Milnor J. Morse Theory (Princeton, 1963)

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M O R S E T H E O R Y

J. Milnor

BY

Based on lecture notes by

M. SPIVAK and R. WELLS



Copyright 1963. by Princeton University Press

All Rights ReservedL. C. Card 63-13729

Printed in the United States of America

CONTENTS

I.

SMOOTH FUNCTIONS ON A MANIFOLD$ 1 . Introduction . . . . . . . . . . . . . . . . . . . . . . .

. Definit ions and Lemmas. . . . . . . . . . . . . . . . . .

$3 . Homotopy Type i n Terms of C ri ti ca l Values . . . . . . . .

$4 . Examples . . . . . . . . . . . . . . . . . . . . . . . . .

$5 . The Morse Inequali t ies. . . . . . . . . . . . . . . . . .

. Manifolds i n Euclidean Space: The Existence of

Non -degenerate Functions . . . . . . . . . . . . . . . . .

$7 . The Theorem on Hyperplane Sections . . . . . . .

PART . A COURSE I N

. Covariant Differentiat ion . . . . . . . . . . . . .

$ 9. The Curvature Tensor . . . . . . . . . . . . . . . .. Geodesics and Completeness . . . . . . . . . . . . .

. THE OF VARIATIONS APPLIED TO GEODESICS

.

§1 2 .

.$14 .

.

§1 6 ..

.

.

The Path Space of a Smooth Mani fold . . . . . . . . . . .

The Energy of a Path. . . . . . . . . . . . . . . . . . .

The Hessian of the Energy Function at a C r i t i c a l P a th . .

Jacobi Fie lds : The Null-space of . . . . . . . . . .

The Index Theorem . . . . . . . . . . . . . . . . . . . .A Fi ni te Dimensional Approximation t o . . . . . . . .

The Topology of the Path Space . . . . . . . . . . .of Non-conjugate Points . . . . . . . . . . . .

1

4

1 2

2 8

43

55

67

70

74

a3

93

Some Relations Between Topology and Curvature . . . . . . lo o

V



CONTENTS

APPLICATIONS TO LIE GROUPS SPACES

. . . . . . . . . . . . . . . . . . . .$20. Symmetric Spaces

$ 2 1 . Lie Groups as Symmetric Spaces 1 1 2

Whole Manifolds of Minimal Geodesics 118

1 2 4

. . . . . . . . . . . . .

. . . . . . . . . .

$23 . The Periodicity Theorem for the Unitary Group . . .$24 . The Theorem for the Orthogonal Group . . . .

THE TYPE OF A UNION

v i

. . . . . . . . 14 9PART I

NON-DEGENERATE SMOOTH FUNCTIONS ON A MANIFOLD.

$ 1 . Introduction.

I n this sect ion we w i l l i l l u s t r a t e by a s p e c i f i c t h e s i t u -

a t i o n that we w i l l i n v e s t i g a t e l a t e r for arb i t r a ry m an ifo lds .

s i d e r a to rus M, tangent to the plane V, as ind ica ted in Diagram 1 .

Let us con-

Diagram

Let f : always denotes the r e al numbers) be the he ight

above the V plane, and let be the se t of a l l po in t s x M such that

a. Then the fol lowing are t rue :

then i s vacuous.

then i s homeomorphic t o a 2-cell .

(3 ) I f < a < then i s homeomorphic t o a cylinder :

a < then t o a compact

manifold of genus one having c i r c l e as boundary:









DEFINITIONS LEMMASI . NON-DEGENERATE FUNCTIONS

completes the induction; and proves 2 . 2 .

Non-degenerate a re i so l a t ed .2 .3

Examples of degenerate cri ti ca l poin ts (f or on

are given below, together with pictur es of t he ir graphs.

7

9

are degenerate, i s the x axis, which i s a sub -manifold of

= The set of c r i t i c a l p o i n t s , a l l of which

(a) The or ig in ( b) .i s a d eg en e ra t e c r i t i c a l The origin i s a degenerate, and

no n- i so la t ed, c r i t i c a l p o i n t .

degenerate , consists of the of the x and axis, which i s

not even a sub-manifold of

We conclude this sect ion wi th a discussion of 1-parameter groups of

The set o f c r i t i c a l p o i n ts , a l l of which are

e n t i a l for more det a i l s .

The reader i s re fe r red t o K. Groups and Differ-

A group of of a manifold M i s a

- R e a l p a r t o f

( 0 .0 )

(x +

i s a d egen era te c r i t i c a l p o i n t (a "monkey saddle").























30I. NON-DEGENERATE FUNCTIONS

rank h rank A - rank i

rank A - rank B + rank j

rank A - rank B + rank C - rank k

...rank A - rank B rank C - rank D .

the last express ion i sa t r i p l e X Y

0.

Now consider the homoiogy exact

Applying this computation t o th e homomorphism

we s ee that

rank - - + .. .

Collecting terms, this means that

which completes the

Applying this subaddit ive function to the s pacesa

C C M

we ob ta in the Morse ineq ua l i t i e s

or

- (M ) . C, +

T hese ine qua l i t i e s a r e de f in i t e ly s harper than the p rev ious ones.

I n f a c t , a d di n g (4 , ) and one obtains and comparingw i th f o r > n one obtains the equali ty ( 3 ) .

A s an i l l u s t r a t i o n of t h e u se of the Morse inequali t ies , suppose

that Then must be zero. Comparing the in eq ua li ti es

and we see that

- C, - C, .

Now suppose that i s al so zero. Then 0, arid a argu-

ment shows th a t

- R, - .

MORSE INEQUALITIES

Subtracting this from t h e e q u a l i t y above we ob ta in the fo l lowing :

5 . 4 . If = = 0 then C, and

= =

( O f course this would a l s o follow from Theorem 3 . 5 . )

Note thatthis corollary enables u s to fin d the homology of complex projective

space (see without use of Theorem 3.5 .

31

.

















46 RIEMANNIAN GEOMETRY

Now suppose t h a t M i s provided w i t h an af f ine connection . ThenDV

v e c t o r f i e l d V along c determines a new vector f i e l d a long c

the c o v a r i a n t d e r iv a t i v e of V. The operation

DV

character ized by the fo l lowing three axioms.

DV DW=

If i s a smooth real valued funct ion on R then

DV.

at

c ) If V i s induced by a v e c t o r f i e l d Y on M , that i s if dc

for each t , then i s e q u a l t o Y=

t h e c o v a r i a n t d e r iv a t i v e o f

v e lo c i t y v e c to r o f

Y i n t h e d i r e c t i o n o f t h e

DVV -0 . 1 .

w h i c h s a t i s f i e s these three c o n d i t i o n s .

There i s one and only one operation

PROOF: Choose a loca l coord inate system for M, a nd l e t

. denote the coord inate s of th e po in t The vector

f i e l d V can be expressed un iquely i n the fo rm

where

su b se t of R), and are t h e s t a n d a r d v e c to r f i e l d s o n t h e c o-

ordinate neighborhood. It follows from ( a ) , an d

,. . . are real valued funct ions on R (oran appropr ia te open

Converse ly , def in ing

that c o n d i t i o n s ( a ) , (b), and (c) are s a t i s f i e d .

by this formula, i t i s n o t d i f f i c u l t v e r i f y

A v e c t o r f i e l d V a lo n g c i s s a i d t o b e a p a r a l l e l v e c to r f i e l d

DVi f the c o v a r i a n t d e r iv a t i v e i s i d e n t i c a l l y z e r o .

a

8. COVARIANT DIFFERENTIATION 4 7

LEMMA Given a curve c and a t a n g e nt v e c t o r

a t t h e p o in t

v e c t o r f i e l d V alon g c which extends

t h e r e i s one and only one parallel

PROOF. The di ff er en ti al equati ons

have so lu t ions

a l l r e l e v a n t v a lu e s of t .

Real Va r i a b l e s ," p .

which are uniquely determined by the i n i t i a l v a lu es

Since these equat ions are l i n e a r , t h e so lu t i o n s c a n b e d ef in e d for

(Compare Graves, "The Theory of Funct ions of

The vector i s sa i d t o be ob ta ined f rom b y p a r a l l e l t r a n s-

l a t i o n a l on g c .

Now suppose that M i s a Riemannian manif old. The inner product

of two vectors w i l l be denoted by .

DEFINITION. A connection on M i s compatible with the Rieman-

n ia n m e t r i c i f p a r a l l e l t r a n s l a t i o n p r e se r ve s i n n e r p r o d u c t s . I n o th e r wor ds ,

fo r any parametrized curve c and any pa ir P, of' p a r a l l e l v e c t o r f i e l d s

a long c , the inner p roduct should be constan t .

LEMMA 8 . 3 . Suppose that the connect ion i s compat ib le wi th

the m e t r i c . L e t V, W b e an y two v e c to r f i e l d s a lo n g c .

Then

PROOF: Choose pa ra ll el vector f i el ds along c which

are orthonormal a t one po in t of c and hence a t every po in t of c. Then

t h e g i v en f i e l d s V an d W be expressed as and respec-

t i v e l y (where = i s a real valued funct ion on . It f o l -

lows that = a n d t h a t

or e

which the proof .





































































1 1 6 APPLICATIONS$2 1 . LI E GROUPS 11 7



some (See Chevalley, of Lie Groups.")

proof.

T h i s completes the

THEOREM 21 .7 ( B o t t ) . L e t G be a compact, simply con-

nected Lie group. Then the loop space has the

homotopy type of a with no odd dimensional

c e l l s , and with only f in i t el y many for each

even value of

Thus the X-th homology groups of i s zero fo r A odd, i s

f r ee abe l i an of f in i t e r ank for X even.

REMARK 1 . T h i s CW-complex w i l l always be inf ini te dimensional . A s

example, i f G i s the group of u n i t quate rnions , then we have seen

that the homology group i s i n f i n i t e c y c l ic a l l even va lues of

REMARK 2. Th i s theorem remains t r u e even for a non-compact group.

In f ac t any connected Lie group cont ains a compact subgroup as deformation

r e t r a c t . ( S e e K. Iwasawa, On some typ es of topological A n n a l s of

Mathematics 50 Theorem 6. )

we have

Ad =

- (Ad (A d .

The li ne ar trans forma tion Ad V i s skew-symmetric; that i s

< Ad > - < > .

This fol lows immediately f rom the identi ty Therefore we can choose

an or thonormal bas is for G so that the m at r ix of Ad V takes the form

PROOF of Choose two poi nt s p and q i n G which are not

conj ugat e alon g any geod esic . By Theorem 17.3 , has the homotopy

type of a with one ce ll of dimension A for each geodes ic f rom

p t o q of index X . By ther e are only f i ni te ly many A-cells for

each Thus i t only remains t o prove that the index X of a geodesic is

always even.

Consider a geodes ic 7 s t a r t i n g a t p w i th ve loc i ty vec to r

vP

According to 6 2 0 . 5 the conjugate points of p on are determined by the

eigenvalues of the l inea r t r ans fo rm at ion

def ined by

.

Defining the homomorphism

Ad V:

It fol lows that the

matr ix

. .

composite li ne ar trans forma tion (Ad (Ad V) ha s

Therefore the non-zero eigenvalues of

occu r i n pa i r s .

- are pos i t ive , and

It follows from 20.5 that the conjugate points of p along a l s o

In ot her words every conjugate point has even m ul t ip l i c i ty .occu r in pa i r s .

Together with the Index Theorem, this implies that the index X of an y

geodesic from p t o q i s even. Th i s completes the proof.





1 2 2 APPLICATIONS 2 2 . MANIFOLDS OF MINIMAL GEODESICS



( 2 ) The index of g a t each c r i t i c a l po in t wh ich l i e s i n the com-

p a c t s e t i s

It follows from 2 2 . 4 that any g which approxim ates

s u f f i c ien t l y c los e ly , the f i r s t and s econd der iva t ives a l s o be ing approx i-

mated, w i l l s a t i s f y ( 2 ) . In fa ct the compact se t can be

covered by fi ni te ly many compact s e t

nate neighborhood.

each of which l ies i n a coordi-

22.4 can then be applied to eachThe function i sThe proof of 22.5 now proceeds as fol lows.

smooth on the compact region

po in t s a r e non-degenerate, with index Hence the manifold

has the homotopy type of w i t h c e l l s of dimension

attached.

c and c r i t i c a l

Now cons ider t he map

0 h: C .

Since < it follows that h i s homotopic within t o

0 h ’ : .

But this l a s t p a i r i s contained in and U can be deformed in to

w i th in M. It fo l low s tha t i s homotopic within t o a map

h ” : - T h i s completes the proof of 2 2 . 5 .

The original theorem, 2 2 . 1 , now can be proved as fol lows. Clear ly

it i s s u f f i c i e n t t o p ro ve t h a t

o

f o r a r b i t r a r i l y l a r g e v a l u e s of c . A s i n the space I nt contains

a smooth manifold In t . . as deformation re tr ac t. The space

of minimal geodesics i s contained i n this smooth manifold.

The energy function E: w hen r es t r i c ted to

I n t a lmos t s a t i s f i e s the hypo thes is of 22.5. The only

d i f f i c u l t y i s t h a t ranges over the int erv al d E < c , in s tead of

the r equ i r ed in te rva l To c o r r e c t this, l e t

F:

be any

Then

F R

s a t i s f i e s the hypo thes i s of 22.5. Hence

d

< Th i s completes the proof.































Milnor J. Morse Theory (Princeton, 1963)

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