Top onogovs Theorem and Applications

by

Wolfgang Meyer

These notes have been prepared for a series of lectures given at the College on

Dierential Geometry at Trieste in the Fall of The lectures center around To

p onogovs triangle comparison theorem critical p oint theory and applications In the

short amount of time available not all the asp ects can be covered We fo cus on those

applications which seem to be most imp ortant and at the same time most suitable

for an exp osition Some basic knowledge in geometry will be assumed It has b een

provided by K Grove in the rst series of these lectures Nevertheless we try to keep

the lectures selfcontained and indep endent as much as p ossible For the result ab out

the sum of Betti numb ers in section a lemma from algebraic top ology is needed A

pro of for this result has b een provided in the app endix

I am indebted to U Abresch for many helpful conversations and also for writing

and typing the app endix

Contents

Review of notation and some to ols

Covariant derivatives

Jacobi elds

Interpretation of curvature in terms of the distance function

The levels ofadistance function

Data in the constant curvature mo del spaces

The Riccati comparison argument

The Top onogov Theorem

Applications of Top onogovs Theorem

An estimate for the number of generators for the fundametal group

Critical p oints of distance functions

The diameter sphere theorem

A critical p oint lemma and a niteness result

An estimate for the sum of Betti numbers

The soul theorem

App endix A top ological Lemma

Review of notation and some to ols

Covariant derivatives

We consider a complete Riemannian M with tangent bundle TM and Rie

mannian metric h i and corresp onding covariant derivative r of Levi Civita which

ector elds is the unique torsion free connection for which h i is parallel ie for anyv

X Y Z on M we have

r Y r X X Y

X Y

and

X hY Z i hr YZi hY r Z i

X X

The last two equations are equivalentto the Levi Civita equation

hr YZi X hY Z i Y hZ X i Z hX Y i

X

hZ X Y i hY Z X i hX Y Z i

If M is an arbitrary manifold and f M M a dierentiable map f T M

TM denotes the dierential of f r naturally extends to a covariant derivative for

vector elds along f For any vector eld A on M and any vector eld Y along f

ie Y M TM satises Y f where TM M denotes the pro jection the

covariant derivative r Y is well dened Due to the fact that r Y dep ends only

A A p

on A and the values of Y in a neighb ourho o d of the p oint p this extension is uniquely

p

determined by requiring the chain rule r X r X for any tangentvector vTM

v f f v

and any vector eld X on M

In a similar way the corresp onding covariant derivative for tensor elds carries over

to a covariant derivative for tensorelds along a map As a consequence one obtains

for example the Cartan structural equations for the Levi Civita connection

r f B r f A f A B

A B

R f A f B Y r r Y r r Y r Y

A B B A AB

where R is the curvature tensor of r A B are vector elds on M and Y is a vector

eld along the map f

For a curve c I M the parameter vector eld on I with resp ect to the parameter

t will be denoted by or D c t c j is the the tangent vector of c at t The

t t

t t

covariant derivative r Y for a vector eld Y along c is abbreviated by Y A parallel

D

t

vector eld Y along c is characterized by the linear dierential equation Y

a geo desic curve by the nonlinear second order equation c For consistency

reasons we avoid the often found notation r c resp r Y for the expressions r c

c c D

t

resp r Y when Y is a vector eld along c The inconsistency of such notation

D

t

b ecomes apparent when c is a singular curve for example a constant curve and Y a

nonconstant vector eld along c If X is a vector eld on M r X r X chain

c D c

t

rule is well dened

The exp onential map exp TM M is determined by the initial value problem for

geo desics If vT M then exp v c where c is the geo desic with initial condition

p

The restriction of exp to the tangent space T M at p is denoted c p and c v

p

by exp Notice that for complete the exp onential map is dened on all of

p

TM by the HopfRinow theorem

For a function f M IR and a vector eld X on M Xf denotes the derivative

of f in direction X The gradientof f is dened via the equation

hgrad fXi Xf

and the Hessian Hess f of f by

Hess f X r grad f

X

Hess f is a selfadjoint endomorphism eld ie hr grad fYi hr grad fXi

X Y

Imp ortant functions on a Riemannian manifold are distance functions or lo cal dis

tance functions from some point in M or from a submanifold of M A lo cal distance

function is a function in an op en subset U of M considered as a Riemannian subman

ifold If p U M and r q dist p q r q dist q p then r q r q

M U U U

r may b e dierentiable in p oints where r fails to b e dierentiable Atypical example

U

arises as follows Let c M b e an injective geo desic segment with initial p oint

p c and without conjugate p oints Then there is a neighborhood U of c

where r is dierentiable However r is not dierentiable in any p oint of the cut lo cus

U

of p For explicit examples lo ok at geo desics on a cylinder

On the set of p oints where a lo cal distance function is dierentiable it satises

jjgrad f jj The gradient lines of any function with this prop erty are geo desics

D E

parametrized by arc length since r grad fX hHess f grad fXi

grad f

hHess fXgrad f i hr grad fgrad f i X hgrad fgrad f i for any vector eld

X

X on M and hence r grad f Therefore the level surfaces of such a function

grad f

are equidistant They are referred to as a family of parallel surfaces

Jacobi elds

Jacobi elds J along a geo desic arise naturally as variational vector elds in one pa

rameter families of geo desic lines and are characterized by the linear second order

dierential equation

J R J c c

If V is a geo desic variation of c ie V I M is dierentiable and

j is a V t ct and t V t s is a geo desic for all s then J tV

t

s

Jacobi eld along c

j J t r r V D j r r V D j r V D D

t D D s t D D t t D s t

t t t s t

z

j r r V D j r r V D

t D D t t D D t

t s s t

z

R V D V D V D j R J c c

s t t t t

Therefore the Jacobi equation is the linearization of the geo desic equation along c

Notice that V can b e written in the following way If p is the curve ps V s and

Y the vector eld along p given by Y s V D j then V t s exp tY s The

t s

initial conditions of the Jacobi eld in terms of p and Y are J p J Y

Y is the initial vector of the geo desic c Any tangentvector u to TM can b e written

as the tangent vector u Y j of a curve s Y j TM Y is a vector eld along

s

the base curve ps Y j If Y and V are dened as ab ove we nd exp u

s

d

exp Y j V D j J Therefore the dierential of the exp onential map is

s

completely determined by Jacobi elds

J w along the For example the Jacobi eld with initial conditions J

geo desic exp tv is obtained from the variation V t s exp tv sw Here ps is

the constant curve Y s v sw J t exp j tw J exp j w This shows

tv p v

that the dierential of the restriction expj is determined by Jacobi elds on M

T M

p

with these initial conditions

Interpretation of curvature in terms of the distance func

tion

Consider two geo desics c c emanating from a p oint p in M c exp v c

exp w v w T M and the distance L distc c in a neighb orho o d of

p o

zero Then the fourth order Taylor formula for L is given by

L jjv w jj hR v ww v i O



When v w this implies for

hR v ww v i

Ljjv w jj O

jjv w jj

For linearly indep endent vectors v w satisfying jjv jj jjw jj this can be rewritten

as

Ljjv w jj K v w hv wi O



where K v w ist the of the plane spanned by v and w Therefore

L grows faster than linear if K and slower than linear if K in a neighborhood

of

To prove we consider the variation

c V t exp t exp

c

w w w p0 p0 p0 v v v

K < 0 K ≡ 0 K > 0

Figure interpretation of sectional curvature

c1 (ε)

c1

T w (ε,t) p0 v aε (t)

c0 E(ε,t)

c0 (ε)

Figure setup for the pro of of

for small values of and t The parameter tangent elds along V are E V D

and T V D The parameter curves a t V t are geo desics connecting the

t

points c and c T is the tangent eld of the geo desics and t E j is a Jacobi

t

eld along a and E j c E j c

tjj is the length of a so that Notice that jja

Ljja tjj jjT jj

t

which is constant in t for xed The derivatives of H L up to the fourth order

are given by

H hr T Tij

D t

D E

H r T T hr T r T i j

D D t

D

E D E D

T j H r T T r T r

t D

D D

D E D E

IV

H r T T hr T r T i r T r T j

D t

D D D D

We will nowevaluate these derivatives at t in order to nd the co ecients for the

T will be used frequently E and r T r Taylor formula The equation r

D D D

t t

during this calculation Also notice that T j since V t p We have

t

E j E j r r

D D

since E j c and E j c From the Jacobi prop erty of E we obtain

r r E R E T T

D D

t t

so that

r r E j

D D t

t t

Hence t E j is a linear vectoreld along the constant curve a Since E j

t

c v E j c w it follows

E j v tw v

t

With this information we can already evaluate H and H

H hr T Tij

D t

D E

H r T T j hr T r T ij

t D D t

D

hr E r E ij

D D t

t t

jjv w jj

Next we show that from

r E j

D t

r r E j

D D t

t

T j r r

t D D

is a consequence of and follows from since

E r E R E T E r r T r r r

D D D D D D

t t

In view of the equations ab ove it suces to show r r r E j for the

D D D t

t t

pro of of For this observe

r r r E r R T EE R T E r T r R T ET

D D D D D D

t t t

The right hand side vanishes at t since r T and T j This suces to

D t

t

nd H

H hr r Tr T ij

D D D t

hr r E r T ij

D D D t

t

hR E T E r T ij hr r Er T ij

D t D D D t

t

from From we get

IV

H hr r r T r T ij

t D D D D

Furthermore

r r r T j r r r E j

D D D t D D D t

t

r R E T E r r r E j

D D D D t

t

R E r E E j r r r E j

D t D D D t

t t

Using and the symmetries of R we nd

IV

v wi hr r r Er T ij H hR v w

D D D D t

t

Since this must also b e constantin t the second term on the right hand side is constant

in t Now

hr r r E r T ij D D hr E r T ij

D D D D t t D D t

t

D D hr E r T ij

t D D t

by using and Therefore

D hr E r T ij hr r E r T ij hr E r r T ij

D D t D D D t D D D t

must b e linear in t But r r E j r r c r r E j

D D D D D D

E r T ij since it r r c and r r T j so that D hr

D D t D D D D t

vanishes at t and at t This proves

IV

H hR v wv wi

Equation now follows from and We leave it to the reader to

verify

V

H h r R v wv wi

v w

V

If H for all choices of v and w then M must be a lo cally symmetric space

since can be used to show that the op erator R R c c is parallel for any

c

geo desic c

The levels of a distance function

In this section we will see that Jacobi elds determine the second fundamental tensor

S of the level surfaces of a lo cal distance function f This will b e used to establish

the Riccati equation for S and a Riccati inequality

We have a natural unit normal vector eld N grad f along the level surfaces of

f The second fundamental tensor S of the levels with resp ect to N is the restriction

of the Hessian of f to the tangent spaces of the levels Su Hess fu r N for

u

tangent vectors u to the levels The derivative S r S in the normal direction is

N

dened by S Y r S Y r SY S r Y for any vector eld Y tangent to

N N N

the levels of f Notice that S Y is again tangent to the levels

Let M be a xed level M f fg after changing f by a constant The other

levels are then given by M f ftg For small values of t the levels M and M

t t

are dieomorphic via the dieomorphism E p exp tNp The dierential of

t

E j can be desrib ed in terms of Jacobi elds Let s ps be a curve in M with

t M o

o

tangent vector v p Then E v J t where J is the Jacobi eld along the

t

geo desic t E p of the geo desic variation V t s E ps J tV D j Its

t t s t

initial conditions are J p v J r N p j r N Sv compare

D p

s

section

The geo desic t V t s is an integral curve of N so that V D j t

t ts

N V t s With this information we obtain J t r V D j r V D j

D s t D t t

t s

r N V j r N j SJ t The second fundamental tensor of the levels now

D t V D t

s s

is determined by

SJ J

Riccati equation for Covariant dierentiation of this equation leads to the imp ortant

S Since is an equation along the geo desic ctVt it reads more precisely

S J J This is useful to remember for the chain rule in r S r S r S

c D c c N

t c

S J S J SJ S J S J Using the S for the computation of J r

c c D

t

Jacobi equation J R JNN we obtain the Riccati equation

S R S

N

where R denotes the curvature op erator R X R X N N in direction N

N N

If there is a lower bound for the sectional curvature K of M then the Riccati

equation leads to a Riccati inequality along the gradient lines c of f Let Y be a

parallel unit vector eld along c tangent to the levels ie hY c i Then by

D E

hSY Y i hR Y N N Y i S Y Y

K Y N jjSY jj

From the assumption K Y N and the Schwarz inequality we obtain the Riccati

inequality

hSY Y i hSY Y i

along c

Data in the constant curvature mo del spaces

Constant curvature mo del spaces are imp ortant in comparison theory b ecause the

geometric quantities in these spaces can b e calculated explicitly

n

n

M denotes the n dimensional hyp erb olic space IH of curvature if the

n

n

p

if euclidian space IR if and the standard sphere S of radius

n

Since hR v uu v i for any pair of orthonormal vectors u v T M wehave R

p u

n

R uu Id on the orthogonal complementof u in T M Therefore the Jacobi

p p

equation and the Riccati equation are rather simple

n

orthogonal to c are given by f Y where Jacobi elds along a geo desic c IR M

Y is a parallel vector eld along c and f IR IR is a solution of the dimensional

Jacobi equation

f f

Let sn and cs b e the solutions of with initial conditions sn sn

and cs cs ie

p

p

sn t t sin

for

p

cs t cos t

sn t t

for

cs t

q

p

sinh sn t jj t

jj

for

q

jj t cs t cosh

Furthermore let

ct tcs tsn t for sn t

The derivatives of these functions are given by

ct sn ct cs cs sn

Furthermore the following elementary equations hold

sn cs

sn a b sn acs bcs asn b

cs a b cs acs b sn asn b

A basis for the Jacobi elds orthogonal to c is given by fsn Y cs Y g where Y

varies over a basis of parallel vector elds orthogonal to c

Notice that the second fundamental tensor of the lo cal distance spheres at distance

r from a xed p oint p in any manifold is determined by equation where J is a

Jacobi eld with initial value J along a normal geo desic emanating from p ie

n

in M by

J r sn r Y r

with Y parallel along c and hY c i Hence

sn r

S Y r Y r ct r Y r

cr

sn r

n

Therefore the principal curvatures of distance spheres in M are equal to ct r

The length of the great circles in the distance spheres is sn r In any manifold

the Hessian of the distance function from a p oint has a zero eigenvalue in the radial

direction For Karchers new pro of of Top onogovs theorem it is convenient to rescale

n

distance function f from the p oint p in M so that all the eigenvalues are equal the

This is achieved by taking md f where

Z

r

cs r for

md r sn tdt

r for

Notice the identity

cs md

From the formula

f hgradf vi gradf f Hess f v md Hess md f v md

sn f Hess f v cs f hgradf vi gradf

it follows that the eigenvalues of Hess md f at a p oint q with f q r are equal

n

to cs f q cs r Using md the lawof cosines in M b ecomes

md c md a bsn asn b cos

where a b c are the lengths of the edges of a geo desic triangle in M and is the

angle opp osite to the edge corresp onding to c Notice that this is a unied formula for

the three classical cases

c a b ab cos

p p p p p

c cos a cos b sin a sin b cos cos

q q q q q

cosh jj c cosh jj a cosh jj b sinh jj a sinh jj bcos

The Riccati comparison argument

A lower curvature b ound in M leads to an imp ortant estimate for the principal

curvatures in distance spheres and hence for the tangential eigenvalues of the Hessian

of the distance function f from a p oint For the mo died distance function md f

this yields an estimate for all the eigenvalues This estimate is the key for Karchers

pro of of Top onogovs theorem and the main reason for intro ducing md The basic

comparison argument is contained in i of the following elementary Lemma and its

Corollary cf K

Lemma Suppose g G are dierentiable functions on some interval satisfying the

Riccati inequalities

g g

G G

i If g r Gr then g r Gr for r r

ii If g r Gr then g r Gr for r r

Pro of From the two Riccati inequalities and we get

R

g G

g G e

from which i and ii follow immediately

The statement ii is useful for estimates involving upp er curvature b ounds K We

are interested mainly in i

p

Corollary If g a IR suppose a if satises g g

and lim g r then

r

g r ct r

Pro of If there is a p oint r a for which g r ct r we can cho ose so

that g r ct r Gr ct r satises the Riccati equation G G

r so that g r Gr on r Then g lim g r lim Gr on

r r

contradicting g

Consider now a normal geo desic segment c with initial point p which do es not

meet the conjugate lo cus of p In a neighborhood U of c we may consider the lo cal

distance function f q dist p q The principal curvatures of the lo cal distance

U

sphere f r at the pointq are denoted by q q From the corollary and

n

we get the estimate

q ct f q

i

q are the eigenvalues of Hess f j corresp onding to eigenvectors tangent to the

i q

distance sphere whereas the radial eigenvalue is zero The hessean of md f satises

values sn f q q i the corresp onding equation and therefore has eigen

i

n in directions tangent to the level r and the eigenvalue cs f q for the

radial direction grad f j This proves the op erator inequality

q

Hess md f cs f Id

Along c this estimate remains true up to the rst conjugate p oint of c which in the

p

case app ears at a distance not farther away than For M M equality

holds in

If f is replaced by g f where is a constant we have Hess g Hess f so

that the tangential eigenvalues of Hess md g jq according to formula are given

by sn g q q and the radial eigenvalue is cs g q The estimate for q ab ove

i i

sn

leads to sn g sn g ct g cs g For small values of

i

sn g

p

and g in the case the Hessian of md g satises consequently

sn

Id Hess md g cs g

sn g

In the case this estimate along c holds up to the rst conjugate p oint

The Top onogov Theorem

The Top onogov comparison theorem app ears to be one of the most powerful to ols

in It is a global generalization of the rst Rauch comparison

theorem The ideas trace back to AD Alexandrow who rst proved the theorem for

convex surfaces Top onogovs pro of of the theorem was technical and contained some

diculties which were resolved in GKM Since then the pro of had b een simplied

considerably by various geometers compare also CE In this lecture series we shall

use an interesting new pro of given byKarcher K In contrast to the previous technique

the Rauch comparison theorem is not used at all It uses the estimate for the Hessian

given in resp and ts nicely into our discussion of distance functions This

do es not mean that our approach is necessarily shorter or more geometric than the

other viable arguments given b efore We certainly encourage the student also to go

through some alternate pro of of Top onogovs basic result in the literature mentioned

ab ove

Denition A geo desic hinge c c in M consists of two non constant geodesic

segments c c with the same initial point making the angle A minimal connection

o

c between the endpoints of c and c is cal led a closing edge of the hinge

o

The length of a geo desic segment c will b e denoted by jcj

et M be a complete Riemannian manifold with sec Theorem Top onogov L

tional curvature K

A Given points p p q in M satisfying p q p q a non constant geodesic c

from p to p and minimal geodesics c from p to q i allparametrizedby

i i

p

arc length Suppose the triangle inequality jcj jc j jc j is satised and jcj

in the case denote the angles at p c c

i i o

c c jcj Then there exists a corresponding comparison triangle p p q

in the model space M with corresponding geodesics c c c which ar e al l minimal

of lengths jc j jc j jcj jcj and

i i

i the corresponding angles satisfy

i i i

ii distq ct distq ct for any t jcj

p

Except for the case when and one of the geodesics has length equal to

the triangle in M is uniquely determined

p

in case and B Let c c be a hinge in M with c minimal and jcj

o o o

c a closing edge Then the closing edge c of any hinge c c in M with

o o

jcj jcj jc j jc j satises

o o

jc j jc j

Remarks

Notice that c need not to be minimal and the case p p is not excluded c

and c have to b e minimal otherwise there are counterexamples

With a little eort statement ii can b e used to show that the length of secants

between c and c are not shorter than the corresp onding secants between c and

i

c provided the segment of c in the cut o triangle is minimal

i

iii distc t cs distc tcs holds as long as cj is minimal

s

iv distc t cs distc tcs holds as long as cj is minimal

sjcj

In the case when c is minimal now any corresp onding secants satisfy j j

j j

For symmetry reasons only iii needs to b e proved

By ii

distq cs distq cs

Connect p cs and q by a minimal geo desic and consider the triangle

s s

p p q with geo desic edges c cj and the corresp onding comparison tri

s s

os

angle p p q in M Using ii for this triangle we obtain

s

distp c t distcsc t

s

The monotonicity relation b etween angle and length of the closing edge of a hinge

and imply in M

c tp cs q p cs q p p c tp p

s s 2 Mκ q˜ M q

c˜1 c1

c˜0 (t) c0 (t)

p˜ 0 c(s)˜ p˜ 1 p0 p1

c(s) = ps

p˜s

Figure sketch for the pro of iii

and then

distc s c t distp c t

s

Inequality iii now follows from and

Statement i is a consequence of ii To prove for example consider the

functions h t distc tct and h t distc t ct for small values

of t By iii we have h h According to of section we have the Taylor

formulas

h t t kc c k O t

h t t kc ck O t

k kc c k and hence so that kc c

The converse implication i ii is also true but more technical to prove

Statement ii carries over to limits in the sense of Gromov for Riemannian spaces

with curvature K where angles cannot b e dened anymore

Part Bis equivalent to Ai This follows immediately from the fact that in M

the length of a closing edge in a hinge with minimal geo desics and the hinge angle

are in a monotone relation Note that B is trivial in the case when the tiangle

inequality is not satised in M For this observe that the triangle inequality in

M is satised since all the corresp onding geo desics in M are minimal

If c is not minimal in B the statement is false For consider in S a hinge

with two geo desics of length making a p ositive angle The end p oints have a

p ositive distance for small However in the corresp onding hinge in S the end

p oints coincide

An anologue of Top onogovs theorem where the lower curvature b ound is replaced

by an upp er curvature b ound is false For example on the sphere S there are

homogeneous metrics Berger metrics with p ositive curvature upp er curvature

b ound and closed geo desics of length However if the sectional curvature

K of M satises K and c c c is a triangle with minimal geo desics and

p

jc j jc j jcj whichiscontained in a ball around p of radius not greater

than the injectivity radius at p then there is a triangle c c c in M with

jc j jc j jcj jcj and This is an immediate consequence of Rauchs

i i

rst comparison theorem

There are generalisations of Top onogovs theorem to a version where the mo del

spaces M are replaced by surfaces of revolution or surfaces with an S action

cf E A U Abresch p ointed out to me that these generalisations can be

handled with the same technique as used in the pro of b elow

Pro of of Theorem By remark ab ove we only have to prove Aii Note that

p

in the case we have diamM by Myers theorem For the case

the pro of is organized in three steps In step we consider the general case for

p p

but we assume diamM and jcj jc j jc j for the case In step

p p

the case diamM and jcj jc j jc j is reduced to step by a

simple limit argument Finallyinstep we show that in the case there are no

p

triangles with circumferece jcj jc j jc j

p

Step For the case we assume diamM and also that the circumference

p

of the triangle satises jcj jc j jc j so that the comparison triangles in M

p

for exists From the triangle inequality jcj jc j jc j we get jcj

p p

Therefore we can cho ose such that diamM and jcj We

rst lo ok at a simple case Supp ose q c jcj Then jcj jc j jc j since c and

c are minimal By the triangle inequalitywe must have jcj jc j jc j Therefore q

divides c into two minimal pieces of length jc j and jc j Consequently equality holds

in ii since the geo desics from q to ct are parts of c If q c jcj we pro ceed as

follows

We consider the distance functions r from q in M r from q in M and dene

ht md r ct

ht md r ct

t ht ht

The idea is to show that cannot have a negative minimum by the use of the Hessian

estimate in section Unfortunately h is not dierentiable in general since r

is not dierentiable beyond the cutlo cus of q This problem is resolved by a lo cal

approximation with a sup erdistance function The argument is slightly dierent in

the cases and

In the case if has a negative minimum in jc j also the function

dened by

tjcjt

tt

jcj

has a negative minimum in jcj

p

and dene tsn t sn on In the case wehave jcj

jcj If has a negative minimum then

has a negative minimum

For the p oint t jcj where or or has a negative minimum we approximate

r by lo cal dierentiable functions in a neighborhood of ct Let be a normal

geo desic from q to ct For small values we dene in some neighb orho o d U

of j j the lo cal sup erdistance functions

x r x distq x r x dist

U

r is dierentiable if U is suciently small Therefore the function

h md r c

is dierentiable in some interval around t and

h t ht h h

Using the estimate for the Hessian we have

h hHess md r j c c i

c

sn

cs r c

sn r c

for small The quantity r ct is b ounded away from zero indep endent of

p

and r ct dist ct from the diameter assumption Observing

we get

h const sn h

with a constant indep endent of Since h h the dierence h h

satises

const sn

Furthermore

t t

by

Case

If has a negative minimum at t then also has a negative minimum at

t but

t t const sn const sn

z

For suciently smallthisisa contradiction

Case

At the p oint t jcj where has a negative minimum we consider and also

by dened

tjcjt

jcj

Then and t t by Therefore also has a lo cal negative

minimum at t But

const sn

jcj

which is a contradiction for small

Case

has a negative minimum we also lo ok at At the p oint t where

o

Again and t t so that has a negative minimum at t

Dierentiate at t to obtain

j t

t

and

j t

t

sn j

t

const sn sn

t t

for suciently small a contradiction

p p

Step Assume now diamM and jcj jc j jc j We

p

cho ose a sequence and lim Then diamM and

i i i i

i

p

jcj jc j jc j By step the theorem holds for the sphere S IR as the com

i

i

i i i

parison space By compactness the sequence of comparison triangles c c c

i

has a subsequence converging to a comparison triangle in S By continuity of the

family of distance functions on the family of spheres S IR statement Aii

now follows for the limit triangle

p

Step Supp ose and jcj jc j jc j We can cho ose such that

p

Then for the comparison triangle in M the geo desics c c c jcj jc j jc j

p

have length and therefore form a great circle The antipodal point q of q has

p

to b e a p ointof csay q ct By step we have distq ct distq ct

p p

contradicting distq ct This completes the pro of

Applications of Top onogovs Theorem

An estimate for the number of generators for the fun

dametal group

As a rst application of Top onogovs theorem we present Gromovs theorem concerning

the numb er of generators for the fundamental group M Since any element of the

fundamental group M with base p oint p of a Riemannian manifold M can be

represented by a geo desic lo op of minimal length at the p oint p it is clear that the

geometry of M should have strong inuence on the structure of M The earliest

result in this direction is Myers theorem cfCE GKM the universal cover of a

compact Riemannian manifold with strictly p ositive Ricci curvature is compact and the

fundamental group nite If the sectional curvature K of a compact even dimensional

manifold is strictly p ositive then by the Synge Lemma cf CE GKM M

or Z dep ending on the orientabilityof M If M is complete noncompact and K

n

then M since M is dieomorphic to IR cf GM Finally if M is complete

noncompact and K then by the soul theorem of Cheeger and Gromoll CG

M contains a lattice group of nite index

Theorem Gromov

n n

i Suppose the sectional curvature of M is nonnegative Then M can be

p

n

generated by N n elements

n

ii If the sectional curvature K of M is bounded from below K and the

n n n

diameter of M is bounded diamM D then M can be generated by

p

n



elements N n coshD

M M Pro of Let G M p be the fundamental group with base point p

denotes the Riemannian universal cover of M The group of covering transformations

G acts on M by We cho ose a p oint x M which covers p and dene for

G the displacement

j j distx x

A minimal geo desic c from x to x pro jects in M to a lo op of minimal length j j

in the homotopy class representing There are only nitely many elements of G

satisfying j j r An innite sequence x of p oints would have a limit p ointinthe

i

compact ball of radius r around zero contradicting the covering prop erty Therefore

G with the prop erty j j minfj jj Gg Inductively wecancho ose an element

we can construct generators of G satisfying j j j j as follows

Supp ose are constructed already and the subgroup generated

k k

G so that j j minfj jj by is not equal to G Then we can cho ose

k k k

Gn g For ij we have j j j j and

k i j

dist x x j j

ij i j j

To prove the last inequality supp ose j j Then has displacement

ij j j

i

j

j j j j and contradicting the choice of

ij j j j

j j

j

For each we cho ose a minimal geo desic c from x to x of length j j For

i i i i i

ij we cho ose a minimal geo desic from x to x of length By Top onogovs

i j ij

theorem the angle c c is b ounded b elowby the angle of a compar

ij i j

ison triangle in M where for i and for ii By the law of cosines

in M

i j ij

cos for

i j

cosh cosh cosh

i j ij

for cos

sinh sinh

i j

The right hand side of is increasing in the variable to see this dierentiate

i

The relation now leads to the estimates

i j ij

i j j

cos for

i

cosh cosh cosh

j j j

cos

cosh

sinh

j

j

cosh D

for

coshD

For the last inequality observe that D by the construction of the generators

i i

of G To see this observe that for any lo op at p in M is homotopic to a

comp osition of lo ops with length D Sub divide the original lo op into segments

of length and then insert minimal connections from the sub division points to

p and their inverses Since in the construction j j is chosen to be minimal in

k

Gn itfollows j j D but was arbitrary Let

k k

for

coshD

arccos for

coshD

then To complete the argument consider the initial vectors v

ij i

c T M Wehave v v In T M there can b e only a nite number

i x i j x

of distinct unit vectors with this prop erty A rough explicit estimate for the maximal

number is obtained as follows The intrinsic balls of radius around the p oints

n

v in the unit sphere S in T M are disjoint Therefore the maximal number N

i x

n

of points v is estimated by the volume of S divided by the volume of a ball of

i

n

radius in S The volume of this spherical ball is estimated b elow by the

volume of a euclidian n ball of radius sin This estimate however can be

impro by the following simple observation The generators satisfy ved by a factor

j j j j Therefore wealsohave v v Hence the volume of the sphere

i i j

i

can be replaced by the volume of the real pro jective space and we obtain

p

n

n n

n



p

vol S

N

n n

n

n

vol B sin cos

sin

q

q

n n n n

The logarithmic convexity of can be used to nd

Inserting the appropriate value for from into the estimate for N nishes the

pro of

The estimates given in the Theorem are never sharp as can be seen by lo oking at

surfaces

Critical p oints of distance functions

Distance functions on a Riemannian manifold M are not dierentiable in general

Despite this fact it is p ossible to develop a critical p oint theory similar to the Morse

theory of a dierentiable function The idea was intro duced by Grove and Shiohama

GS for the pro of of their diameter sphere theorem cf theorem Subsequently it

has b een rened by Gromov G in connection with his niteness result for the sum of

the Betti numbers cf theorem These applications deal with distance functions

from a p oint in M Recently Grove and Petersen have generalized the concept for

distance functions from closed subsets of a manifold in particular from the diagonal

in M M This leads to an interesting niteness result concerning the number of

homotopytyp es of Riemannian manifolds cf GP

Denition Let A be a closed subset of M Consider the distance function dist

A

from A denedby dist q distA q Apoint q M is said to beacritical p oint for

A

dist if for any vector v T M there is a distance minimizing geodesic c from q to A

A q

satisfying

hv c i

A noncritical point is cal led a regular point

For p oints q A this condition is equivalentto v c Instead of referring

to a critical or regular p ointfor dist we also shall say that q is critical or regular for

A

A Notice that any point q A is critical for A

In the following examples let Afpg

Examples

Then y i Consider the at cylinder S IR IR and p x y z x

p and q x y z are the only critical p oints for p

q cut locus of p

p

Figure cylinder

ii Consider the at torus T IR Z Z and p Then the only critical

points for p are p q q q

According to the denition a p oint q is regular for A if the initial vectors for all

minimal geo desic from q to A are contained in an op en half space of T M ie there

q

is avector v T M such that

q

hc vi

for any minimal geo desic c from q to A Of course equivalently we have a vector

w v such that

hc wi

for any minimal geo desic c from q to A q2 q3 q2 cut locus q2

cut locus q3 p q1 q1 p

q2 q3 q2 q1

flat torus torus of revolution

Figure critical p oints and cut lo cus for a p oint p in tori

Lemma lo cal existence of gradientlike vector elds Let M be complete

and A a closed subset of M Then for any regular point q of dist there is a unit

A

vectoreld X on some open neighborhood U of q such that

hX c i

q

for any q U and any minimal geodesic c from q to A

Denition A unit vector eld X on U satisfying is cal led a gradientlike

vector eld for dist

A

Pro of Since q is a regular p ointwecancho ose a unit vector X T M with hX c i

q q q

for any minimal geo desic c from q to A Extend X to an arbitrary smo oth vector

q

eld on some op en neighb orho o d of q Then X satises condition on a suciently

small ball U around q Otherwise there would be a sequence of points q converging

i

A satisfying hX c i A limiting to q and minimizing geo desics c from q to

q i i i

i

geo desic c of c would be a minimal geo desic from q to A with hX c i con

i q

tradicting the choice of X

q

Corollary existence of global gradientlike vector elds As above let M

be complete and A a closed subset of M Then

a The set of regular points of dist is open

A

b On the open set U of r egular points there exists a gradientlike vector eld for

dist

A

Pro of a is obvious from For the pro of of b we p oint out that lo cal vector elds

of the lemma can b e glued together by means of a partition of unity to obtain a vector

eld X on U satisfying This is a consequence of the following observation If

v v are unit vectors in a euclidian vector space satisfying hv wi then any

m i

P P

m m

convex linear combination v v satises hv wi

i i i i

i i

Now we can take X Xk X k

The following lemma contains an imp ortant monotonicity prop erty for gradientlike

vector elds

Lemma Let M be complete A a closed subset U an open subset of M and X

a gradientlike vector eld for dist on U the ow of X and the ow of X

A

Then

a dist is strictly decreasing along any integral curve of X

A

b On any compact subset C of U the decreasing rate is control led by a Lipschitz

constant There is a constant such that

dist q t dist q t

A A

as long as q t C for Equivalently we have

dist q t dist q t

A A

as long as q t C for

Pro of It suces to proveb First notice that X satises the inequality hX c i

q

for some any q C and any minimal geo desic c from q to A Other

wise there would be sequences q C and minimal geo desics c from q to A with

i i i

lim hX c i By compactness there would be a limit point q C and a

i q i

i

contradicting Con minimal limiting geo desic from q to A with hX c i

q

sider now the function ht dist q t We construct an upp er supp ort function

A

h for h as follows Let p A with distp q t dist q t and cho ose a min

A

imal geo desic c M from q t to p For a xed let

ht distcjcj q t h is dierentiable in a neighborhood of t and sat

ises ht distp q t dist q t ht and ht ht The derivative

A

E D

j hc X q t i at t is given by h t grad dist j

qt cjcj qt

t

hence h t Such a supp ort function exists at any t therefore condition

follows easily

As an immediate consequence we have

Corollary Any local maximum point q of dist is a critical point for A

A

n

Corollary Let M be complete and B B p r a bal l of radius r around the

point p M Suppose there are no critical points of dist in B

p

Then B is a topological nsubmanifold of M

ve to show that B is lo cally euclidian Consider a vector eld Pro of We only ha

X on the set of regular points with prop erty and the ow of X For a

given p oint q B let Q b e a lo cal ndim submanifold through q which is transver

sal to X for example the image under the exp onential map of a neighborhood V

of the origin in the nplane orthogonal to X in T M By the inverse map

q q

ping theorem we can assume that V and are chosen such that j

Q

is a lo cal dieomorphism Since t dist q t is strictly decreasing we have

p

dist q dist q dist q Therefore by continuity we can assume

p p p

that dist q dist q dist q for q Q after shrinking Q if neces

p p p

sary Nowany integral curve of X through a p oint q of Q meets exactly one p ointof

B dist r by the monotonicity prop erty The map from Q to B dened by the

p

pro jection along the integral curves is a homeomorphism onto its image

Corollary Let M bea complete noncompact manifold and suppose that for some

point p M al l the critical points of dist are contained in a bal l B B p r

p

Then M is homeomorhpic to the interior of a compact manifold with boundary

Pro of Let X be a gradientlikevector eld on the set of regular p oints with ow

Then F B M dened by F q t q t maps B homeo

morphic onto M n B For this the prop erties and kX k are imp ortant

Hence M is homeomorphic to B F B B F B with b oundary

F B fg B

Corollary Suppose that there is no critical point of dist in B p r nfpg

p

Then B p r is contractible

Pro of An easy exercise

Denition An isotopy of M in the topological category is a homotopy G

M M such that p Gp is a homeomorphism from M onto a subset of

M for any and p Gp is the identity map of M

If B and B are subsets of M we say that the isotopy G moves B into B provided

GB fg B

Corollary Isotopy Lemma Given a complete manifold M a point p M

r r and an open neighborhood U of the annulus A B p r n B p r

Assume that there are no critical points of dist in A

p

Then there is an isotopy of M which is the identity on M n U and which moves B p r

into B p r

Pro of Using a partition of unity one can construct a vector eld X on M which is

gradientlike on some neighborhood W of A with W U and X j

M nU

If r we can cho ose such that holds on the compact set A Then

r r the isotopy G dened by Gq q t moves B p r into for t

B p r where again is the ow of X

If r we consider F B M F q t q t and use on the

domain of this homeomorphism onto a subset of M an isotopy induced from a defor

t

mation of into for instance Gt ln e

The elementary corollaries ab ove demonstrate that gradientlike vector elds can

be used for deformations in the same way as gradient vector elds in standard Morse

theory However all these deformation arguments are useless unless one can get addi

tional information on the set of critical p oints In standard Morse theory the Morse

Lemma is an imp ortant to ol for this purp ose Unfortunately there is no analogue of

the Morse Lemma available In fact one cannot saymuch ab out the change of top ology

of B p r dist r when r passes a critical level

p

In the presence of a lower curvature b ound however Top onogovs comparison theo

rem can b e used to obtain additional information ab out critical p oints leading to rather

strong conclusions In contrast to standard Morse theory the information obtained on

the set of critical p oints is more of a global nature For the pro of of one has

to consider not just a single distance function but all the distance functions from the

various p oints of M

The diameter sphere theorem

One of the famous results in Riemannian geometry is the pinching sphere theorem

cf GKM CE which can be stated as follows

n

Theorem Rauch Berger Klingenb erg Suppose M is complete simply

connected and the sectional curvature K satises

K

Then M is homeomorphic to the standard sphere

One of the essential steps in the pro of of this theorem is to show that the injectivity

p

radius of the exp onential map and hence the diameter of M is where

is the minimum of the sectional curvatures on M Grove and Shiohama have

generalized the pinching sphere theorem to the diameter sphere theorem b elow by

replacing the upp er curvature b ound by this lower b ound for the diameter The pro of

is a nice application of critical p oint theory and of Top onogovs theorem

n

Theorem GroveShiohama Let M bea complete manifold with K

n

p

and diamM Then M is homeomorphic to S

Pro of After rescaling the metric we can assume K and diamM Let p q be

two p oints of maximal distance in M distp q diamM By corollary q is critical

for p We show that q is uniquely determined by p Supp ose q q are two p oints

satisfying distp q diamM Cho ose minimal geo desics c from q to q and c from

i i

q to p Since q is critical for p c can b e chosen suchthat c c

i

and jcj Consider the corresp onding Then jc j jc j diamM

comparison triangle c c c in the standard sphere with corresp onding angle and

By the law of cosines edge lengths jcj jc j jc j Then by

in S we have

sin sin cos cos cos cos cos cos

and hence ie q q

Next we show that p and q are the only critical points for p more precisely Let

q q q M q q p and c M b e a minimal geo desic of length from

q to q Then for any minimal geo desic c from q to p we have hc c i

ie the vector v c T M can be used to dene the op en half space for the

q



regularity of q To show this cho ose a minimal geo desic c from q to p and let

jc j jc j By the uniqueness of q q we have and

For the geo desic triangle c c c with angle c c we

consider the corresp onding comparison triangle in S with corresp onding angle

Then hc c i cos cos and by the law of cosines

cos cos cos

cos

sin sin

since

Now let be suciently small such that exp j and also exp j are lo cal

B p B q

dieomorphisms The vector eld X grad dist j satises condition

p

B pnfpg

in section The regularity argumentabove shows that X grad dist j

q B qnfq g

satises this condition as well Therefore one can construct a gradientlikevector eld

X on M nfp q g which coincides with X on B p nfpg and with X on B q nfq g

and kX k The ow of X satises on all of M n B p B q Hence

all the integral curves of X have nite lengths and extend continuously to the end

points p and q For a unit vector v T M the integral curve t exp v t

p v

is dened on an interval where is the length of Since X is dierentiable

v v v

the function v is dierentiable Let F t v t F v p F v q

v v v

for t and kv k Then the map G tv F t v maps the closed unit ball B

n

of T M onto M inducing a homeomorphism from the quotientspace BB S to

p

M

With a slight mo dication of the reparametrisation of in the pro of ab ove the

v

map G can be made smo oth in the interior B of B However there is no information

ab out the twist of the map near q

The diameter sphere theorem may also be viewed as a diameter pinching theorem

for manifolds of p ositive curvature The quantity

diamM

minK

M

is invariant under scalings of the metric By Myers theorem we have Ac

M

n n

If cording to the diameter sphere theorem M is homeomorphic to S if

M

n n

M is isometric to the sphere S This rigidity result was originally obtained

M

by Top onogov as an application of the triangle comparison theorem cf CE but it

also follows from the more general theorem of Cheng Cg which has b een discussed in

the rst part of this lecture series

If one relaxes the assumption in the pinching theorem to K then there

is the rigidity theorem of Berger cf CE In view of this result one also should exp ect

a rigidity theorem if one assumes K and diamM In fact Gromoll and Grove

cf GG have obtained a corresp onding result

Under the given hyp othesis either

a M is homeomorphic to a sphere or

b M has the cohomology ring of the Cayley plane or

m

c M is isometric to one of the following spaces with their standard metrics CPl

d

n

z z z z g S where the or IHP CPl fz z

d d d d

n

thogonal representation of M on IR is reducible

The pro of is somewhat technical for our exp osition

A critical point lemma and a niteness result

The critical p oint lemma b elow was one of the basic observations which lead Gromov

to the niteness theorem in the next section Its pro of is a simple application of

Top onogovs theorem used twice The given estimate is somewhat stronger than in

Gromovs original lemma It was also used by Abresc h A

Lemma critical point lemma Let M becomplete and p q q M q p

i

and assume q is critical for p Furthermore let c be minimal geodesics from p to q

i i

of length and c c

i

a If the sectional curvature satises K then

cos

b If K and diamM D then

D cothD cos

Pro of Let c b e a minimal geo desic from q to q of length Since q is critical for p

there is a minimal geo desic c from q to p of length such that c c

Using Top onovss theorem part B for this hinge and the law of cosines we get

for K

cosh cosh cosh for K

Consider now the geo desic triangle c c c and the corresp onding triangle c c

Then c with the same edge length in the comparison space IR resp ectively M



the angle comparison theorem A i leads to c c or eqivalently

cos cos Applying again the law of cosines we can nish the argument

cos for K

and

cosh



cosh cosh

cosh cosh cosh

cosh



cos

sinh sinh sinh sinh

tanh coth coth D coth D

for K

Corollary

n

a Given a complete manifold M with K and a constant L Then there

are only nitely many critical points q q for the distance function dist

k p

satisfying

dist q L dist q

p i p i

p

n

If L then k n

b For manifolds with K and diamM D the same statement holds for

p

n

L D coth D

Remark

By reversing the indexing of the p oints q we also have at most n critical p oints

i

satisfying

dist q dist q

p i p i

L

if L is chosen as sp ecied in the corollary

Pro of of corollary We consider the case K and leave the simple mo di

cation for b to the reader Connect p and q by minimal geo desics c of lengths

i i i

Then L for ij By the critical p oint lemma the angles c c

i j ij i j

j

satisfy cos or equivalently arccos There are only nitely

ij ij

L L

i

many vectors in T M with this condition compare also the pro of of theorem If

p

p p

n n

and k n If L then L wehave

n ij ij

n

p

where arcsin By the ball packing argument due to Abresch cf A

n

n

part I I there are at most n vectors in IR making a pairwise angle

n

n

Corollary Let M be a complete noncompact manifold with K p M

Then al l critical points of dist are contained in some bal l of nite radius around p

p

As a consequence we obtain the following

n

Theorem Gromov Let M beacomplete noncompact manifold with K

Then M is homeomorphic to the interior of a compact manifold with boundary hence

M is of nite top ological type

Pro of Since the critical p oints of dist are contained in some ball of nite radius

p

corollary applies

Recent examples of Sha and Yang show that a similar result do es not hold for

manifolds with p ositive Ricci curvature cf SY However if Ric and in addition



n

K and the diameter growth of Bp r is of the order or then the same

conclusion as in the theorem holds cf AG

The ab ove theorem may b e viewed as a weak version of the much more subtle

soul theorem of Cheeger and Gromoll by which M contains a compact totally geo desic

submanifold S such that M is dieomorphic to the normal bundle of S in M cf

CG CE and section

An estimate for the sum of Betti numbers

In this section H M denotes the singular homology of M with co ecients in some

th

Betti number of M with resp ect to F is given by b M arbitrary eld F The k

k

dim H M For a compact nmanifold and by theorem also for complete n

F k

P

n

manifolds M of nonnegative curvature we have b M dim H M

k F

k

By an ingeniously designed Morse theory for distance functions Gromov G ob

tained the following result

Theorem Gromov

a There is a constant C n such that any complete nmanifold M of nonnegative

curvature satises

dim H M C n

F

b Given D and and n then thereisaconstant C D n such that any

complete nmanifold with sectional curvature K and diamM D satises

dim H M C D n

F

In his pap er G Gromov indicated that a similar theorem as a holds for manifolds

with assymptotically nonnegative curvature U AbreschAgave the precise denition

of assymptotically nonnegative curvature for which such a theorem can be proved

He also rened Gromovs metho d and develop ed the necessary to ols to obtain the

following result

R

c Let IR IR be a decreasing function satisfying rr dr then

there is a constant C n such that

dim H M C n

F

n

for any complete Riemannian manifold M with sectional curvatures K r

r

at distance r from a given point p M

Remarks

The lower b ound for the sectional curvature cannot b e replaced byalower b ound

for the Ricci curvature Sha and Yang recently have constructed metrics of p os

itive Ricci curvature on the connected sum of an arbitrary number of copies of

n m

S S cf SY In SY they also gave complete noncompact examples with

p ositive Ricci curvature of innite homology typ e

Under the hyp othesis in the theorem one cannot exp ect niteness for the number

of homotopy typ es Here the lens spaces and also the simply connected Wallach

examples AW should b e observed

However Grove and Petersen have shown that there are only nitely many homo

topytyp es of compact manifolds if in addition to the lower curvature b ound and

the upp er diameter b ound one assumes alower b ound for the volume cf GP

The metho ds for the pro of of a and b are essentially the same For the pro of

of c Abresch had to develop a more general version of Top onogovs triangle

comparison theorem compare remark in section Though the pro of of c is

somewhat more technical the rened metho d of Abresch leads to a simplied

pro of of a and b It also gives a b etter estimate for the constants C n and

C n D than in G

For reasons of exp osition we concentrate on the pro of of a using the rened version

due to Abresch So we assume K for the remainder of this section unless stated

otherwise We also will x the constant

p

n

L

as determined in corollary

It is convenient to use the the following notation in connection with metric balls

If B is a ball of radius r around p then B denotes the concentric ball of radius r

around p

In contrast to standard Morse theory one cannot estimate the dimension of the

homology of the sublevels of distance functions ie of metric balls directly since the

intersection of a ball with the cutlo cus can be rather complicated As a replacement

for this part of the Morse theory Gromov intro duces the concept of content

Denition Let Y X beopen subsets of M The content of Y in X is dened

as the rank of the inclusion map on the homology level

contY X rkH Y H X

The content of a metric bal l B in M is dened as

contB contB B

The contentof B is a measure for howmuch of its homology survives after the inclusion

y contractible ball B By corollary and map into B Clearly contB for an

the Isotopy Lemma for suciently large balls B there is an isotopy of M which

moves M into B Therfore there is a map f M M such that the induced map f

is the identityon H M and f M B B M Hence contB contM M

dim H M

F

The strategy for the pro of now consists in showing that the content of any metric

ball and hence of M is b ounded by a constant C n For this purp ose Gromov in

tro duces the concepts of corank and compressibility for metric balls with the following

prop erties

i Either a ball of content is incompressible or it can be deformed into an

incompressible smaller ball of at least the same content and of at least the same

corank

ii The corank is b ounded by a constant k n

iii If a ball B of radius r and of corank k is incompressible then any ball of radius

r

with center in B has corank at least k

L

iv A ball with maximal corank has content

Now the pro of is based on a reverse induction over the corank By i only incom

pressible balls need to be considered Supp ose that the content of any ball of corank

k is b ounded by a n Let B be an incompressible ball of radius r and corank k

k

r

Then B is covered by balls B of radius such that the concentric balls

i n

L

B are disjoint The maximal number N of these balls can be estimated from ab ove

i

by the BishopGromov volume comparison argument It dep ends only on n Using

prop erty iii and the induction assumption a top ological argument stemming from a

generalized MayerVietoris sequence for nested coverings then is used to show that the

n

content of B is b ounded by a n N completing the induction argument

k

We start intro ducing the concept of compressibility which essentially corresp onds

to compr essibil ity used by Abresch with the xed value

Denition Abal l B of radius r in M is cal led compressible if there isabal l B

of radius r r around some point in B such that there is an isotopy of M which

is xed outside B and which moves B into B Briey we say that B is compressible

into B when these conditions hold

If B is compressible into B then B B B and the pairs B B and B B

are homotopically equivalent Therefore it is clear that

contB contB

Consequently for each ball B of content there is an incompressible ball B

B B such that contB contB For this observe that the injectivity radius

B is b ounded b elow by some constant and if a ball can on the compact ball

be compressed successively into a nal ball of radius then it must have content

since the balls are contractible

Lemma Suppose B is an incompressible bal l of radius r Then for any point

p B there must be a critical point q for the distance function dist in the compact

p

r annulus A B p r n B p

p

Pro of Supp ose for some p oint p B there is no critical point in A Let B

p

B p r Then we have inclusions B B p r and B p r B By the isotopy

lemma there is an isotopy of M which is xed outside B moving B p r and

hence B into B contradicting the incompressibilityof B

Denition Given p M r Let k p be the maximal number of critical

r

points q j k p for dist satisfying

j r p

dist q Lr and dist q dist q

p j p j p j

L

The corank of the bal l B B p r is dened as

corankB inf fk p j p B g

r

Note that k p n and therefore corank B n by the choice of L If B is

r

compressible into B then corankB corank B

As an immediate consequence of the previous Lemma we have

r

Corollary Suppose B B p r is incompressible and r

L

a If p B then k p corankB

r

b If p B and B B p r then corankB corankB

Pro of For a let q j k p k be critical p oints with dist q Lr and

j r p j

dist q dist q Since B is incompressible there is a critical p oint q for p

p j p j k

L

in the annulus A as in lemma Thus Lr r dist q r dist q

p p k p j

L

Now the k p points q satisfy the condition for the denition of k p hence a

r j r

follows

B B Now k p corank B for For b observe the inclusions B

r

L

any p B

As a consequence a ball of maximal corank must have content If B has maximal

corank then b ecause of b in the lemma B must b e compressible into a ball of radius

r with the same maximal corank and at least the same content This pro cedure can

k

b e rep eated k times until one reaches a ball B of radius r which is smaller than the

B Then contB and hence contB injectivity radius on the compact ball

These are the basic ingredients from critical p oint theory We now turn to the

covering arguments

Lemma Given an ndim Riemannian manifold M of nonnegative Ricci curva

ture a bal l B of radius r and a covering of B by bal ls B B of radius r

N

with center in B such that the corresponding bal ls B B are disjoint Then

N

r

n

N

Pro of Cho ose i such that the ball B with center p has the smallest volume

i

among all the given balls The ball B around p of radius r contains all the B

i

Therefore

n

r volB

N

B vol

i

where the last inequality is the BishopGromov estimate for the volume of concentric

balls which has b een discussed in the rst series of these lectures given by K Grove

compare also K for a pro of

Since the pro of of theorem will b e based on reverse induction over the corank

we intro duce the following notation

Let k n be the maximal corank of metric balls For k k we denote by B

k

the set of balls having corank k

The top ological information for the induction step is contained in the next lemma

Lemma Suppose cont B is bounded by a constant a for any B B Further

k k

more let B B be incompressible Then

k

n

contB a N

k

n n

where N L

r

Pro of Cho ose a covering of B by balls B B of radius r such that

N n

L

n n

B B are disjoint Then by lemma N L For j n

N

j j j

j

we also consider the coverings B B where B B The radii of all

i

i

N

j

r

corankB k these balls are By corollary we have corankB

i

L

j

a Using the result on the nested coverings in corollary of the hence contB

k

i

app endix we obtain

n

X X

n n n n n

cont B B cont B B B B

i i i i i i

i i i i

j

By the choice of the radii and the triangle inequalitywehave inclusions B B

i

j j

B B and therefore

i i

 

n

B B B B

i i

i i

and

n n n n

n n

B B B B B B

i i i i i i

S S

n

B B The rst chain of inclusions implies contB cont and from

i i

i i

the second we conclude that the content of any of the intersections is b ounded by

n

contB a Since the number of terms in the sum on the right hand side is

k

i

n

b ounded by N compare in the app endix the pro of is complete

Pro of of Theorem a Reverse induction over the corank F or B B wehave

k

contB Assume now that contB a n for any B B Let B B If B

k k k

is compressible and contB then B can be compressed into a ball of at least

the same content and of at least the same corank Therefore we can assume that B

n

is incompressible Now lemma applies and we get contB a n N Since

k

k n we get recursively



n n

dim H M contM N

F

p

n n n n

Using L an explicit rough where N L L

estimate for C n is given by

  

n n n

C n

Remarks

Note that the exp onent in the estimate for C n is a p olynomial of order in n

Gromovs original constant dep ended double exp onentially on n The reason for

this improvement due to Abresch is the choice of L the mo dication of corank

and compressibility to eliminate one of Gromovs critical point lemmas which

all together gave a b etter estimate for the corank and nally the improvement

of the estimate in the inductive lemma where Gromov uses the estimate

N

contB a

k

The estimate for the constant C n still seems to b e far away from reality Known

examples of nmanifolds with nonnegativecurvature all have a sum of Betti num

n

b ers

The soul theorem

This nal section is devoted to the soul theorem cf GM CG

n

Theorem Cheeger Gromol l Let M be a complete noncompact manifold of

nonnegative curvature K Then there is a compact total ly geodesic submanifold S in

M such that M is dieomorphic to the normal bund le S of S If K then M

n

is dieomorphic to IR

We rst intro duce afew basic concepts which are needed for the pro of

Denition A nonempty subset C of M is cal led totally convex if for arbitrary

points p q C any geodesic with endpoints p and q is contained in C

Denition A ray in M is a normal geodesic c M for which any nite

segment is minimal For a ray c M we dene the halfspaces B respectively

c

H by

c

B B ctt

c

t

H M n B

c c

where B ctt is the open metric bal l of radius t around ct

Note that in a complete noncompact manifold M for any p M there exists a ray

c M with initial p oint c p For a sequence q M with lim p q

i i

i

and normal minimal geo desics c from p to q any limiting geo desic c obtained

i i

from a convergent subsequence of c will be a ray emanating from p c has an

i i

accumulation p ointinthe compact unit sphere in T M

p

The basic observation ab out the halfspaces H is the following

c

Lemma If M is complete noncompact of nonnegative sectional curvature then

H is total ly convex for any ray in M

c

Pro of Supp ose H is not totally convex ie there is a geo desic c M with

c

endp oints c c H but c s B for some s Then q c s B ct t

c c

for some t and hence q B ctt for any t t by the triangle inequality In

fact setting

t distq ct

we have

distq ct distq ct distct ct

o

t t t t

for t t

Let c s b e a p ointon c which is closest to ct Further consider the restriction

t

t t t

c c j and a minimal geo desic c from c s to ct Since c c s

t t

s

t

t t t

is the closest p oint to ct on c we have c c Furthermore jc j

t t

distc ct distc s ct distq ct t and jc j jc j Consider

t

t t

Using Top onogovs theorem part B with comparison space now the hinge c c

IR and the lawof cosines we obtain

t t t

dist c s ct dist c ct jc j jc j jc j t

t

Furthermore distc ct t since c H M n B Therefore t jc j t

c c

which for large values of t is a contradiction

Wenowxapoint p M For a ray c M we also consider the restriction

c cj Let

t

t

C H

t c

t

c

where the intersection is taken over all the rays c emanating from p

Lemma C is a compact total ly convex set for al l t moreover

t

and a C C for t t

t t

 

C fq C j distq C t t g

t t t

  

in particular

C fq C j distq C t t g

t t t

  

S

b C M

t

t

c p C

Pro of Clearly C is totally convex and closed and p C If some C were not compact

t t t

it would contain a ray c C starting from p use the same argument as for the

t

existence of rays in a noncompact manifold Now ct C for t t contradicting

t

the denition of C Statementcisobvious from the construction of C The pro of of

t t

a and b now is an exercise using only the deniton of C and the triangle inequality

t

cf CE

Note that the interior of C is nonempty for t This is not true for C in

t

general as can b e seen on the parab oloid of revolution in IR If p is the umbilic p oint

of the parab oloid then C fpg

⎫ ⎬ Ct ⎭

p

Figure parab oloid

The C provide an expanding ltration of M by compact totally convex sets Our next

t

goal is to construct minimal totally convex sets by a contraction pro cedure which will

be used to nd a soul S For this imp ortant part of the pro of we also need the lo cal

concept of convexity

Denition A subset A of M is cal led strongly convex if for any q q A there

is a unique minimal geodesic from q to q which is contained in A

Recall that there is a continuous function r M the convexity radius such

that for any p M any op en metric ball B which is contained in B p r p is strongly

convex cf GKM

Denition We say that a subset C of M is convex if for any p C there is a

number p rp such that C B p p is strongly convex

Note that a totally convex set is convex and connected Also the closure of a convex

set is again convex

Let C be a connected nonempty convex subset of M For l n we may

l l

consider the collection fN g of smo oth l dim submanifolds of M such that N C

S

k k

Let k denote the largest integer such that fN g is nonempty and N N C

N Lemma N is a smooth connected total ly geodesic submanifold of M and C

N C is a topological manifold with possibly empty bounary N N n N Moreover

Pro of outline The full details are technical therefore we only give the main

k

idea CG CE Let p N and p as in the denition ab ove Then p N for

p of p in some Therefore we can cho ose a neighborhood U N B p

N and p such that exp j is a dieomorphism onto a neighborhood

U

T of p in M where U fv TU j kv k g TM is the tub e in the

normal bundle U of U To prove that N is a submanifold it suces to show

that N T U Supp ose q N T n U C T n U Let q be the closest

U otherwise we get a contradiction to the invertibility point to q in U Then q

of exp j close to q The minimal geo desic from q to q then is orthogonal to U

U

By the choice of p the exp onential map in the ball of radius around q is

invertible Therefore all the unique minimal geo desics from q to q for q in some

neighborhood U of q are transversal to U and are contained in C The conical set

fexp tu j u M kuk q exp u U t g then is a k dimensional

q

submanifold in C whichcon tradicts the denition of k From the existence of T and

the convexityof C it follows that N is totally geo desic For the remaining statements

we refer to CG and CE

Denition Let C be a convex subset of M The tangent cone to C at a point

p C is by denition the set

v

T C fv T M j exp t N for some trpg fg

p p

kv k

Clearly if p N intC then T C T N The following lemma contains all the

p p

technical information ab out T C we need

p

Lemma tangent cone lemma Let C M be convex and p C

a Then T C nfg is contained in an open halfspace of T M

p p

b Suppose that there exists q int C and a minimal normal geodesic c d C

from q to p such that jcj distq C Then

g T C nfg fv T C j v c d

p p

where T C is the subspace of T M spanned by T C

p p p

Pro of a T C is convex in T M since C is convex If T C nfg is not contained in

p p p

an op en halfspace of T M then T C must b e a linear subspace of T M of dimension

p p p

the the dimint C and hence p is an interior p oint of C For b and the details of

analysis of convex sets we refer to CG

The following lemma is the key for constructing the soul of M via a contraction

pro cedure

Lemma contraction lemma Suppose M has nonnegative sectional curva

ture and C M is a closed total ly convex subset with C We set

a a max

C C fp C j distp C ag C

a

C

Then

a

a C is closed and total ly convex

max

b dim C dim C

max

c If K then C is a point

This is a corollary of the following more general lemma

Lemma Under the assumptions of lemma dist M IR is a

C

concave function ie for any normal geodesic c which is contained in C we have

ct t ct ct

If the sectional cuvature satises K then the strict inequality holds in

Pro of It is sucient to show that for any p oint cs of c there is a number

such that cs is b ounded ab oveby a linear function hs on s s satisfying

hs cs d Let c be a distance minimizing normal geo desic of length d

s

from cs to C and c c s Then we can take

s

hsd s s cos

To show hs cs we consider the three cases Note that

we only have to consider points s s

Let E denote the parallel unit vector eld along c Case with E c s

s

By the second comparison theorem of Rauch there is a number such that the

length of the curve c t exp s s E t has length jc j d jc j for s s

s s s

c d C hence The geo desic c s exp s s E d is orthogonal to c at q

s s

c T C by lemma so that ct int C for t q Therefore cs

q

jc jd d s s cos

s

Case Let E c be the unique unit vector in the convex cone spanned

s

by c s and c and extend it to the parallel vector eld E along c Dene c as

s s s

in the rst case to obtain

jc j d

s

Applying the hinge version of Top onogovs theorem or just Rauch I to the hinge with

geo desics t exp tE t s s cos and t cs t with angle

one obtains

dist cs exps s cos E s s cos

Combining and the inequality cs d s s cos follows

Case Cho ose the p oint c t on c such that distcsc d

s s s s

distcsc t and a normal minimal geo desic a from c t to cs

s s s s s

Then a c t Further E denotes the parallel vector eld along c j

s s s s

t d

s

with E t a The curve c t exp ja j E t t t d is of length

s s s s s

jc jd t for s s if is suciently small As b efore distcsC jc j

s s s

thus

distc sC d t

s

Applying the hinge version of Top onogovs theorem or just Rauch I to the hinges

cj c j resp ectively c j a we obtain ja j s s t

s s s t t s s

s s

s s

t s s cos resp ectively s s ja j t hence

s s

s

t s s cos

s

From and the estimate cs hs follows

The discussion of the strict inequality in the case k is left to the reader

Pro of of the soul theorem Let p M and consider the ltration of M by compact

totally convex sets C as in lemma If C let S C If C appli

t

cation of the contraction lemma to the compact totally convex set C gives us a

max

compact totally convex set C of dimension dim C Rep eating this pro cedure

leads us in a nite number n of steps to a compact totally convex set S C with

submanifold of dim S n and S In particular S is a compact totally geo desic

M

Wenow showthat M is dieomorphic to the normalbundle S The dieomorphism

is constructed by means of the ow of a gradientlikevector eld of dist Let q M n S

S

Then q C for some t or q int C By the contraction lemma wehave either

t

a max

q C for some a or q int C Rep eating this argument a nite number of

times we nd a compact totally convex set C such that q C and S int C Any

geo desic from q to S has its initial tangentvector in the tangentcone T C Hence all

q

such initial vectors are contained in an op en half space of T M compare lemma

q

Therefore dist has no critical p oints on M n S Cho ose such that expj is a

S

S

dieomorphism onto the tub e around S Here S fv TS j kv k g Then

X grad dist is a gradientlikevector eld on exp S n S suchthat hX j c i

S q q

for the unique minimal normal geo desic c from q to S Therefore one can

q

construct a global gradientlike vector eld X on M n S such that hX c i for

q

any distance minimizing geo desic from q to S and X X j for q exp S Let

q q

be the ow of X Dene F S M as follows F v expv for kv k

v t for v S and t Then F is a dieomorphism and F tv exp

as follows easily by using

Remarks

A soul of M is not uniquely determined in general as can be seen by lo oking at

cylinders However any two souls of M are isometric cf S and Y

If co dimS then exp j is an is an between S with its

S

standard at bundle metric and M cf CG

In general the normal bundle S need not to b e trivial Furthermore M is not

k

lo cally isometric to a pro duct S IR in general By the Top onogov splitting

theorem cf CG however any line in M splits o isometricallyso that M is

k k

isometric to M IR where IR carries the standard at metric and M do es not

contain any lines This even holds for manifolds of nonnegative Ricci curvature

cf CG EH More generally Strake St has shown the following Supp ose

k k

the holonomygroupof S is trivial then M is isometric to S IR where IR

carries a metric of nonnegative curv ature For further results in this context we

also refer to ESS

For a discussion on the structure of the fundamental group see CG

There is no analogue of the soul theorem for complete op en manifolds of p ositive

Ricci curvature cf the examples in GM SY and B but compare also the

result in AG

App endix A top ological Lemma

m

Theorem Nested Coverings Let B B B i N be a

i i i

S

j

j

N

for family of nested open subsets in a topological space X and let X B

i

i

j m Then

p

rk H X H X

p p

p

X X

pk pk pk pk

rk H B B H B B

pk pk

i i i i

k k

i i

k

k

for p m here H stands for singular homology with coecients in some

p

arbitrary eld F

Pro of Let

M

j j j

j j

U

S X F F and A B C S B

p

i i

p pq p

q

i i

q

j

stand for the the groups of singular simplices which are ne wrt the covering of X

j j

j

j

by the B Whenever q homomorphisms C C which commute

pq

i

pq pq

L

j j

with the dierentials of the singular chain complexes C C can b e dened in

p

q pq

j j

the manner of Cech homology one adds up the inclusions S B F B

i i

p

q

d

j j j j

F with sign Dening similarly maps B S B B

p

i i i

p

q

j j

C A one obtains on each level j separately a long exact sequence of chain

p p

complexes the generalized MayerVietoris sequence BT pp

j j

j

q 

j j j

j

C C A C

q

q

This sequence is natural wrt the inclusion maps j j

j j j j

   

C C

q q q

j j j j

   

A A

j j

j

 

j j j

 

For q we set A With im and dene as the restriction of

q

pq pq q

this shorthand the generalized MayerVietoris sequence splits naturally into short exact

sequences of chain complexes

j

j j

A A C

q

q q

Taking the corresp onding long exact homology sequences leads in particular to both

the commutative diagrams with exact rows

H C H A H A

p p p

q q q

p

p

pq

pq

  

p

p p

H A H A H C

p p p

q

q q

pp pp

pq pq

  

p

p p

H A H A H C

p p p

q

q q

when p m and

H A H A H C

q q q

q q

  

H C H A H A

q q q

j j

else Here the vanishing o ccurs already on the chain level A C

q q

When applying standard diagram chasing techniques and yield the following

estimates resp ectively

p pp p

rk rk

pq pq pq

p

pp

rk rk for p m

pq

pq

rk rk

q q

By induction we conclude that

p p

X X

pkpk kk

p

rk rk rk

pkqk kpq k pq

k k

for p m Setting q to this inequality sp ecializes in the presence of formulae

and precisely to the claim in Theorem

n

Corollary Nested Coverings Let B B B i N be a

i i i

n

family of nested open subsets in an n dimensional topological manifold M Then

n

rk H B H B

i i

i i

n

X X

nk nk nk nk

B H B B H B rk

i i i i

k k

i i

k

k

where H is again singular homology with coecients in some arbitrary eld F

Remark The number of terms on the rhs of is

n n

k

X X

N

N

n

A

N N

k

k

k k

Pro of of the Corollary Since we are dealing with op en subsets in an n dimensional

n

manifold M H vanishes unless p n Therefore

p

n

H B rk H B

i i

i i

n

X

n

rk H B H B

p p

i i

p

i i

n

X

np

n

rk H B H B

p p

i

i

p

i i

Each term on the rhs can be estimated separately by applying Theorem to the

np

n

nested op en sets B B i N With this shift in the indexing in

i

i

mind m n n p one gets slightly sharp er than

n

rk H B H B

i i

i i

n nk

X X X

nk nk nk nk

rk H B B H B B

i i i i

k k

i i

k

k

thus proving the Corollary

References

A Abresch U Lower Curvature Bounds Top onogovs Theorem and Bounded

e

Top ology II I Ann scient Ec Norm Sup serie t

AG Abresch U Gromoll D Op en manifolds with nonnegative Ricci curvature

AW Alo S Wallach N R An innite family of distinct manifolds admitting

p ositively curved Riemannian structures Bull Amer Math So c

B Berard Bergery L Quelques exemples de varietes riemanniennes completes

non compactesa courbure de Ricci p ositive C R Acad Sc Paris t Serie

o

I n

BT Bott R Tu L W Dierential Forms in Algebraic T op ology Graduate Texts

in Mathematics Springer

CE Cheeger J Ebin DG Comparison Theorems in Riemannian Geometry

American Elsevier New York

CG Cheeger J Gromoll D On the structure of complete manifolds of nonneg

ative curvature Ann of Math

CG Cheeger J Gromoll D The splitting theorem for manifolds of nonnegative

Ricci curvature J Di Geom

Cg Cheng S Y Eigenvalue comparison theorem and its geometric applications

Math Z

E Elerath D An Improved Top onogov Comparison Theorem for Nonnegativly

Curved Manifolds J Di Geom

EH Eschenburg J Heintze E An elementary pro of of the CheegerGromoll

splitting theorem Ann Glob Analysis and Geometry

ESS Eschenburg J Schro eder V Strake M Curvature at innity of op en

nonnegatively curved manifolds J Di Geom

GG Gromoll D Grove K Rigidity of p ositively curved manifolds with large

diameter Seminar on Dierential Geometry Ann of Math Studies Princeton

University Press

GKM Gromoll D Klingenb erg W MeyerW Riemannsche Geometrie im

Grossen Lecture Notes Springer

GM Gromoll D Meyer W On complete manifolds of p ositive curvature Ann

of Math

GM Gromoll D Meyer W Examples of complete manifolds with p ositive Ricci

curvature J Di Geom

G Gromov M Almost at manifolds J Di Geom

G Gromov M Curvature diameter and Betti numb ers Comment Math Helv

GS Grove K Shiohama A A generalized sphere theorem Ann of Math

GP Grove K Petersen P Bounding homotopy typ es by geometry Ann of

Math

K Karcher H Riemannian Comparison Constructions Preprint Bonn

SY Sha JiPing Yang DaGang Examples of manifolds of p ositive Ricci curva

ture J Di Geo

SY Sha JiPing Yang DaGang Positive Ricci Curvature on the Connected

n m

Sums of S S Preprint

S Sharafutdinov V A Convex sets in a manifold of negative curvature Math

Zametki

St Strake M A splitting theorem for op en nonnegatively curved manifolds

Manuscripta Math

W Wallach N L Compact homogeneous Riemannian manifolds with strictly

p ositivecurvature Ann of Math

Y Yim J W Distance nondecreasing retraction on a complete op en manifold

of nonnegative curvature Preprint University of Pennsylvania