπM (X) u± .X). Then the chart changings look as follows:

’1

T uβ —¦ (T u± )’1 : T u± (πM (U±β )) = u± (U±β ) — Rn ’

’1

’ uβ (U±β ) — Rn = T uβ (πM (U±β )),

((T uβ —¦ (T u± )’1 )(y, Y ))(f ) = ((T u± )’1 (y, Y ))(f —¦ uβ )

= (y, Y )(f —¦ uβ —¦ u’1 ) = d(f —¦ uβ —¦ u’1 )(y).Y

± ±

= df (uβ —¦ u’1 (y)).d(uβ —¦ u’1 )(y).Y

± ±

= (uβ —¦ u’1 (y), d(uβ —¦ u’1 )(y).Y )(f ).

± ±

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

8 Chapter I. Manifolds and Lie groups

So the chart changings are smooth. We choose the topology on T M in such

a way that all T u± become homeomorphisms. This is a Hausdor¬ topology,

since X, Y ∈ T M may be separated in M if π(X) = π(Y ), and in one chart if

π(X) = π(Y ). So T M is again a smooth manifold in a canonical way; the triple

(T M, πM , M ) is called the tangent bundle of M .

∞

1.8. Kinematic de¬nition of the tangent space. Consider C0 (R, M ), the

space of germs at 0 of smooth curves R ’ M . We put the following equivalence

∞

relation on C0 (R, M ): the germ of c is equivalent to the germ of e if and only

if c(0) = e(0) and in one (equivalently each) chart (U, u) with c(0) = e(0) ∈ U

d d

we have dt |0 (u —¦ c)(t) = dt |0 (u —¦ e)(t). The equivalence classes are called velocity

vectors of curves in M . We have the following mappings

u

∞ ∞

g

e

C0 (R, M )/ ∼ C0 (R, M )

ee

ee β

ue u

ev0

±

w M,

TM πM

∞

d

where ±(c)(germc(0) f ) = dt |0 f (c(t)) and β : T M ’ C0 (R, M ) is given by:

β((T u)’1 (y, Y )) is the germ at 0 of t ’ u’1 (y + tY ). So T M is canonically

identi¬ed with the set of all possible velocity vectors of curves in M .

1.9. Let f : M ’ N be a smooth mapping between manifolds. Then f induces a

linear mapping Tx f : Tx M ’ Tf (x) N for each x ∈ M by (Tx f.Xx )(h) = Xx (h—¦f )

for h ∈ Cf (x) (N, R). This mapping is linear since f — : Cf (x) (N, R) ’ Cx (M, R),

∞ ∞ ∞

given by h ’ h —¦ f , is linear, and Tx f is its adjoint, restricted to the subspace

of derivations.

If (U, u) is a chart around x and (V, v) is one around f (x), then

u’1 ),

(Tx f. ‚ui |x )(v j ) = j j

‚ ‚ ‚

‚ui |x (v —¦ f) = ‚xi (v —¦ f —¦

‚ ‚ j‚

Tx f. ‚ui |x = j (Tx f. ‚ui |x )(v ) ‚v j |f (x) by 1.7

‚(v j —¦f —¦u’1 ) ‚

(u(x)) ‚vj |f (x) .

= ‚xi

j

‚ ‚

So the matrix of Tx f : Tx M ’ Tf (x) N in the bases ( ‚ui |x ) and ( ‚vj |f (x) ) is just

the Jacobi matrix d(v —¦ f —¦ u’1 )(u(x)) of the mapping v —¦ f —¦ u’1 at u(x), so

Tf (x) v —¦ Tx f —¦ (Tx u)’1 = d(v —¦ f —¦ u’1 )(u(x)).

Let us denote by T f : T M ’ T N the total mapping, given by T f |Tx M :=

Tx f . Then the composition T v —¦ T f —¦ (T u)’1 : u(U ) — Rm ’ v(V ) — Rn is given

by (y, Y ) ’ ((v —¦ f —¦ u’1 )(y), d(v —¦ f —¦ u’1 )(y)Y ), and thus T f : T M ’ T N is

again smooth.

If f : M ’ N and g : N ’ P are smooth mappings, then we have T (g —¦ f ) =

T g —¦ T f . This is a direct consequence of (g —¦ f )— = f — —¦ g — , and it is the global

version of the chain rule. Furthermore we have T (IdM ) = IdT M .

If f ∈ C ∞ (M, R), then T f : T M ’ T R = R — R. We then de¬ne the

di¬erential of f by df := pr2 —¦ T f : T M ’ R. Let t denote the identity function

on R, then (T f.Xx )(t) = Xx (t —¦ f ) = Xx (f ), so we have df (Xx ) = Xx (f ).

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

1. Di¬erentiable manifolds 9

1.10. Submanifolds. A subset N of a manifold M is called a submanifold, if for

each x ∈ N there is a chart (U, u) of M such that u(U © N ) = u(U ) © (Rk — 0),

where Rk — 0 ’ Rk — Rn’k = Rn . Then clearly N is itself a manifold with

(U © N, u|U © N ) as charts, where (U, u) runs through all submanifold charts as

above and the injection i : N ’ M is an embedding in the following sense:

An embedding f : N ’ M from a manifold N into another one is an injective

smooth mapping such that f (N ) is a submanifold of M and the (co)restricted

mapping N ’ f (N ) is a di¬eomorphism.

If f : Rn ’ Rq is smooth and the rank of f (more exactly: the rank of its

derivative) is q at each point of f ’1 (0), say, then f ’1 (0) is a submanifold of Rn

of dimension n ’ q or empty. This is an immediate consequence of the implicit

function theorem.

The following theorem needs three applications of the implicit function theo-

rem for its proof, which can be found in [Dieudonn´, I, 60, 10.3.1].

e

Theorem. Let f : W ’ Rq be a smooth mapping, where W is an open subset

of Rn . If the derivative df (x) has constant rank k for each x ∈ W , then for each

a ∈ W there are charts (U, u) of W centered at a and (V, v) of Rq centered at

f (a) such that v —¦ f —¦ u’1 : u(U ) ’ v(V ) has the following form:

(x1 , . . . , xn ) ’ (x1 , . . . , xk , 0, . . . , 0).

So f ’1 (b) is a submanifold of W of dimension n ’ k for each b ∈ f (W ).

1.11. Example: Spheres. We consider the space Rn+1 , equipped with the

xi y i . The n-sphere S n is then the subset

standard inner product x, y =

{x ∈ Rn+1 : x, x = 1}. Since f (x) = x, x , f : Rn+1 ’ R, satis¬es df (x)y =

2 x, y , it is of rank 1 o¬ 0 and by 1.10 the sphere S n is a submanifold of Rn+1 .

In order to get some feeling for the sphere we will describe an explicit atlas

for S n , the stereographic atlas. Choose a ∈ S n (˜south pole™). Let

x’ x,a a

u+ : U+ ’ {a}⊥ ,

U+ := S n \ {a}, u+ (x) = 1’ x,a ,

x’ x,a a

u’ : U’ ’ {a}⊥ ,

U’ := S n \ {’a}, u’ (x) = 1+ x,a .

From an obvious drawing in the 2-plane through 0, x, and a it is easily seen that

u+ is the usual stereographic projection. We also get

|y|2 ’1

u’1 (y) = for y ∈ {a}⊥

2

|y|2 +1 a + |y|2 +1 y

+

and (u’ —¦ u’1 )(y) = y

|y|2 . The latter equation can directly be seen from a

+

drawing.

1.12. Products. Let M and N be smooth manifolds described by smooth at-

lases (U± , u± )±∈A and (Vβ , vβ )β∈B , respectively. Then the family (U± — Vβ , u± —

vβ : U± — Vβ ’ Rm — Rn )(±,β)∈A—B is a smooth atlas for the cartesian product

M — N . Clearly the projections

pr1 pr2

M ←’ M — N ’’ N

’ ’

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

10 Chapter I. Manifolds and Lie groups

are also smooth. The product (M — N, pr1 , pr2 ) has the following universal

property:

For any smooth manifold P and smooth mappings f : P ’ M and g : P ’ N

the mapping (f, g) : P ’ M — N , (f, g)(x) = (f (x), g(x)), is the unique smooth

mapping with pr1 —¦ (f, g) = f , pr2 —¦ (f, g) = g.

From the construction of the tangent bundle in 1.7 it is immediately clear

that

T (pr1 ) T (pr2 )

T M ← ’ ’ T (M — N ) ’ ’ ’ T N

’’ ’’

is again a product, so that T (M — N ) = T M — T N in a canonical way.

Clearly we can form products of ¬nitely many manifolds.

1.13. Theorem. Let M be a connected manifold and suppose that f : M ’ M

is smooth with f —¦ f = f . Then the image f (M ) of f is a submanifold of M .

This result can also be expressed as: ˜smooth retracts™ of manifolds are man-

ifolds. If we do not suppose that M is connected, then f (M ) will not be a

pure manifold in general, it will have di¬erent dimension in di¬erent connected

components.

Proof. We claim that there is an open neighborhood U of f (M ) in M such that

the rank of Ty f is constant for y ∈ U . Then by theorem 1.10 the result follows.

For x ∈ f (M ) we have Tx f —¦ Tx f = Tx f , thus im Tx f = ker(Id ’Tx f ) and

rank Tx f + rank(Id ’Tx f ) = dim M . Since rank Tx f and rank(Id ’Tx f ) can-

not fall locally, rank Tx f is locally constant for x ∈ f (M ), and since f (M ) is

connected, rank Tx f = r for all x ∈ f (M ).

But then for each x ∈ f (M ) there is an open neighborhood Ux in M with

rank Ty f ≥ r for all y ∈ Ux . On the other hand rank Ty f = rank Ty (f —¦ f ) =

rank Tf (y) f —¦ Ty f ¤ rank Tf (y) f = r. So the neighborhood we need is given by

U = x∈f (M ) Ux .

1.14. Corollary. 1. The (separable) connected smooth manifolds are exactly

the smooth retracts of connected open subsets of Rn ™s.

2. f : M ’ N is an embedding of a submanifold if and only if there is an

open neighborhood U of f (M ) in N and a smooth mapping r : U ’ M with

r —¦ f = IdM .

Proof. Any manifold M may be embedded into some Rn , see 1.15 below. Then

there exists a tubular neighborhood of M in Rn (see [Hirsch, 76, pp. 109“118]),

and M is clearly a retract of such a tubular neighborhood. The converse follows

from 1.13.

For the second assertion repeat the argument for N instead of Rn .

1.15. Embeddings into Rn ™s. Let M be a smooth manifold of dimension m.

Then M can be embedded into Rn , if

(1) n = 2m + 1 (see [Hirsch, 76, p 55] or [Br¨cker-J¨nich, 73, p 73]),

o a

(2) n = 2m (see [Whitney, 44]).

(3) Conjecture (still unproved): The minimal n is n = 2m ’ ±(m) + 1, where

±(m) is the number of 1™s in the dyadic expansion of m.

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

2. Submersions and immersions 11

There exists an immersion (see section 2) M ’ Rn , if

(1) n = 2m (see [Hirsch, 76]),

(2) n = 2m ’ ±(m) (see [Cohen, 82]).

2. Submersions and immersions

2.1. De¬nition. A mapping f : M ’ N between manifolds is called a sub-

mersion at x ∈ M , if the rank of Tx f : Tx M ’ Tf (x) N equals dim N . Since the

rank cannot fall locally (the determinant of a submatrix of the Jacobi matrix is

not 0), f is then a submersion in a whole neighborhood of x. The mapping f is

said to be a submersion, if it is a submersion at each x ∈ M .

2.2. Lemma. If f : M ’ N is a submersion at x ∈ M , then for any chart

(V, v) centered at f (x) on N there is chart (U, u) centered at x on M such that

v —¦ f —¦ u’1 looks as follows:

(y 1 , . . . , y n , y n+1 , . . . , y m ) ’ (y 1 , . . . , y n )

Proof. Use the inverse function theorem.

2.3. Corollary. Any submersion f : M ’ N is open: for each open U ‚ M

the set f (U ) is open in N .

2.4. De¬nition. A triple (M, p, N ), where p : M ’ N is a surjective submer-

sion, is called a ¬bered manifold. M is called the total space, N is called the

base.

A ¬bered manifold admits local sections: For each x ∈ M there is an open

neighborhood U of p(x) in N and a smooth mapping s : U ’ M with p—¦s = IdU

and s(p(x)) = x.

The existence of local sections in turn implies the following universal property:

RR

M

RT

R

u

p

w

f

N P

If (M, p, N ) is a ¬bered manifold and f : N ’ P is a mapping into some further

manifold, such that f —¦ p : M ’ P is smooth, then f is smooth.

2.5. De¬nition. A smooth mapping f : M ’ N is called an immersion at

x ∈ M if the rank of Tx f : Tx M ’ Tf (x) N equals dim M . Since the rank is

maximal at x and cannot fall locally, f is an immersion on a whole neighborhood

of x. f is called an immersion if it is so at every x ∈ M .

2.6. Lemma. If f : M ’ N is an immersion, then for any chart (U, u) centered

at x ∈ M there is a chart (V, v) centered at f (x) on N such that v —¦ f —¦ u’1 has

the form:

(y 1 , . . . , y m ) ’ (y 1 , . . . , y m , 0, . . . , 0)

Proof. Use the inverse function theorem.

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

12 Chapter I. Manifolds and Lie groups

2.7 Corollary. If f : M ’ N is an immersion, then for any x ∈ M there is

an open neighborhood U of x ∈ M such that f (U ) is a submanifold of N and

f |U : U ’ f (U ) is a di¬eomorphism.

2.8. De¬nition. If i : M ’ N is an injective immersion, then (M, i) is called

an immersed submanifold of N .

A submanifold is an immersed submanifold, but the converse is wrong in gen-

eral. The structure of an immersed submanifold (M, i) is in general not deter-

mined by the subset i(M ) ‚ N . All this is illustrated by the following example.

Consider the curve γ(t) = (sin3 t, sin t. cos t) in R2 . Then ((’π, π), γ|(’π, π))

and ((0, 2π), γ|(0, 2π)) are two di¬erent immersed submanifolds, but the image

of the embedding is in both cases just the ¬gure eight.

2.9. Let M be a submanifold of N . Then the embedding i : M ’ N is an

injective immersion with the following property:

(1) For any manifold Z a mapping f : Z ’ M is smooth if and only if

i —¦ f : Z ’ N is smooth.

The example in 2.8 shows that there are injective immersions without property

(1).

2.10. We want to determine all injective immersions i : M ’ N with property

2.9.1. To require that i is a homeomorphism onto its image is too strong as 2.11

and 2.12 below show. To look for all smooth mappings i : M ’ N with property

2.9.1 (initial mappings in categorical terms) is too di¬cult as remark 2.13 below

shows.

2.11. Lemma. If an injective immersion i : M ’ N is a homeomorphism onto

its image, then i(M ) is a submanifold of N .

Proof. Use 2.7.

2.12. Example. We consider the 2-dimensional torus T2 = R2 /Z2 . Then the

quotient mapping π : R2 ’ T2 is a covering map, so locally a di¬eomorphism.

Let us also consider the mapping f : R ’ R2 , f (t) = (t, ±.t), where ± is

irrational. Then π —¦ f : R ’ T2 is an injective immersion with dense image, and

it is obviously not a homeomorphism onto its image. But π —¦ f has property

2.9.1, which follows from the fact that π is a covering map.

2.13. Remark. If f : R ’ R is a function such that f p and f q are smooth for

some p, q which are relatively prime in N, then f itself turns out to be smooth,

p

see [Joris, 82]. So the mapping i : t ’ tq , R ’ R2 , has property 2.9.1, but i is

t

not an immersion at 0.

2.14. De¬nition. For an arbitrary subset A of a manifold N and x0 ∈ A let

Cx0 (A) denote the set of all x ∈ A which can be joined to x0 by a smooth curve

in N lying in A.