MORSE THEORY BY
J. Milnor Based on lecture notes by M. SPIVAK and R. WELLS
ANNALS OF MATHEMATICS STUDIES PRINCETON UNIVERSITY PRESS
Annals of Mathematics Studies Number 51
MORSE THEORY BY
J. Milnor Based on lecture notes by
M. SPIVAK and R. WELLS
PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS
Copyright © 1963, © 1969, by Princeton University Press All Rights Reserved L.C. Card 63-13729 ISBN 0-691-08008-9
Third Printing, with corrections
and a new Preface, 1969 Fourth Printing, 1970 Fifth Printing, 1973
Printed in the United States of America
PREFACE This book gives a present-day account of Marston Morse's theory of the calculus of variations in the large.
However, there have been im-
portant developments during the past few years which are not mentioned. Let me describe three of these R. Palais and S. Smale nave studied Morse theory for a real-valued function on an infinite dimensional manifold and have given direct proofs of the main theorems, without making any use of finite dimensional approximations.
The manifolds in question must be locally diffeomorphic
to Hilbert space, and the function must satisfy a weak compactness condition.
M
As an example, to study paths on a finite dimensional manifold
one considers the Hilbert manifold consisting of all absolutely con-
tinuous paths
w: (0,11 - M with square integrable first derivative. Ac-
counts of this work are contained in R. Palais, Morse Theory on Hilbert Manifolds, Topology, Vol. 2 (1963), pp. 299-340; and in S. Smale, Morse Theory and a Non-linear Generalization of the Dirichlet Problem, Annals of Mathematics, Vol. 8o (1964), pp. 382_396. The Bott periodicity theorems were originally inspired by Morse theory
(see part IV).
However, more elementary proofs, which do not in-
volve Morse theory at all, have recently been given.
See M. Atiyah and
R. Bott, On the Periodicity Theorem for Complex Vector Bundles, Acts,
Mathematica, Vol. 112 (1964), pp. 229_247, as well as R. Wood, Banach Algebras and Bott Periodicity, Topology, 4 (1965-66), pp. 371-389. Morse theory has provided the inspiration for exciting developments in differential topology by S. Smale, A. Wallace, and others, including a proof of the generalized Poincare hypothesis in high dimensions.
I
have tried to describe some of this work in Lectures on the h-cobordism theorem, notes by L. Siebenmann and J. Sondow, Princeton University Press, 1965.
Let me take this opportunity to clarify one term which may cause confusion.
In §12 I use the word "energy" for the integral v
vi
PREFACE 1
2
E = s u at II at 0
along a path w(t).
V. Arnol'd points out to me that mathematicians for
the past 200 years have called E the "action"integral. This discrepancy in terminology is caused by the fact that the integral can be interpreted, in terms of a physical model, in more than one way. Think of a particle P which moves along a surface M during the time interval 0 < t < 1.
'Tie action of the particle during this time interval
is defined to be a certain constant times the integral E.
If no forces
act on P (except for the constraining forces which hold it within M), then the "principle of least action" asserts that E will be minimized within the class of all paths joining w(0) to w(1), or at least that the first variation of E will be zero.
Hence P must traverse a geodesic.
But a quite different physical model is possible.
Think of a rubber
band which is stretched between two points of a slippery curved surface. If the band is described parametrically by the equation x = w(t), 0 < t < 1, then the potential energy arising from tension will be proportional to our integral E (at least to a first order of approximation).
For an
equilibrium position this energy must be minimized, and hence the rubber band will describe a geodesic.
The text which follows is identical with that of the first printing except for a few corrections.
I am grateful to V. Arnol'd, D. Epstein
and W. B. Houston, Jr. for pointing out corrections.
J.W.M.
Los Angeles, June 1968.
CONTENTS
PREFACE
v
NON-DEGENERATE SMOOTH FUNCTIONS ON A MANIFOLD
PART I.
Introduction . . . . . . . . . . . . . . . §2. Definitions and Lemmas . . . . . . . . . . §3. Homotopy Type in Terms of Critical Values §1.
§4.
Examples.
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§5. The Morse Inequalities . . . . . . . . . . . . . §6. Manifolds in Euclidean Space: The Existence of Non-degenerate Functions . §7.
PART II.
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The Lefschetz Theorem on Hyperplane Sections.
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A RAPID COURSE IN RIEMANNIAN GEOMETRY §8.
Covariant Differentiation .
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§9.
The Curvature Tensor .
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§10. Geodesics and Completeness .
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PART III. THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS §11.
The Path Space of a Smooth Manifold
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§12.
The Energy of a Path .
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§13.
The Hessian of the Energy Function at a Critical Path .
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§14.
Jacobi Fields:
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§15.
The Index Theorem .
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§16.
A Finite Dimensional Approximation to
§17.
The Topology of the Full Path Space
§18.
Existence of Non-conjugate Points
§19.
Some Relations Between Topology and Curvature
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The Null-space of .
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CONTENTS
APPLICATIONS TO LIE GROUPS AND SYMMETFIC SPACES
PART IV.
§20.
Symmetric Spaces
§21.
Lie Groups as Symmetric Spaces
§22.
Whole Manifolds of Minimal Geodesics
§23.
The Bott Periodicity Theorem for the Unitary Group
§24.
The Periodicity Theorem for the Orthogonal Group.
APPENDIX.
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THE HOMOTOPY TYPE OF A MONOTONE UNION
viii
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log 112
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149
PART I
NON-DEGENERATE SMOOTH FUNCTIONS ON A MANIFOLD.
§1.
Introduction.
In this section we will illustrate by a specific example the situation that we will investigate later for arbitrary manifolds. sider a torus
M,
tangent to the plane
Let us con-
as indicated in Diagram 1.
V,
Diagram 1. Let
f: M -f R (R always denotes the real numbers) be the height
above the V plane, and let f(x) < a.
Ma
be the set of all points
x e M
such that
Then the following things are true: (1)
If
a < 0 = f(p),
(2)
If
f(p) < a < f(q),
then Ma
is homeomorphic to a 2-cell.
(3)' If
f(q) < a < f(r),
then Ma
is homeomorphic to a cylinder:
(4)
If
Ma
then
f(r) < a < f(s), then Ma
is vacuous.
is homeomorphic to a compact
manifold of genus one having a circle as boundary:
1
I.
2
(5)
If
NON-DEGENERATE FUNCTIONS
then
f(s) < a,
Ma'
is the full torus.
In order to describe the change in Ma of the points
f(p),f(q),f(r),f(s)
a passes through one
it is convenient to consider homotopy
type rather than homeomorphism type. (1)
as
In terms of homotopy types:
is the operation of attaching a 0-cell.
homotopy type is concerned, the space
Ma,
For as far as
f(p) < a < f(q), cannot be dis-
tinguished from a 0-cell:
Here
means "is of the same homotopy type as." 1-cell:
(2) -* (3)
is the operation of attaching a
(3) - (4)
is again the operation of attaching a 1-cell:
(4) -. (5)
is the operation of attaching a 2-cell.
The precise definition of "attaching a k-cell" can be given as follows.
Let Y be any topological space, and let
ek
=
{x ERk : 1xII < 1)
be the k-cell consisting of all vectors in Euclidean k-space with length < 1.
P.
INTRODUCTION
3
The boundary
ek will be denoted by
Sk-1.
=
If
(x E Rk g:
Sk-1
IIxII =
1)
-+ Y is a continuous map then
Y .gek (Y with a k-cell attached by
g)
is obtained by first taking the topologi-
cal sum (= disjoint union) of Y and with
x E Sk-1
point and let
g(x) E Y. 60 = S-1
and then identifying each
ek,
To tale care of the case
k = 0
let
be a
eo
be vacuous, so that Y with a 0-cell attached is
just the union of Y and a disjoint point. As one might expect, the points topy type of
Ma'
p,q,r
At
both zero. that
x2
-
p we can choose
f = constant y 2.
-x2 - y2,
so that
(x,y)
and at
q
and
f.
If we choose any coordinate
near these points, then the derivatives
(x,y)
at which the homo-
s
changes, have a simple characterization in terms of
They are the critical points of the function. system
and
r
(Tx
and y are
f = x2 + y2, so that
at
so
s
f = constant +
Note that the number of minus signs in the expression for
each point is the dimension of the cell we must attach to go from Ma Mb,
where
a < f(point) < b.
at
f
to
Our first theorems will generalize these
facts for any differentiable function on a manifold.
REFERENCES For further information on Morse Theory, the following sources are extremely useful. M. Morse,
"The calculus of variations in the large," American
Mathematical Society, New York, 1934. H. Seifert and W. Threlfall,
"Variationsrechnung its Grossen,"
published in the United States by Chelsea, New York, 1951. R. Bott,
The stable homotopy of the classical groups, Annals of
Mathematics, Vol. 70 (1959), pp. 313-337. R. Bott,
Morse Theory and its application to homotopy theory,
Lecture notes by A. van de Ven (mimeographed), University of Bonn, 1960.
4
NON-DEGENERATE FUNCTIONS
I.
Definitions and Lemmas.
§2.
The words "smooth" and "differentiable" will be used interchange-
ably to mean differentiable of class
manifold M at a point smooth map with
p e M
(x',...,xn)
is zero.
of
a
J
this means that
_ ... = of
(P)
axn
(p) = 0
.
Ma
f
A critical point
f.
x e M
the set of all points
such that
p
f(x) < a.
then it follows from the implicit
is a smooth manifold-with-boundary.
is a smooth submanifold of
f- I(a)
p
of
is not a critical value of
function theorem that
A
if the induced map
is called a critical value of
f(p)
We denote by Ma If
f
M.
If we choose a local coordinate system
in a neighborhood U
ax
is a
TNq.
is called a critical point of
The real number
g: M -+ N
be a smooth real valued function on a manifold
f
f*: TMp -T Rf(p)
If
TMp.
then the induced linear map of tangent spaces
g,: TMp
Now let point
will be denoted by
p
g(p) = q,
will be denoted by
The tangent space of a smooth
C".
The boundary
M.
is called non-degenerate if and only if the
matrix a2f (p))
axiaxj
is non-singular.
It can be checked directly that non-degeneracy does not
depend on the coordinate system.
This will follow also from the following
intrinsic definition. If
functional then
v
p f**
is a critical point on
TMp,
of
f
called the Hessian of
and w have extensions v and w
f**(v,w) = vp(w(f)),
we define a symmetric bilinear f
at
p.
If
to vector fields.
v,w c TMp We let *
is, of course, just v. We must show that VP this is symmetric and well-defined. It is symmetric because where
vp(w(f)) - wp(v(f)) _ [v,w]p(f) = 0 where
[v,wl
Here w(f)
is the Poisson bracket of
and
w,
denotes the directional derivative of
and where [v,w] (f) = 0 f
in the direction w.
DEFINITIONS AND LEMMAS
§2.
since
p
has
f
as a critical point. f**
Therefore
is independent of
w = E bj a-.Ip
of
while
v,
w. a
is a local coordinate system and
(x1,...,xn)
If
It is now clearly well-defined since
is symmetric.
is independent of the extension v
vp(w(f)) = v(w(f))
wp (v(f))
5
w = E bj aj
we can take
where
bj
v = E a. p, 1 axi
now denotes a con-
ax
ax
Then
stant function.
f**(v,w) = v(w(f))(p) = v(E b af) =
respect to the basis
I
aX p
,...,
a
Z
ij
i axj
f
i b i ax
(p)
a
axn p
We can now talk about the index and the nullity of the bilinear functional tor space
ff* on
The index of a bilinear functional
TMp.
on a vec-
H,
is defined to be the maximal dimension of a subspace of V
V,
on which H is negative definite; the nullity is the dimension of the nullspace, i.e., the subspace consisting of all
for every w e V. point of
f
index of
f**
The point
if and only if on
TMp
p f**
v E V such that
H(v,w) = 0
is obviously a non-degenerate critical
on TMp
has nullity equal to
will be referred to simply as the index of
The Lemma of Morse shows that the behaviour of described by this index.
at
f
p
The
0.
f
Before stating this lemma we first prove the
following:
LEMMA 2.1.
Let
borhood V
of
f 0
be a
C°° function in a convex neigh-
in Rn,
with n
f(x1,...,xn) _
f(o) = 0.
Then
xigi(x1,...,xn) i=t
for some suitable
C" functions
defined in V,
gi
with
gi(o) = )f (o). i
PROOF:
df(tx1,...,txn)
f(xt,...,xn) = J
f0
dt =
1 I 0 i=1
of
tx1,...,txn) xi dt i
1
Therefore we can let
gi(x1...,xn) = f 0
(tx1,...,txn) dt i
at
p.
can be completely
I.
6
NON-DEGENERATE FUNCTIONS
Let p be a non-degenerate (Lemma of Morse). Then there is a local coordinate f.
LEMMA 2.2
critical point for system
(y1,...,yn)
for all
= 0
yi(p)
f - f(p)
+
... + (yn)2
is the index of
where
U,
must be the index of
(z1,...,zn),
(y% 2 + (yk+1)2 -
at
f
p.
We first show that if there is any such expression for
PROOF: X
in a neighborhood U of p with and such that the identity
(Y1)2- ...
-
holds throughout
then
i
at
f
f,
For any coordinate system
p.
if
f(q) = f(p) - (z1(q))2- ... - (ZX(q))2 + (z11+1(9.))2 + ... + (Zn(q))2 then we have
2
if i = j < x if i = J> X
0
otherwise ,
-2
f
(p)
=
az1 azj
which shows that the matrix representing
a
f*,*
,
,
with respect to the basis
is
Ip,...,
azn IP
Therefore there is a subspace of
TMp
of dimension
tive definite, and a subspace V of dimension n-X definite.
on which
If there were a subspace of f**
where
l
where
f**
f**
is nega-
is positive
of dimension greater than
TMp
X
were negative definite then this subspace would intersect V,
which is clearly impossible.
Therefore
X
is the index of
We now show that a suitable coordinate system Obviously we can assume that
p
f**.
(y1,...,yn)
is the origin of Rn and that
exists.
f(p) = f(o) = 0.
By 2.1 we can write
xi gg(x1,...,xn)
f(x1,...,xn) _
J= for
(x1,...,xn)
1
in some neighborhood of
0.
Since
critical point:
go (o) = a (0) = 0
0
is assumed to be a
DEFINITIONS AND LEMMAS
§2.
7
we have
Therefore, applying 2.1 to the gj
n gj(x1)...,xn) _
xihij(x1) ...,xn)
i=1 for certain smooth functions
It follows that
hij.
n xixjhij(x1,...,xn)
f(x1) ...,xn) _ i,j=1
We can assume that and then have
and
hij = Fiji
since we can write
hij = hji,
f = E xixjhij
hij = '-2(hij+ hji),
Moreover the matrix (hij(o))
.
2
is equal to
( 2
and hence is non-singular.
(o)),
ax
aX7
There is a non-singular transformation of the coordinate functions which gives us the desired expression for
borhood of
f,
in a perhaps smaller neigh-
To see this we just imitate the usual diagonalization proof
0.
(See for example, Birkhoff and MacLane, "A survey of
for quadratic forms.
The key step can be described as follows.
modern algebra," p. 271.)
Suppose by induction that there exist coordinates a neighborhood f
throughout
of
U1
so that
0
+ (u1)2 + ... + (uY_1)2 +
where the matrices
U1;
a linear change in the last Let
n-r+1
be a smooth, non-zero function of borhood
U2 C U1
vi=ui
uiujHij(u1,...,un)
i,j>r
of
are symmetric. After
(Hij(u1,...,un))
coordinates we may assume that
denote the square root of
g(u1,...,un)
in
u1, ...,un
u1,...,un
1Hr,r(u1,...,un)I.
This will
throughout some smaller neigh-
Now introduce new coordinates
0.
o.
Hr,r(o)
vl,...,vn by
fori#r
vr(u1)...,un) = g(u1,...,un)[ur, +
uiHir(u1,...,un)/Hr,r,(u1,...sun)]'
i> r
It follows from the inverse function theorem that
v1, ...,vn will serve as
coordinate functions within some sufficiently small neighborhood It is easily verified that f =
i
f
can be expressed as
+ (vi)2 +
i,j>r
vivjHij(v1,...,vn)
U3
of
0.
I.
8
throughout
U3.
NON-DEGENERATE FUNCTIONS
This completes the induction; and proves Lemma 2.2.
COROLLARY 2.3
Non-degenerate critical points are isolated.
Examples of degenerate critical points (for functions on R and R2)
are given below, together with pictures of their graphs.
(a)
f(x) = x3.
The origin
is a degenerate critical point.
(b)
F(x) = e- 1/X2 Sin2(,/X)
The origin is a degenerate, and non-isolated, critical point.
(c)
f(x,y) = x3 - 3xy2 = Real part of (o.o)
(x + iy)3.
is a degenerate critical point (a "monkey saddle").
§2.
(d)
f(x,y) = x2.
DEFINITIONS AND LEMMAS
The set of critical points, all of which
are degenerate, is the
(e)
f(x,y) = x2y2.
9
x
axis,
which is a sub-manifold of R 2.
The set of critical points, all of which are
degenerate, consists of the union of the not even a sub-manifold of
x and y axis, which is
R2.
We conclude this section with a discussion of 1-parameter groups of diffeomorphisms.
The reader is referred to K. Nomizu,"Lie Groups and Differ-
ential Geometry;'for more details.
A 1-parameter group
of diffeomorphisms of a manifold M is a
map
W: R x M - M
C00
10
I.
NON-DEGENERATE FUNCTIONS
such that 1)
for each
a 1-parameter group
pt+s = Ipt
cp
on M as follows.
a vector field X f
t,s c R we have
for all
Given
cpt: M -+ M defined by
°
onto itself,
''s
of diffeomorphisms of
M we define
For every smooth real valued function
let
f((Ph(q.))
Xq(f) =h lim - o This vector field X
M
is a diffeomorphism of
= (p(t,q)
cpt(q) 2)
t E R the map
h
- f(q)
is said to generate the group
p.
LEMMA 2.4.
A smooth vector field on M which vanishes outside of a compact set K C M generates a unique 1parameter group of diffeomorphisms of M. Given any smooth curve
PROOF:
t - c(t) E M it is convenient to define the velocity vector c TMc(t)
by the identity
(f)
= h
yme fc(t+hh-fc(t)
(Compare §8.)
Now let
be a 1-parameter group of diffeomorphisms, generated by the vector field X. Then for each fixed
q
the curve
t -" pt(9) satisfies the differential equation
dcpt(q)
pct with initial condition dcpt(_q)
(f)
rlim
Q
=
where
p = cpt(q).
cpe(q)
= q.
Xrot(q)
This is true since
f (Tt+h(q)) - f(cwt(9))
lim h
h
o
f (roh(p) )
- f(p)
X(f) p
But it is well known that such a differential equation,
locally, has a unique solution which depends smoothly on the initial condition.
(Compare Graves, "The Theory of Functions of Real Variables," p. 166.
Note that, in terms of local coordinates
u1,...,un,
the differential equa-
i
tion takes on the more familiar form:
dam- =
x1(ul,...,un),
i = 1,...,n.)
DEFINITIONS AND LEMMAS
§2.
there exists a neighborhood U and a
Thus for each point of M number
11
so that the differential equation
s > 0
dpt(q) - = Xcpt(q),
has a unique smooth solution for
= q
CPO(q)
q e U,
It! < e.
The compact set K can be covered by a finite number of such neighborhoods numbers
e.
e0 > 0
Let
U.
Setting
denote the smallest of the corresponding
pt(q) = q
for
tial equation has a unique solution q e M.
for
(pt(q)
It, <
E0
and
for all
Further-
This solution is smooth as a function of both variables.
more, it is clear that
Therefore each such
cpt
is a diffeomorphism.
be expressed as a multiple of t = k(so/2) + r
Nt = 9PE0/2
°
cpE /2 0
only necessary to replace ke /2 0
is defined for all values of
t.
for
cpt
Any number
Iti > so.
plus a remainder
e0/2
with k > 0,
where the transformation
1tj,1s1,1t+s1 < eo.
providing that
q>s
cpt+s = Wt °
It only remains to define
If
it follows that this differen-
q # K,
r
with
can
t
Irk < eo/2
set
CPEo/2 °
...
is iterated
by
(p_E /2
°
cpe0/2 k
'Pr
°
times.
iterated
If -k
k < 0 times.
it is
Thus
(pt
0
It is not difficult to verify that
well defined, smooth, and satisfies the condition
(pt+s = Tt G
(ps
Tt
is
This
completes the proof of Lemma 2.4 REMARK: cannot be omitted.
The hypothesis that X vanishes outside of a compact set For example let
M be the open unit interval
and let X be the standard vector field
d _ff
on
M.
generate any 1-parameter group of diffeomorphisms of
(0,1) C R,
Then X does not M.
12
NON-DEGENERATE FUNCTIONS
I.
Homotopy Type in Terms of Critical Values.
§3.
Throughout this section, if manifold
f
is a real valued function on a
we let
M,
Ma = f-1(- .,a] = (p e M : f(p) < a) THEOREM 3.1.
on a manifold f-1[a,b],
Let
.
be a smooth real valued function
f
Let a < b and suppose that the set consisting of all p e M with a < f(p) < b, M.
is compact, and contains no critical points of f. Then Ma is diffeomorphic to Mb. Furthermore, Ma is a deformation retract of Mb, so that the inclusion map Ma Mb is a homotopy equivalence.
The idea of the proof is to push Mb nal trajectories of the hypersurfaces
Ma
down to
f = constant.
along the orthogo-
(Compare Diagram 2.)
Diagram 2.
Choose a Riemannian metric on
M;
and let < X,Y > denote the
inner product of two tangent vectors, as determined by this metric. gradient of
f
is the vector field grad f
The
on M which is characterized
by the identity*
<X, grad f> = X(f) (= directional derivative of
for any vector field
X.
This
vector field grad f vanishes precisely at the critical points of
f.
If
f
along
X)
In classical notation, in terms of local coordinates gradient has components E gij f
*
j
au3
u1,...,un,
the
HOMOTOPY TYPE
§3.
c: R
M is a curve with velocity vector grad f
d
-
'dE
13
note the identity
d(dam
p: M -+ R be a smooth function which is equal to
Let
1/ < grad f, grad f > throughout the compact set f-1[a,bl; and which vanishes outside of a compact neighborhood of this set.
Then the vector field
X,
defined by Xq = p(q) (grad f)q
Hence X
satisfies the conditions of Lemma 2.4.
generates a 1-parameter
group of diffeomorphisms
CPt: M - M. q E M consider the function
For fixed
f-1[a,bl,
lies in the set
df(cpt(q))
dt
_<
dcpt(q) d
f(cpt(q)).
t
If Wt(q)
then
, grad f > = < X, grad f > = +
1
.
Thus the correspondence
t - f(Wt(q.)) is linear with derivative +1
as long as
f(cpt(q))
Now consider the diffeomorphism Ma
diffeomorphically onto
Mb.
cpb_a: M
lies between M.
a
and
b.
Clearly this carries
This proves the first half of
3.1.
Define a 1-parameter family of maps
rt: Mb . Mb by
rt(q) = Then m'
re
is the identity, and
is a deformation retract of REMARK:
r1
J
q
if f(q) < a
''t(a-f(q))(q)
if a < f(q) < b
is a retraction from Mb
Mb.
The condition that
to
ma.
Hence
This completes the proof. f-1[a,b]
is compact cannot be omitted.
For example Diagram 3 indicates a situation in which this set is not compact. The manifold
M does not contain the point
formation retract of
Mb.
p.
Clearly Ma
is not a de-
14
I.
NON-DEGENERATE FUNCTIONS
Diagram 3. Let f: M-+ R be a smooth function, and let p be a non-degenerate critical point with index X. Setting f(p) = c, suppose that f-1[c-e,c+el is compact, and contains no critical point of f other then p, for the set some s > 0. Then, for all sufficiently small e, Mc-e with a %-cell attached. MC+e has the homotopy type of THEOREM 3.2.
The idea of the proof of this theorem is indicated in Diagram 4, for the special case of the height function on a torus.
The region
Mc-e
= f-1(-co,c-e1
is heavily shaded.
We will introduce a new function
coincides with the height function
borhood of
p.
f
except that
F: M -+ R which
F < f
in a small neigh-
Thus the region F-1will consist of
gether with a region H near
p.
In Diagram 4,
shaded region.
MC+E
Diagram it.
Mc-e
to-
H is the horizontally
§3.
HOMOTOPY TYPE
Choosing a suitable cell
15
a direct argument
e% C H,
Mc-EU ex
ing in along the horizontal lines) will show that Mc-E
retract of
U H.
of
is a deformation
Finally, by applying 3.1 to the function F Mc-e
we will see that
region F-1[c-E,c+E1
(i.e., push-
and the
u H is a deformation retract
This will complete the proof.
Mc+E.
in a neighborhood U
u1,.... un
Choose a coordinate system
of
p
so that the identity f = c - (u1)2- ... - (uX)2 + (ut`+1)2+... + (un)2
holds throughout
Thus the critical point
U.
p will have coordinates
u1(p) = ... = un(p) = 0 Choose
e > 0
sufficiently small so that The region
(1)
.
points other than
is compact and contains no critical
f-1[c-E,c+E1
p.
The image of U under the diffeomorphic
(2)
(u...... un):
imbedding
U --'Rn
contains the closed ball. ((u1,...,un):
Now define
E (u')2 < 2E)
to be the set of points in U with
eX
1
...
X 2 + (u)< e
and
X+1(u)2+ u=
... = un = 0.
The resulting situation is illustrated schematically in Diagram 5.
Diagram 5.
16
NON-DEGENERATE FUNCTIONS
I.
The coordinate lines represent the planes u1
respectively;
= ... = ux = 0
... = un = 0
u11+1=
and
the circle represents the boundary of the
ball of radius; and the hyperbolas represent the hypersurfaces M0-e
The region
and f-1(c+e).
is heavily shaded; the region
is heavily dotted; and the region
zontal dark line through p Note that
e
Mc-e
is attached to
formation retract of
X
f-1[c,c+el
The hori-
e
is precisely the boundary
as required.
f-1[c-e,cl
is lightly dotted.
represents the cell
M°-e
n
f-1(c-e)
We must prove that
ex,
so that
Mc-eu eX
e
is a de-
Mc+e
F: M -'f R
Construct a new smooth function
as follows.
Let
µ:R-pR be a C"
function satisfying the conditions µ(o) > E
µ(r) -1 <
where
dµ
µ'(r) =
neighborhood
.
U,
for r > 2e
0
µ'(r) < 0
for all
F coincide with
Now let
r,
outside of the coordinate
f
and let
F = f - µ((u1)2+. ..+(u2 + 2(uX+1 )2+...+2(un)2 within this coordinate neighborhood.
It'is easily verified that
well defined smooth function throughout
F
is a
M.
It is convenient to define two functions
t,11: U--i
[0,oo)
by _
(u1)2 + ... + (uX ) 2 {u?'+1) 2 +
Then
f = c -
t
+ q;
... +
(un) 2
so that:
F(q) = c - (q) + q(q) - u(t(q) + 2q(q)) for all
q e U.
ASSERTION 1.
The region F-1(-oo,c+el
coincides with the region
Mc+e = f-1(- oo,c+el. PROOF:
Outside of the ellipsoid
g
+ 2q < 2e the functions
f
and
U. HOMOTOPY TYPE F
coincide.
17
Within this ellipsoid we have F < f = c-g+q < c+ 2g+q < c+e
This completes the proof.
The critical points of
ASSERTION 2.
are the same as those of
F
f.
Note that
PROOF:
aF Tq-
1
- 2µ'(g+2q) > 1
.
Since dg +
dF
where the covectors
and
dg
dq
dq
are simultaneously zero only at the origin,
has no critical points in U
it follows that F
By Assertion 1 together
Now consider the region F-1[c-e,c+e].
with the inequality F < f
other than the origin.
we see that
F-1[C-e,C+e] C f-1[C-e,c+e] Therefore
this region is compact.
except possibly
But
p.
F(p) =c-µ(o)
F
It can contain no critical points of
contains no critical points.
F-1[c-e,c+e]
.
Together with 3.1 this
proves the following. ASSERTION 3.
The region F-1is a deformation retract of
Mc+e
It will be convenient to denote this region F-1(-.,c-e] Mc-e
u H;
where
In the terminology of
REMARK:
described as
H denotes the closure of
Mc-e
that the manifold-with-boundary
by
Mc-e
Smale, the region Mc-e v H is
with a "handle" attached. MC-e
F-1(-co,c-El
-
It follows from Theorem
U H is diffeomorphic to
fact is important in Smale's theory of differentiable manifolds.
Mc+e
3.1
This
(Compare
S. Smale, Generalized Poincare's conjecture in dimensions greater than four, Annals of Mathematics, Vol. 74 (1961), pp. 391-4o6.)
18
I.
NON-DEGENERATE FUNCTIONS
Now consider the cell
e
X
t(q) < e, Note that
eX
q with
consisting of all points
= 0.
TI(q)
is contained in the "handle"
H.
In fact, since
< 0,
we have
F(q) < F(p) < c-e but
f(q) > c-e
for
q e ex.
Diagram 6.
The present situation is illustrated in Diagram 6. Mc-E
is heavily shaded;
the handle
and the region F-1[c-e,c+e3 ASSERTION I+.
PROOF:
Mc-e
H is shaded with vertical arrows;
is dotted. u ex
lows.
rt
Mc-e
is a deformation retract of rt:
A deformation retraction
Mc-e
Me-E
u H -
indicated schematically by the vertical arrows in Diagram 6. let
The region
be the identity outside of
U;
and define
rt
u
u H.
H is
More precisely
within U as fol-
It in necessary to distinguish three cases as indicated in Diagram 7. CASE 1.
Within the region
g < e
let
rt
correspond to the trans-
formation
(u1,...,un) _ (u1,...,ux,tu%+1,...,tun)
.
HOMOTOPY TYPE
§3.
19
CASE 2
CASE 2
Diagram 7. Thus
is the identity and
r1
fact that each rt equality
Tq_
maps
r0
maps the entire region into
The
e
into itself, follows from the in-
F-1(-oo,c-el
> 0.
CASE 2.
Within the region
e < g < n + e
let
correspond to
rt
the transformation (u1,...Iun)
where the number
-+ (ul,...,ux,stux+l,...,Stun)
is defined by
st e [0,11
St = t + Thus
( 1 -t)
is again the identity, and
r1
hypersurface
f-1(c-e).
remain continuous as with that of Case
1
g
g
be the identity.
r0
-+ e,
when
g
stu1
Note that this definition coincides
-' 0.
= e.
q + e < g (i.e., within
MC-e)
let
This coincides with the preceeding definition when
= q + e. This completes the proof that
of
maps the entire region into the
The reader should verify that the functions
CASE . Within the region rt
1 /2
F-1(-co,c+el.
MC-eu ex
is a deformation retract
Together with Assertion 3, it completes the proof of
Theorem 3.2. REMARK 3.3.
critical points
More generally suppose that there are
pi,...,pk with indices
x1,...,xk
in
f-
k non-degenerate 1(c).
Then a x
similar proof shows that
MC+e
has the homotopy type of
x
MC-eu a 1u...u e k
I. NON-DEGENERATE FUNCTIONS
20
REMARK .4. Mc
the set
A
simple modification of the proof of 3.2 shows that
is also a deformation retract of
Mc+e.
In fact
Mc
is a
deformation retract of F-1which is a deformation retract of (Compare Diagram 8.) Mc-E ,
et`
Combining this fact with 3.2 we see easily that
is a deformation retract of
Diagram 8:
Mc
Mc+e
MCI.
is heavily shaded, and
F-1[c,c+e]
is dotted.
THEOREM 3.5. If f is a differentiable function on a manifold M with no degenerate critical points, and if each Ma is compact, then M has the homotopy type of a CW-complex, with one cell of dimension X for each critical point of index X.
(For the definition of
CW-complex see J. H. C. Whitehead, Combin-
atorial Homotopy I, Bulletin of the American Mathematical Society, Vol. 55, (1949), pp. 213-245.)
The proof will be based on two lemmas concerning a topological
space X with a cell attached.
(Whitehead) Let Wo and cp1 be homotopic maps from the sphere et' to X. Then the identity map of X extends to a homotopy equivalence
LEMMA 3.6.
k:Xue%-*XueX 'P0
CP 1
HOMOTOPY TYPE
§3.
Here
k by the formulas
Define
PROOF:
k(x) = x
for x E X
k(tu) = 2tu
for
0 < t < 2
u E
k(tu) = CP2_2t(u)
for
< t < 1,
u E
and
and
denotes the homotopy between
9t
product of the scalar
Wo
with the unit vector
t
f:
e -+ X
Xu
u
W1
is defined by similar formulas. compositions Thus
k
21
kF
and
fk
0
91; u.
to
denotes the
A corresponding map
ex
It is now not difficult to verify that the
are homotopic to the respective identity maps.
is a homotopy equivalence. For further details the reader is referred to, Lemma 5 of J. H. C.
Whitehead,
On Simply Connected 4-Dimensional Polyhedra, Commentarii Math.
Helvetici, Vol. 22 (1949), pp. 48-92.
LEMMA 3.7.
Let W: e1''-+ X be an attaching map. Any homotopy equivalence f: X -+ Y extends to a homotopy equivalence
F
X uT eX -+ Y ..f(p eX.
:
(Following an unpublished paper by P.Hilton.)
PROOF:
Define
F
by the conditions
Let
g: Y -r X
FIX
=
Fled'
= identity
f
be a homotopy inverse to
.
G: Y
gfp
fq)
by the corresponding conditions Since
gfp
and define
f
GAY
is homotopic to
=
W,
e identity.
g,
it follows from 3.6 that there is
a homotopy equivalence k: X
e
u
gfp
W
We will first prove that the composition kGF: X u ex W
is homotopic to the identity map.
X u ex T
22
NON-DEGENERATE FUNCTIONS
I.
Let
ht
be a homotopy between
specific definitions of
k, G, and
F,
gf
note that
kGF(x) = gf(x)
for x E X,
kGF(tu) = 2tu
for'O < t < 2
u e
11F(tu)
for 2 < t < 1,
u e
h2-2t(P(u)
Using the
and the identity.
The required homotopy
q
T
: X sex
X.e
is now defined by the formula qT(x)
,2T
u
,
1
gT(tu) =
F
o < t <
for
to
gT(tu)
Therefore
for x e X ,
hT(x)
=
12
for
h2-2t+TW(u)
< t < t,
u e
has a left homotopy inverse.
The proof that F
is a homotopy equivalence will now be purely
formal, based on the following. ASSERTION.
If
a map F
right homotopy inverse
R
(or
PROOF:
L)
has a left homotopy inverse
then F
R,
is a 2-sided
L and a
is a homotopy equivalence; and
homotopy inverse.
The relations LF
ti
identity,
FR
ti
identity,
imply that
L-' L(FR)
_
(LF)R ti R.
Consequently RF ti IF ti identity ,
which proves that R
is a 2-sided inverse.
The proof of Lemma 3.7 can now be completed as follows.
The rela-
tion kGF ti identity
asserts that
F
has a left homotopy inverse; and a similar proof shows that
G has a left homotopy inverse. Step 1.
Since
inverse, it follows that
k(GF)
identity, and
(GF)k a identity.
k
is known to have a left
§3.
Step 2.
inverse, it follows that
G
is known to have a left
F
has
(Fk)G ti identity.
F(kG) a identity, and
Since
F
also, it follows that
23
G(Fk) a identity, and
Since
Step 3.
HOMOTOPY TYPE
kG
as left inverse
This completes the
is a homotopy equivalence.
proof of 3.7. PROOF OF THEOREM 3.5.
f: M - R.
values of
Ma
is compact.
a
c1,c21C31...
Let
The sequence
Ma
The set
ci > a.
has no cluster point since each a < c1.
Suppose OW-complex. Mc+e
By Theorems 3.1, 3.2, and 3.3,
has
%
MC-cu e1 u...u e3(c) for certain maps
the homotopy type of
q)1,.
'J (c)
T1 e
(ci)
is of the homotopy type of a
X
when
be the critical
c1 < c2 < c3 < ...
is vacuous for
Ma
and that
be the smallest
c
Let
is small enough, and there is a homotopy equivalence
We have assumed that there is a homotopy equivalence
h:
h': Ma
K,
Mc-e - Ma.
where K
is a OW-complex.
Then each
h'
o h o Tj
is homotopic by cellular approximation to
a map
>Vj: 6 J
Then K
u
e
u...u
ei(c)
Mc+e,
is a OW-complex, and has the same homotopy
by Lemmas 3.6, 3.7.
By induction it follows that each Ma' OW-complex.
K.
''j (c)
`1
type as
-+ (), j-1) - skeleton of
has the homotopy type of a
If M is compact this completes the proof.
If M is not com-
pact, but all critical points lie in one of the compact sets proof similar to that of Theorem 3.1 shows that the set retract of
M,
Ma
Ma,
then a
is a deformation
so the proof is again complete.
If there are infinitely many critical points then the above construction gives us an infinite sequence of homotopy equivalences Mat C Ma2 C Ma3 C
f K1
each extending the previous one.
I C K2
...
f C K3
C ...
,
Let K denote the union of the Ki in the
direct limit topology, i.e., the finest possible compatible topology, and
24
let
NON-DEGENERATE FUNCTIONS
I.
g: M -r K be the limit map.
groups in all dimensions.
Then
g induces isomorphisms of homotopy
We need only apply Theorem 1 of Combinatorial
homotopy I to conclude that
g
is a homotopy equivalence.
[Whitehead's
theorem states that if M and K are both dominated by CW-complexes, then any map M -+ K which induces isomorphisms of homotopy groups is a homotopy
Certainly K is dominated by itself.
equivalence.
To prove that
M is
dominated by a CW-complex it is only necessary to consider M as a retract This completes the proof
of tubular neighborhood in some Euclidean space.] of Theorem 3.5.
We have also proved that each Ma has the homotopy type
REMARK.
of a finite
CW-complex, with one cell of dimension
point of index
X
in
(Compare Remark 3.4.)
Ma.
This is true even if
a
X
for each critical
is a critical value.
EXAMPLES
§4.
§4.
25
Examples.
As an application of the theorems of §3 we shall prove: (Reeb). If M is a compact manifold and f is a differentiable function on M with only two critical points, both of which are non-degenerate, then M is homeomorphic to a sphere.
THEOREM 4.1
PROOF: This follows from Theorem 3.1 together with the Lemma of Morse (§2.2). points. If
s
The two critical points must be the minimum and maximum f(p) = 0
Say that
is the mimimum and
is small enough then the sets
closed n-cells by §2.2.
But
Me
Ms = f_1[0,e)
between M and
M1-E
is the maximum. f_1[1-e,11
by §3.1.
f-1[1-e,1),
are
Thus
matched
It is now easy to construct a homeomorphism
Sn.
REMARK 1. degenerate.
M1-E
and
1
and
is homeomorphic to
M is the union of two closed n-cells, along their common boundary.
f(q) =
The theorem remains true even if the critical points are
However, the proof is more difficult. (Compare Milnor, Differ-
ential topology, in "Lectures on Modern Mathematics II," ed. by T. L:.Saaty (Wiley, 1964), pp. 165-183; Theorem 1'; or
R. Rosen, A weak form of the
star conjecture for manifolds, Abstract 570-28, Notices Amer. Math Soc., Vol. 7 (1960), p. 380; Lemma 1.) REMARK 2.
It is not true that
M must be diffeomorphic to
Sn with
its usual differentiable structure.(Compare: Milnor, On manifolds homeomorphic to the 7-sphere, Annals of Mathematics, Vol. 64 (1956), pp. 399-405. In this paper a 7-sphere with a non-standard differentiable structure is
proved to be topologically
S7 by finding a function on it with two non-
26
NON-DEGENERATE FUNCTIONS
I.
degenerate critical points.)
As another application of the previous theorems we note that if an n-manifold has a non-degenerate function on it with only three critical points then they have index
0, n and
n/2 (by Poincare duality), and the
manifold has the homotopy type of an n/2-sphere with an n-cell attached. See J. Eells and N. Kuiper, Manifolds which are like projective planes, Inst. des Hautes Etudes Sci., Publ. Math. 14, 1962. Such a function exists for example on the real or complex projective plane.
Let CPn be complex projective n-space. equivalence classes of EIzj12
=
(n+1)-tuples
of complex numbers, with
(z0,...,zn)
Denote the equivalence class of (z0,...,zn) by (z0:z1:...:zn).
1.
Define a real valued function
on CPn by the identity
f
f(z0:z1:...:zn) = I where
We will think of CPn as
c0,c1,...,cn
cj1zjl2
are distinct real constants.
In order to determine the critical points of following local coordinate system.
with
z0
0,
Let
Zj zo
and set I Za I
Then
U0
be the set of
consider the (zo:z1:...:zn)
= x+ iyj U0 - R
x1,y1,...,xn,yn:
are the required coordinate functions, mapping U0 the open unit ball in R 2ri.
f,
diffeomorphically onto
Clearly Izol2
zjI2 = xj2 + yj2
=
1
- E (xj2 + yj2)
so that
f = c0 j=1
throughout the coordinate neighborhood f
within
U.
U0.
Thus the only critical point of
lies at the center point p0 = (1:0:0:...:0)
of the coordinate system.
At this point
index equal to twice the number of
j
f
with
is non-degenerate; and has cj < co,
Similarly one can consider other coordinate systems centered at the points p1
= (0:1:0: ...:0),...,Pn = (0:0:...:0:1)
§4.
It follows that index of
f
at
EXAMPLES
27
p0,pi,...,pn are the only critical points of pk is equal to twice the number of
Thus every possible even index between
0
and
2n
j
with
f.
The
cj < ck.
occurs exactly once.
By Theorem 3.5:
C Pn has the homotopy type of a CW-complex of the form e° u e2 u e4 u...u e2n
It follows that the integral homology groups of CPn are given by Fii(CPn;Z)
Z l 0
for i = 0,2,4,...,2n for other values of i
28
NON-DEGENERATE FUNCTIONS
I.
The Morse Inequalities.
§5.
In Morse's original treatment of this subject, Theorem 3.5 was not
The relationship between the topology of M and the critical
available.
points of a real valued function on M were described instead in terms of a collection of inequalities.
This section will describe this original
point of view. DEFINITION: the integers.
S
S(X,Y) + S(Y,Z).
Let
S
be a function from certain pairs of spaces to
If equality holds,
as coefficient group, let
RX(X,Y) = Xth Betti number of = rank over (X,Y)
S(X,Z) <
is called additive.
S
As an example, given any field F
for any pair
we have
is subadditive if whenever XD Y: )Z
F
(X,Y)
HX(X,Y;F)
of
such that this rank is finite.
RX
is subadditive, as
is easily seen by examining the following portion of the exact sequence for (X,Y,Z):
HX(X,Z) - HX(X,Y) -+ ...
... -+ HX(Y,Z)
The Euler characteristic
X(X,Y)
is additive, where
X(X,Y) _
E (-1)X RX(X,Y).
LEMMA 5.1.
Then
Let
S be subadditive and let
S(Xn,XO)<
S(Xi,Xi_,).
If
S
X0C...C Xn.
is additive then
equality holds. 1 PROOF:
the case
n = 2
Induction on
n.
For
n = 1, equality holds trivially and
is the definition of [sub] additivity. n-1
If the result is true for
n - 1,
then
S(Xn_1,XO) <
}
Therefore
S(Xn,X0) < S(Xn_1,Xa) + S(Xn,Xn_1) < 2; S(Xi,X1_1)
S(Xi,Xi_1).
and the result
1
is true for Let
n.
S(X,o) = S(X).
S(Xn) <
(1)
with equality if
Taking X. = 0 n
S
is additive.
in Lemma 5.1, we have
S(Xi,Xi-1)
§5.
THE MORSE INEQUALITIES
Let M be a compact manifold
and
29
a differentiable function
f
on M with isolated, non-degenerate, critical points. be such that
contains exactly
Mai
a1 <...< ak
Let
critical points, and
i
Mak
M.
=
Then e% 1 Mai-1)
= H (Mai-1
H*(Mat Mat-1)
where
is the index of the
critical point, X.
), .
H*(e 1,e 1)
by excision,
coefficient group in dimension otherwise.
0
Applying (1) to
0
=
Mo a C...C M ak = M with k
Xi
S = R.
we have
R),(Ma1,Ma1-1)
= CX; i=1
where
C.
denotes the number of critical points of index
formula to the case
S = X
X.
Applying this
we have
k X(Ma1,Ma1-1)
x(M) =
= C0 - C1 + C2 -+...+ Cn
i=1
Thus we have proven:
THEOREM 5.2 (Weak Morse Inequalities). If Cx denotes the number of critical points of index X on the compact mani-
fold M then RX(M) < CX
,
and (-1)X CX
E (-1)" RX(M)
Slightly sharper inequalities can be proven by the following argument.
LEMMA 5.3.
The function
S.
is subadditive, where
SX(X,Y) = RX(X,Y) - Rx_1(X,Y) + Rx_2(X,Y) - +...+ R0(X,Y)
PROOF:
Given an exact sequence
h+A' B
Ck... ...-+D-+0
of vector spaces note that the rank of the homomorphism h plus the rank of
i
is equal to the rank of
A.
Therefore,
30
NON-DEGENERATE FUNCTIONS
I.
rank h
=
rank A - rank i
=
rank A - rank B + rank
j
=
rank A - rank B + rank
C
=
rank A - rank B + rank
C - +...+ rank D
Hence the last expression is
> 0.
- rank k
.
Now consider the homology exact sequence
Applying this computation to the homomorphism
of a triple X J Y J Z.
Hx+1 (X,Y)
a H,,(Y,Z)
we see that rank 3 = RX(Y,Z) - RX(X,Z) + RX(X,Y) - Rx_1(Y,Z) + ... > o Collecting terms, this means that S,(Y,Z) - S (X,Z) + SX(X,Y) > o which completes the proof. Applying this subadditive function
S1,
to the spaces
0 C Mat C Mat C...C Mak
we obtain the Morse inequalities:
k
X(Mi'Mi-1) a a = CX - CX-1 +-...+ Co
Sx(M) < i=1
or
(40
R>(M) - RX_1(M) +-...+ Ro(M) < CX - CX_1+ -...+ Co. These inequalities are definitely sharper than the previous ones.
In fact, adding
with
(4,_1)
(4x) and (4,_1), one obtains (2x);
for x> n
one obtains the equality
and comparing
(40
(3).
As an illustration of the use of the Morse inequalities, suppose that
C,+1 = 0.
(4.)
and
Then R,+1
(41,+1),
must also be zero.
we see that
RX - R)6_1 +-...+ Ro
Now suppose that
Comparing the inequalities
C.-1
=
is also zero.
C% - C1'_1
-...± Co
Then R._1 = 0,
.
and a similar argu-
ment shows that Rx-2 - Rx-3 +-...± Ro
=
Cx-2
-
Cx-3 +-...± Co
.
§5.
THE MORSE INEQUALITIES
31
Subtracting this from the equality above we obtain the following:
COROLLARY 5.4.
If
Cx+1 = CX_1 = 0
then R. = C.
and
R)L+1 = RX_1 = 0.
(Of course this would also follow from Theorem 3.5.)
Note that
this corollary enables us to find the homology groups of complex projective space (see §4) without making use of Theorem 3.5.
32
NON-DEGENERATE FUNCTIONS
I.
§6.
Manifolds in Euclidean Space.
Although we have so far considered, on a manifold, only functions which have no degenerate critical points, we have not yet even shown that In this section we will construct many func-
such functions always exist.
tions with no degenerate critical points, on any manifold embedded in Rn.
In fact, if for fixed p c Rn define the function lip-gII2.
It will turn out that for almost all
LP: M - R by LL(q) _ the function Lp
p,
has
only non-degenerate critical points.
Let M C Rn be a manifold of dimension bedded in Rn.
Let
N = ((q,v): q e M,
v
differentiably embedded in R2ri.
Let
differentiably em-
perpendicular to M at
It is not difficult to show that N
vector bundle of
k < n,
N C M x Rn be defined by
(N
q).
is an n-dimensional manifold
is the total space of the normal
M.)
E: N -+ Rn be
E(q,v) = q + v.
(E
is the "endpoint" map.)
E(4,v)
DEFINITION. µ
if
nullity
e = q + v µ > 0.
e E Rn is a focal point of where
(q,v) E N and the Jacobian of
The point
a focal point of
(M,q)
e
with multiplicity
(M,q)
E
at
(q,v)
will be called a focal point of M if
for some
has e
q e M.
Intuitively, a focal point of M is a point in Rn where nearby normals intersect.
We will use the following theorem, which we will not prove.
is
§6.
MANIFOLDS IN EUCLIDEAN SPACE
33
THEOREM 6.1 (Sard). If M1 and M2 are differentiable manifolds having a countable basis, of the same dimension, then the image of the M2 is of class C1, and f: M1
set of critical points has measure A critical point of
in
0
M2-
is a point where the Jacobian of
f
f
is
For a proof see de Rham, "Variotes Differentiables," Hermann,
singular.
Paris, 1955, p. 10. COROLLARY 6.2.
PROOF:
E: N -s Rn.
is
M.
N
We have just seen that
is a focal point iff x
x
the point
For almost all x e Rn,
not a focal point of
is an n-manifold.
The point x
is in the image of the set of critical points of
Therefore the set of focal points has measure
0.
For a better understanding of the concept of focal point, it is convenient to introduce the "second fundamental form" of a manifold in Euclidean space.
We will not attempt to give an invariant definition; but will make
use of a fixed local coordinate system. Let
u1,...,uk be coordinates for a region of the manifold M C R.
Then the inclusion map from M
to Rn determines
n
smooth functions
k x1(u 1 ,...,uk ),...,xn(u '...,u) 1
These functions will be written briefly as x(u1,...,uk) (x1,...,xn).
To be consistent the point
where x =
q e M C Rn will now be denoted by
q.
The first fundamental form associated with the coordinate system is defined to be the symmetric matrix of real valued functions
(gij) The second fundamental form
(3ui
_
j)
-
.
on the other hand, is a symmetric matrix (ij)
of vector valued functions.
2-'
at a point of M can The vector au aukk be expressed as the sum of a vector tangent to M and a vector normal to M. It is defined as follows.
2-
Define T..
x
to be the normal component of au
au
v which is normal to M at q the matrix
(v auJ
)
_
(v i.1j
.
Given any unit vector
34
I.
NON-DEGENERATE FUNCTIONS
at q in the direction
M
can be called the "second fundamental form of
v." It will simplify the discussion to assume that the coordinates have been chosen to that
evaluated at q,
gi3,
Then the eigenvalues of the matrix curvatures
K1,...,KK
of
( v
M at -1
is the identity matrix.
are called the principal
in the normal direction v.
The re-
of these principal curvatures are called the princi-
ciprocals pal radii of curvature. is singular.
Of course it may happen that the matrix (v .7
)
Ki will be zero; and hence
In this case one or more of the
the corresponding radii
ii
will not be defined.
Kit
Now consider the normal line
where
consisting of all q + tv,
2
M at q
v is a fixed unit vector orthogonal to LEMMA 6.3.
The focal points of (M q) along f are precisely the points _q q+ Ki-1 _V, where 1 < i< k, Ki / o. v Thus there are at most, k focal points of (M,q) along f,
each being counted with its proper multiplicity. Choose
PROOF:
along the manifold so that w1,...;wn_k nal to each other and to t1,...Itn-k) to-k)
k
1
n-k vector fields
are unit vectors which are orthogo-
We can introduce coordinates
M.
N C M x e as follows.
on the manifold
k
1
w1(u ,...,u ),...,wn_k(u ,...,u )
Let
(u1,...,uk,
(u1,...,uk,t1,..
correspond to the point
n-k \
to wa(u1,...,uk)1 e N
(x(u1,...,uk), a=1
Then the function
-. Rn
E:N gives rise to the correspondence (ll ,...,U ,t ,...,t n-k ) 1
k
1
e'
1
k
X(u ,...,u )
+
J
a -+ 1 k wa(u ,...,u )
t
,
with partial derivatives
aB
aX
aul
au1
+
to awa aul
ae a
a
Taking the inner products of these n-vectors with the linearly independent
MANIFOLDS IN EUCLIDEAN SPACE
§6.
vectors
axl,
axk, w1'...;wn_k we will obtain an nxn matrix whose
au au rank equals the rank of the Jacobian of
This
35
E
at the corresponding point.
nxn matrix clearly has the following form ax au
aX
y
awa
+
to
auj
ax
w 1 1
to awa
\1
aul 0 /
(
TIT
a
S
identity
0
matrix
Thus the nullity is equal to the nullity of the upper left hand block. Using the identity
o=
(W 7'T 1
ax a
=
auk
aWa
ax
au1
auk
x
+ Wa
2)u1auj
we see that this upper left hand block is just the matrix
-. ta wa
( gig
Iii
a Thus:
ASSERTION 6.4. q + tv µ
is a focal point of (M,q)
with multiplicity
if and only if the matrix
(gig - tv Iii )
(*)
is singular, with nullity
Now suppose that lar if and only if
-f
more the multiplicity
(gij)
P.
is the identity matrix.
is an eigenvalue of the matrix (v µ
Then (*) is singu)
Further-
is equal to the multiplicity of - as eigenvalue.
This completes the proof of Lemma 6.3.
Now for fixed p E Rn let us study the function
Ip
=
f:
M-R
where f(x(u1,...,uk))
=
Ij-X(u1,...,uk)
- pII2 =
x x
-
2x
p + p p
.
We have of aul
Thus at
f
q
= 2 ax
TIT
(x - p)
has a critical point at q if and only if q - p is normal to M
36
NON-DEGENERATE FUNCTIONS
I.
The second partial derivatives at a critical point are given by 2f_
2( ax aul
=
au au
Setting p = x + tv,
,
ax
x
+
auk
.
lX
3
a
as in the proof of Lemma 6.3, this becomes
2(gi. - tv au auk
Therefore:
The point q e M is a degenerate critical point f = L5- if and only if p is a focal point of (M,q). The nullity of q as critical point is equal to the multiplicity of p as focal point. LEMMA 6.5. of
Combining this result with Corollary 6.2 to Sari's theorem, we immediately obtain:
(all but a set of
THEOREM 6.6. For almost all p e Rn measure 0) the function
LL: M -+R has no degenerate critical points.
This theorem has several interesting consequences. there exists a dif-
On any manifold M
COROLLARY 6.7.
ferentiable function, with no degenerate critical points,
for which each Ma
PROOF:
is compact.
This follows from Theorem 6.6 and the fact that an n-dimen-
sional manifold M can be embedded differentiably as.a closed subset of R2n+1
(see Whitney, Geometric Integration Theory, p. 113).
A differentiable manifold has the homotopy type of
APPLICATION 1. a CW-complex.
This follows from the above corollary and Theorem 3.5.
On a compact manifold M
APPLICATION 2.
X
there is a vector field
such that the sum of the indices of the critical points of X
x(M),
the Euler characteristic of
any differentiable function
f
M.
This can be seen as follows:
on M we have
is the number of critical points with index the vector field grad
f
equals
at a point where
X.
f
x(M) = E (-1)x Cx
But
(-1)x
has index
x.
for
where
C.
is the index of
MANIFOLDS IN EUCLIDEAN SPACE
§6.
37
It follows that the sum of the indices of any vector field on M is equal to
because this sum is a topological invariant
x(M)
(see Steen-
rod, "The Topology of Fibre Bundles," §39.7). The preceding corollary can be sharpened as follows.
Let
k > 0
be an integer and let K C M be a compact set. COROLLARY 6.8.
Any bounded smooth function
f: M -+ R can
g which
be uniformly approximated by a smooth function
has no degenerate critical points. Furthermore g can be chosen so that the i-th derivatives of g on the compact set K uniformly approximate the corresponding derivatives
of
f, for i < k.
(Compare M. Morse,
The critical points of a function of n vari-
ables, Transactions of the American Mathematical Society, Vol. 33 (1931), pp. 71-91.) PROOF:
Choose some imbedding
h: M -+ Rn of M as a bounded sub-
set of some euclidean space so that the first coordinate the given function
f.
Let
c
be a large number.
p
=
(-C+e1,e2,..., En)
(-C,0,...,o) E Rn
close to
degenerate;
so that the function
g
(x) is non-degenerate.
c
is large and the
C2
2C
1
f
M -+ R is non-
LP:
A short computation shows that
g(x) = f(x) + I hi(x)2/2c -
Clearly, if
Choose a point
and set g(x)
Clearly
is precisely
h1
ei
ei2/2c
eihi(x)/c + 1
1
are small,
then
g will approximate
as required.
The above theory can also be used to describe the index of the function
at a critical point.
LEMMA 6.9. (Index theorem for
Lp.)
The index of
Lp
at a non-degenerate critical point q e M is equal to the number of focal points of (M,q) which lie on the segment from q to p; each focal point being counted with its multiplicity.
38
I.
NON-DEGENERATE FUNCTIONS
An analogous statement in Part III (the Morse Index Theorem) will be of fundamental importance.
The index of the matrix
PROOF:
a2
au au
2(gij - tv
is equal to the number of negative eigenvalues.
ij)
Assuming that
(gij)
the identity matrix, this is equal to the number of eigenvalues of (v which are
>
.
7
is
lid)
Comparing this statement with 6.3, the conclusion follows.
THE LEFSCHETZ THEOREM
§7.
39
The Lefschetz Theorem on Hyperplane Sections.
§7.
As an application of the ideas which have been developed, we will prove some results concerning the topology of algebraic varieties.
These
were originally proved by Lefschetz, using quite different arguments.
The
present version is due to Andreotti and Frankel*. THEOREM 7.1. If M C Cn is a non-singular affine algebraic variety in complex n-space with real dimension 2k,
then for
Hi(M;Z) = 0
i > k.
This is a consequence of the stronger:
A complex analytic manifold M of complex bianalytically embedded as a closed subset of Cn has the homotopy type of a k-dimensional CW-complex.
THEOREM 7.2.
dimension
k,
The proof will be broken up into several steps.
quadratic form in k
Q(z1,...,zk) = I bhj
If we substitute xh + iyh
for
obtain a real quadratic form in
zh,
zhzi
ASSERTION 1. -e
If
e
.
and then take the real part of
bhj(xh+iyh)(x3+iy3)
is an eigenvalue of
Q'
with multiplicity
is also an eigenvalue with the same multiplicity
PROOF.
The identity
the quadratic form change of variables.
Q'
Q we
2k real variables:
Q'(x1,...,xk,yt,...,yk) = real part of
then
First consider a
complex variables
Q(iz1,...,izk)
=
can be transformed into
µ.
-Q(z1,...,zk) -Q'
µ,
shows that
by an orthogonal
Assertion 1 clearly follows.
11
See S. Lefschetz, "L'analysis situs et la geometrie algebrique," Paris, and A. Andreotti and T. Frankel, The Lefschetz theorem on hyperplane sections, Annals of Mathematics, Vol. 69 (1959), pp. 713-717. 1921+;
4o
NON-DEGENERATE FUNCTIONS
I.
Now consider a complex manifold M which is bianalytically imbedded as a subset of Cn.
Let
be a point of
q
The focal points of
ASSERTION 2.
M.
along any normal line
(M,q)
occur in pairs which are situated symmetrically about
In other words if
q + tv
2
q.
then q - tv
is a focal point,
is a
focal point with the same multiplicity.
PROOF.
Choose complex coordinates z1(q) = ... = zk(q)
borhood of
q
so that
determines
n
complex analytic functions
Let
The inclusion map M-+ Cn
= 0.
a = 1,
wa(z1 ,...,zk ),
WC,
for M in a neigh-
z1,...,zk
..,n.
be a fixed unit vector which is orthogonal to M at
v
q.
Consider
the Hermitian inner product
wa(z1,...,zk)va
wava = of
w
and
v.
This can be expanded as a complex power series
I wa(z1,...,zk)va = constant + Q(z1,...,zk) + higher terms, where
Q denotes a homogeneous quadratic function.
ish since
v
is orthogonal to
Now substitute system for
M;
(The linear terms van-
M.)
xh + iyh
so as to obtain a real coordinate
zh
for
and consider the real inner product
w v = real part of
wava
.
This function has the real power series expansion
xk
w- v = constant + Q1(x1
Clearly the quadratic terms
M Q'
at
q
v.
occur in equal and opposite pairs. and
k
higher terms.
determine the second fundamental form of
Q'
in the normal direction
along the line through q
1
,Y
q + v
By Assertion 1
the eigenvalues of
Hence the focal points of
(M,q)
also occur in symmetric pairs.
This
proves Assertion 2.
We are now ready to prove 7.2. squared-distance function
Choose a point
p e Cn
so that the
THE LEFSCHETZ THEOREM
§7.
41
Y M -+ R Since M is a closed subset of Cn,
has no degenerate critical points.
it
is clear that each set Ma = is compact.
Now consider the index of
at a critical point
Lp
ing to 6.9, this index is equal to the number of focal points of
which lie on the line segment from p
to
buted symmetrically about
and
q.
(M,q)
2k
But there are at most
q.
and
focal points along the full line through p
Accord-
q.
Hence at most
k
and these are distri-
q;
of them can lie between p
q. Thus the index of
Lp
at
q
is
It follows that M has the
< k.
which completes the proof
homotopy type of a OW-complex of dimension < k;
of 7.2. COROLLARY 7.3
Let V be an algebraic variety
(Lefschetz).
k which lies in the complex projective space CPn. Let P be a hyperplane in CPn which contains the singular points (if any) of V. Then the inclusion map of complex dimension
V n P-V induces isomorphisms of homology groups in dimensions less than
Furthermore, the induced homomorphism
k-1.
Hk_1(V n P;Z)
Hk-l(V;Z)
is onto. PROOF.
Using the exact sequence of the pair
clearly sufficient to show that
H1,(V,V n P;Z) = 0
for
(V,V n P)
it is
r < k-1.
But the
Lefschetz duality theorem asserts that
Hr,(V,V n P;Z) = But V -(V n P) CPn - P.
H2k-r(V -(V
n P) ;Z)
is a non-singular algebraic variety in the affine space
Hence it follows from 7.2 that the last group is zero for This result can be sharpened as follows: THEOREM 7.4 (Lefschetz). Under the hypothesis of the preceding corollary, the relative homotopy group Trr(V,V n P)
is zero for
r < k.
r < k-1.
42
I.
PROOF.
borhood U
NON-DEGENERATE FUNCTIONS
The proof will be based on the hypothesis that some neigh-
of V n P
can be deformed into V n P
within V.
This can be
proved, for example, using the theorem that algebraic varieties can be triangulated.
In place of the function where
LP: V - V n P -+ R we will use
ro
f(x)
for x e V n P
t/Lp(x)
Since the critical points of the critical points of
homotopy type of
Lp
f have index >
no degenerate critical points with Ve = f-1[o,e]
f: V -+ R
for
x j P.
< k it follows that
have index 2k - k = k.
s < f < co.
The function
f
has
Therefore V has the
with finitely many cells of dimension > k
attached.
Choose r-cube.
e
small enough so that
Then every map of the pair
VE C U.
(Ir,Ir)
Let
into
Ir
denote the unit
(V,V n P)
can be deform-
ed into a map (Ir,Ir) - (V-,V n P) C (U,V n P) since
proof.
r < k,
and hence can be deformed into V n P.
,
This completes the
PART II
A RAPID COURSE IN RIEMANNIAN GEOMETRY
§8.
Covariant Differentiation
The object of Part II will be to give a rapid outline of some basic concepts of Riemannian geometry which will be needed later.
For more infor-
mation the reader should consult Nomizu, "Lie groups and differential geoMath. Soc. Japan, 1956; Helgason, "Differential geometry and sym-
metry.
metric spaces," Academic Press, 1962; Sternberg, "Lectures on differential geometry," Prentice-Hall, 1964; or Laugwitz, "Differential and Riemannian geometry," Academic Press, 1965.
Let M be a smooth manifold. DEFINITION.
p E M
An affine connection at a point
which assigns to each tangent vector Xp E TMp
is a function
and to each vector field Y
a new tangent vector
Xp - Y E TMp of Y in the direction Xp.
called the covariant derivative
quired to be bilinear as a function of Xp
and
Y.
This is re-
Furthermore, if
f: M- R is a real valued function, and if
fY
denotes the vector field
(fY)q = f(q)Yq 'then
h
is required to satisfy the identity
Xp I- (fY)
=
(Xpf)Yp + f(p)Xp
I- Y
.
Note that our X F Y coincides with Nomizu's Vxy. The notation is intended to suggest that the differential operator X acts on the vector field Y.
44
RIEMANNIAN GEOMETRY
II.
(As usual,
denotes the directional derivative of
Xpf
in the direction
f
XP.)
(or briefly a connection) on M is a
A global affine connection
function which assigns to each p e M an affine connection
at
Fp
p,
satisfying the following smoothness condition.
and Y are smooth vector fields on M
X
If
1)
then the vector
defined by the identity
field X F Y,
(X FY)p=Xp FpY
,
must also be smooth. Note that: (2)
X F Y is bilinear as a function of
(3)
(fX)
(4)
(X F (fY) = (Xf)Y + f(X F Y)
F Y = f(X F Y)
and Y
X
,
Conditions (1), (2), (3), (4)
can be taken as the definition of
a connection.
In terms of local coordinates
neighborhood U C M,
the connection
valued functions ik field
on
U.
on
ul,...,un
is determined by
F
as follows.
U,
defined on a coordinate
Then any vector field X
Let
n3
smooth real
ak denote the vector
on U can be expressed
au uniquely as
X
=
k xkak
k=1
where the field
(5)
xk
ai F ai
are real valued functions on
U.
In particular the vector
can be expressed as
ai F a _
rij
k
These functions i, determine the connection completely on In fact given vector fields
X
x'ai
=
expand X F Y by the rules (2), (3), (4);
(6)
XFY
=
k
i
and Y
=
yJaj
one can
yielding the formula
xiyki)ak
U.
§8.
where the symbol
yki
45
COVARIANT DIFFERENTIATION
stands for the real valued function
y'i
aiyk +
=
ij y j
Conversely, given any smooth real valued functions ik one can define X F Y by the formula (6). the conditions
The result clearly satisfies
one can define the covariant derivative of
F
a vector field along a curve in
First some definitions.
M.
A parametrized curve in M is a smooth function M.
assigns to each
U,
(1), (2), (3), (4), (5).
Using the connection
numbers to
on
A vector field V along the curve
c
c
from the real
is a function which
t e R a tangent vector Vt E TMc(t)
This is required to be smooth in the following sense: tion
For any smooth func-
on M the correspondence
f
t
tf
V
should define a smooth function on R. As an example the velocity vector field vector field along
c
do
Tdt
of the curve is the
which is defined by the rule
UE Here
(if
-
c*
d
T£
denotes the standard vector field on the real numbers, and c*:
TRt
TMc(t)
denotes the homomorphism of tangent spaces induced by the map Diagram 9.) de
Diagram 9
c.
(Compare
46
II.
RIEMANNIAN GEOMETRY
Now suppose that M is provided with an affine connection.
any vector field V along
determines a new vector field
c
called the covariant derivative of
DV
Then
along
c
The operation
V.
DV ffu
V
is characterized by the following three axioms.
a)
D(V+W)
Tf-
=
ur+D(fV)
c)
V+f
_
DV
.
If V is induced by a vector field Y on Vt
Yc(t)
for each
then
t,
DV
that is if
M,
is equal to
_CTE
F Y
(= the covariant derivative of Y in the direction of the velocity vector of
c)
There is one and only one operation V -+ DV which satisfies these three conditions. LEMMA 8.1.
PROOF:
u1(t),. .,un(t)
field V
Choose a local coordinate system for denote the coordinates of the point
and let The vector
c(t).
can be expressed uniquely in the form
V
where
M,
v1,...,vn
subset of R),
I v4j
=
are real valued functions on R and
(or an appropriate open
a1,...1an are the standard vector fields on the co-
ordinate neighborhood.
It follows from (a), (b), and (c)
that
i dam-
k
= ri vi, ak
+
3
i,j
Conversely, defining DV by this formula, it is not difficult to verify that conditions (a), (b), and (c)
A vector field V along if the covariant derivative
CT:E
are satisfied. c
is said to be a parallel vector field
is identically zero.
§8.
47
COVARIANT DIFFERENTIATION
Given a curve c and a tangent vector V0 there is one and only one parallel c(o), vector field V along c which extends Vo. LEMMA 8.2.
at the point
The differential equations
PROOF.
dv k
du' r k vi ij
+
i,j have solutions vk(o).
0
=
which are uniquely determined by the initial values
vk(t)
Since these equations are linear, the solutions can be defined for
all relevant values of
p. 152.)
Real Variables,"
The vector Vt lation along
(Compare Graves, "The Theory of Functions of
t.
is said to be obtained from V0 by parallel trans-
c.
Now suppose that M
Xp, Yp will be denoted by
of two vectors
DEFINITION.
The inner product
is a Riemannian manifold.
A connection
< Xp, Yp >
.
on M is compatible with the Rieman-
F
nian metric if parallel translation preserves inner products. for any parametrized curve along
c,
and any pair
c
P, P'
In other words,
of parallel vector fields
the inner product < P,P' > should be constant.
Suppose that the connection is compatible with
LEMMA 8.3.
Let V, W be any two vector fields along
the metric.
c.
Then
E < V,W >
_
+
Choose parallel vector fields
PROOF:
are orthonormal at one point of
c
v1
lows that < V,W >
_
I v1Pi
>, v'w'
DW T£
dv1
Therefore +
which completes the proof.
_
which
c
Then
c.
wiPj
and
respecIt fol-
and that
Pi,
_dT ,W>
along
= < V,Pi > is a real valued function on R).
DV
<
P1,...,Pn
and hence at every point of
the given fields V and W can be expressed as tively (where
.
d
i
dwJ _
wi + v1 d
i\ )
P
=
d
48
RIEMANNIAN GEOMETRY
II.
For any vector fields vector XP e TMp: COROLLARY 8.4.
Xp Choose a curve
PROOF.
<Xp F Y,YY>
_
c
on M and any
Y,Y'
+
.
whose velocity vector at
t = 0
is
Xp;
and apply 8.3.
A
DEFINITION 8.5.
connection
is called symmetric if it satis-
F
fies the identity*
(X F Y) - (Y F X) (As usual,
[X,Y]
denotes the poison bracket
two vector fields.) since
=
[ai,ajl = 0
[X,Y]
.
[X,Y]f = X(Yf) - Y(Xf)
Applying this identity to the case X = ai,
of
Y = aJ,
one obtains the relation rij - rjk
Conversely if ik = rji
=
o.
then using formula (6) it is not difficult to
verify that the connection
F
is symmetric throughout the coordinate neigh-
borhood.
LEMMA 8.6. (Fundamental lemma of Riemannian geometry.) A Riemannian manifold possesses one and only one symmetric connection which is compatible with its metric.
(Compare Nomizu p. 76, Laugwitz p. 95.) PROOF of uniqueness.
and setting
Permuting
Applying 8.4 to the vector fields
< aj,ak >
=
gjk one obtains the iaentity
ai gjk
=
< ai F apak > + < aj,ai F ak >
i,j,
and
k
ai,a3,ak,
.
this gives three linear equations relating the
*
The following reformulation may (or may not) seem more intuitive. Define The "covariant second derivative" of a real valued function f along two
vectors XpYp to be the expression Xp(Yf) - (Xp F Y)f
where Y denotes any vector field extending Yp.
It can be verified that this expression does not depend on the choice of Y. (Compare the proof of Lemma 9.1 below.) Then the connection is symmetric if this second derivative is symmetric as a function of Xp and Yp.
COVARIANT DIFFERENTIATION
§8.
49
three quantities
< i F aj,akKaj
aj>
k'ai>, and
(There are only three such quantities since
ai F 3j
=
F ai
aj
.)
These
equations can be solved uniquely; yielding the first Christoffel identity ai F aj,ak >
=
2
(aigjk + ajgik - akgij)
The left hand side of this identity is equal to
rij
gkk
.
Multiplying
.
P
by the inverse (g'
)
of the matrix
this yields the second Christof-
(gQk)
fel identity r j
2 \ai gjk + aj gik - ak gij) gk¢
=
k Thus the connection is uniquely determined by the metric. Conversely, defining
by this formula, one can verify that the
rl
resulting connection is symmetric and compatible with the metric.
This
completes the proof.
An alternative characterization of symmetry will be very useful Consider a
later.
"parametrized surface" in
s: By a vector field V along (x,y)
a R2
M:
that is a smooth function
R2 - M
is meant a function which assigns to each
s
a tangent vector V(x,Y) c TM5(x,Y)
As examples, the two standard vector fields tor fields
s, .
Jx and y ;
and
along
s* -jV
s.
yo,
give rise to vec-
and called the "velocity vector fields" of
For any smooth vector field V along and
and TV
x These will be denoted briefly by s
are new vector fields, constructed as follows.
Ty-
s.
the covariant derivatives
For each fixed
restricting V to the curve x
s(x,Y0)
one obtains a vector field along this curve. respect to
x
is defined to be ()(x
the entire parametrized surface
Y) 'o
Its covariant derivative with This defines
-DV
along
s.
As examples, we can form the two covariant derivatives of the two
II.
50
vector fields
as cTx
and
as
3y
RIEMANNIAN GEOMETRY The derivatives
simply the acceleration vectors of suitable and
the mixed derivatives
LEMMA 8.7.
PROOF.
and compute.
If
D
D
as
Tx
and
D as
are
cTy cTy
coordinate curves.
However,
cannot be described so simply.
the connection is symmetric then
D as ax ay
=
D as ay ax
Express both sides in terms of a local coordinate system,
§9.
THE CURVATURE TENSOR
§9.
The Curvature Tensor
The curvature tensor R
of an affine connection
extent to which the second covariant derivative
metric in
and
i
Given vector fields
j.
51
measures the
F
ai F () j
F Z)
is sym-
define a new vector field
X,Y,Z
R(X,Y)Z by the identity*
R(X,Y) Z
=
-X F (Y F Z) + Y F (X F Z) + [X,Y1 F Z
The value of R(X,Y)Z at a point p E M depends only on the vectors Xp,Yp,Zp at this point p and not on their values at nearby points. Furthermore the correspondence R(Xp,Yp)Zp Xp,Yp,Zp LEMMA 9.1.
from
to
TMp x TMp x TMp
is tri-linear.
TMp
Briefly, this lemma can be expressed by saying that R is a "tensor."
PROOF:
If
X
Clearly R(X,Y)Z
is replaced by a multiple
is a tri-linear function of then the three terms
fX
X,Y,
and
Z.
-X F (Y F Z),
Y F (X F Z), [X,Y] F Z are replaced respectively by i) - fX F (Y F Z) , ii)
iii)
(Yf) (X F Z)
+ fY F (X F Z)
- (Yf)(X F Z) + f[X,Y1
F Z
Adding these three terms one obtains the identity
R(fX,Y) Z
fR(X,Y) Z
=
Corresponding identities for Y and
Z
.
are easily obtained by similar
computations.
Now suppose that
X
xiai,
=
Y
=
y'aj
,
and
Z
=
zk)k.
Then
R(X,Y)Z
=
R(xiai,yjaj)(zk) k) i i k x y z R(ai)aj)ak
*
Nomizu gives R the opposite sign. Our sign convention has the advantage that (in the Riemannian case) the inner product < Ro hl)i)aj,ak coincides with the classical symbol Rhijk
52
RIEMANNIAN GEOMETRY
II.
p
Evaluating this expression at (R(X,Y)Z)p
one obtains the formula
x1(p)yj(p)zk(p)(R(ai,aj)ak)p
=
xi,yJ,zk at
which depends only on the values of the functions
and
p,
This completes the proof.
not on their values at nearby points.
Now consider a parametrized surface
s: R2,M Given any vector field V along
one can apply the two covariant dif-
s.
ferentiation operators -& and
D to
In general these operators will
V.
not commute with each other.
LEMMA 9.2.
- R( TX_,Ty) V
Tx rTy V
cTy 3x V
Express both sides in terms of a local coordinate system,
PROOF:
and compute, making use of the identity
a1 F (ai F ak) - ai F (ai F ak)
R(ai,ai )ak
=
.
[It is interesting to ask whether one can construct a vector field P
along
which is parallel, in the sense that
s
P
=
-
P
0,
=
_5_X_
and which has a given value
at the origin.
P(0 0)
In general no such
,
vector field exists. then
P
can be constructed as follows.
field along the fixed
x0
However, if the curvature tensor happens to be zero Let
For each
be a parallel vector field along the curve
let
P(xo,y)
having the right value for P
be a parallel vector
x-axis, satisfying the given initial condition.
y - s(x0,y)
Clearly
P(x,o)
y = 0.
,
This defines
and ZX P
is identically zero;
P
everywhere along
Now the identity
D D
3yTxP implies that y zx- P = 0.
D D
TxF
=
as as R( \3x'T P
=
0
In other words, the vector field D P
parallel along the curves
y
s(x0,y)
s.
is zero along the x-axis.
is
THE CURVATURE TENSOR
§9.
Since
(3Dx P)(x
o
0)
=
this implies that
0,
and completes the proof that
P
-
53
x P
is identically zero;
is parallel along
Henceforth we will assume that
s.]
M is a Riemannian manifold, pro-
vided with the unique symmetric connection which is compatible with its metric.
In conclusion we will prove that the tensor
satisfies four
R
symmetry relations.
The curvature tensor of a Riemannian manifold
LEMMA 9.3. satisfies: (1)
R(X,Y)Z + R(Y,X)Z = 0
(2) (3) (4)
R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0
+ = 0 _
PROOF:
definition of
The skew-symmetry relation (1) follows immediately from the R.
Since all three terms of (2)
are tensors, it is sufficient to
prove (2) when the bracket products zero.
and
[X,Y], [X,ZI
[Y,Z1
are all
Under this hypothesis we must verify the identity
X F (Y F Z)
+
Y F (X F Z)
Y F (Z F X) Z F (X FY)
+
Z F (Y F X)
+
X F (Z FY)
=
o .
But the symmetry of the connection implies that
Y FZ - Z FY =
[Y,ZI
=
0
.
Thus the upper left term cancels the lower right term. maining terms cancel in pairs.
Similarly the re-
This proves (2).
To prove (3) we must show that the expression < R(X,Y)Z,W > skew-symmetric in
Z
and W.
that
for all X,Y,Z. R(X,Y)Z,Z >
is
This is clearly equivalent to the assertion
Again we may assume that
=
0
[X,YI
=
so that
0,
is equal to
- X F (Y F Z) + Y F (X F Z),Z>
.
54
II.
RIEMANNIAN GEOMETRY
In other words we must prove that the expression
is symmetric in X Since
and
Y.
[X,Y]
and =
Y. 0
the expression YX < Z,Z > is symmetric in X
Since the connection is compatible with the metric, we have
X < Z,Z >
2 <X F Z,Z >
=
hence
YX = 2 + 2 <X F Z,Y F Z> But the right hand term is clearly symmetric in X
Y F (X F Z),Z > is symmetric in X
and
and
Y.
Therefore
Y; which proves property (3).
Property (4) may be proved from (1), (2), and (3) as follows.
Formula (2) asserts that the sum of the quantities at the vertices
of shaded triangle W is zero.
Similarly (making use of (1) and (3)) the
sum of the vertices of each of the other shaded triangles is zero.
Adding
these identities for the top two shaded triangles, and subtracting the identities for the bottom ones, this means that twice the top vertex minus twice the bottom vertex is zero.
This proves (4), and completes the proof.
§10.
GEODESICS AND COMPLETENESS
§10.
Geodesics and Completeness
55
Let M be a connected Riemannian manifold. A parametrized path
DEFINITION.
I
y: where
I
-+ M,
acceleration vector field UE
identity
d£
shown that the length constant along
y.
af,
uu
Thus the velocity
is identically zero.
Tdf
,
2 < UE
=
H
,
dy
_
0
of the velocity vector is
_H
Introducing the arc-length function
s(t)
=
\
DD
IIdt + constant
This statement can be rephrased as follows:
The parameter
geodesic is a linear function of the arc-length.
ally equal to the arc-length if and only if
Idyll
t
The parameter =
t-+ y(t) e M determines
The equation
n
along a t
is actu-
1.
In terms of a local coordinate system with coordinates a curve
if the
denotes any interval of real numbers, is called a geodesic
smooth functions
u1,...,un
u1(t),...,un(t).
-aE UE for a geodesic then takes the form
d2uk +
r k
un) dul duJ
(u1
-
o
i,J=1
The existence of geodesics depends, therefore, on the solutions of a certain system of second order differential equations.
More generally consider any system of equations of the form
-11
=
dt2
Here u stands for
(ul,...,un)
and
FI
stands for an
n-tuple of
functions, all defined throughout some neighborhood U of a point (u1,v1) a R
en
C°°
56
II.
RIEMANNIAN GEOMETRY There exists a
EXISTENCE AND UNIQUENESS THEOREM 10.1.
and a number neighborhood W of the point (u1,-v1) e > o so that, for each (u0V0) E W the differential equation
dt2(u'
at)
has a unique solution t -; u(t) which is defined for and satisfies the initial conditions Iti < e,
y
u0,
u(0)
v0
- (0)
Furthermore, the solution depends smoothly on the initial conditions. In other words, the correspondence
(u0,v0,t) -' u(t) from W x (-e,e)
to
is a
R n
C°° function of all
2n+1 variables. i
PROOF:
vi = 3T this system of
Introducing the new variables
second order equations becomes a system of
n
2n first order equations:
du =
dv _df
V
=(u,V)
The assertion then follows from Graves, "Theory of Functions of Real Variables," p. 166.
(Compare our §2.4.)
Applying this theorem to the differential equation for geodesics, one obtains the following.
For every point p0 on a Riemannian M there exists a neighborhood U of p0 and a number e > 0 so that: for each p c U and each tangent vector v E TMp with length < e there is a unique geodesic LEMMA 10.2.
manifold
M
7v:
satisfying the conditions
7 (o) PROOF.
=
p,
-- (0) dYv
=
v
If we were willing to replace the interval
(-2,2)
by an
arbitrarily small interval, then this statement would follow immediately from 10.1.
To be more precise; there exists a neighborhood U
of
p0
and
§10.
numbers 11vII < 51
E1,e2 > 0
GEODESICS AND COMPLETENESS
57
for each p E U and each v E TMp with
so that:
there is a unique geodesic
(-252,252) - M
yv:
satisfying the required initial conditions.
To obtain the sharper statement it is only necessary to observe that the differential equation for geodesics has the following homogeneity property.
Let
c
If the parametrized curve
be any constant.
t
-F y(t)
is a geodesic, then the parametrized curve
t - y(ct) will also be a geodesic.
Now suppose that Iti < 2
is smaller than
e
5152.
Then if
ilvil < e
and
note that
and
IIv/5211 < 51 Hence we can define
yv(t)
to be
le2tl <252
yv/e2(e2t)
.
This proves 10.2.
.
This following notation will be convenient.
Let
V E TMq
be a
tangent vector, and suppose that there exists a geodesic y:
M
[0,11
satisfying the conditions y(o)
Then the point exponential
=
q,
-(o)
y(1) E M will be denoted by of the tangent vector
v.
=
v.
expq(v)
The geodesic
and called the y
can thus be des-
cribed by the formula y(t)
=
expq(ty)
.
The historical motivation for this terminology is the following. If M is the group of all n x n unitary matrices then the tangent space TMI at the identity can be identified with the space of n x n skew-Hermitian matrices. The function
expl:
TMI
-. M
as defined above is then given by the exponential power series expl(A)
=
I + A + 2 A2 +
A3 + ... 1
RIEM&NNIAN GEOMETRY
II.
58
Lemma 10.2 says that
is not defined for large vectors
expq(v)
In general,
defined at all,
expq(v)
However, if
v.
is always uniquely defined.
The manifold M is geodesically complete
DEFINITION.
is defined for all
is defined providing that IIvjj is small enough.
expq(v)
q E M and all vectors
v E TMq.
expq(v)
if
This is clearly equiva-
lent to the following requirement:
For every geodesic segment to extend
[a,b]
yo:
to an infinite geodesic
70
y:
R-M
We will return to a study of completeness
.
after proving some local results.
Let TM be the tangent manifold of (p,v)
if
with p E M,
(u1,...,un)
ai
consisting of all pairs
the following
TM
q E U can be expressed uniquely as I
au
We give
v E TMp.
M,
COO
structure:
is a coordinate system in an open set U C M then every
tangent vector at where
- M it should be possible
t1 a1 +...+ tnan,
Then the functions uconstitute
q
TU C TM.
a coordinate system on the open set
Lemma 10.2 says that for each p E M
the map
(q,v) - expq(v)
is defined throughout a neighborhood V of the point more this map is differentiable throughout
V.
Now consider the smooth function F: F(q,v) (p,o)
=
(q, expq(v)).
is non-singular.
Further-
(p,o) E TM.
V - M x M defined by
We claim that the Jacobian of F
at the point
In fact, denoting the induced coordinates on
U x U C M x M by
F(
u
F*
T
a
Thus the Jacobian matrix of F
u 2),
we have
a
a +u2
_
a
a
=
au?
at
(p,o)
has the form
( o
I 1
,
and
hence is non-singular.
It follows from the implicit function theorem that F
neighborhood V'
of
maps some
(p,o) E TM diffeomorphically onto some neighborhood
§10.
of
p
such that
(q,v)
and such that
59
We may assume that the first neighborhood
(p,p) e M X M.
of all pairs
GEODESICS AND COMPLETENESS
belongs to a given neighborhood
q
Choose a smaller neighborhood W
livII < s.
consists
v,
of
p
U'
of
so that
Then we have proven the following.
F(V') D W x W.
For each p e M there exists a neighborhood W and a number e > 0 so that: Any two points of W are joined by a unique (1) geodesic in M of length < e. LEMMA 10.3.
(2)
points.
This geodesic depends smoothly upon the two 0 < t < 1, is the t -r exp q1(tv),
(I.e., if
then the pair (q1,v) e geodesic joining q1 and q2, TM depends differentiably on (g1,g2).) For each q e W the map expq maps the open (3) e-ball in TMq diffeomorphically onto an open set
Uq) W. REMARK.
With more care it would be possible to choose W
so that
the geodesic joining any two of its points lies completely within W. ptre J. H. C. Whitehead,
Convex regions in the geometry of paths,
Com-
Quarter-
ly Journal of Mathematics (Oxford) Vol. 3, (1932), pp. 33-42. Now let us study the relationship between geodesics and arc-length.
THEOREM 10.4.
Let W and
e
be as in Lemma 10.3.
Let
[0,1]- M
y:
be the geodesic of length < e joining two points of W, and let w: [0,1]- M be any other piecewise smooth path joining the same two points.
Then, 1
1
SlI
IIdt<
0
SII
Udt
0
where equality can hold only if the point set
w([o,1])
coincides with y([0,1]).
Thus
y
is the shortest path joining its end points.
The proof will be based on two lemmas. be as in 10.3.
Let
q = y(o) and let
Uq
II. RIEMANNIAN GEOMETRY
6o
In Uq,
LEMMA 10.5.
the geodesics through
q
are
the orthogonal trajectories of hypersurfaces expq(v)
Let
PROOF:
v e TMq,
:
constant}
=
IIvii
denote any curve in TMq with
t -+ v(t)
11v(t)II
1.
We must show that the corresponding curves
t -+ expq(rov(t) ) in
Uq,
are orthogonal to the radial geodesics
0 < ro < e,
where
r -+ expq(rv(t0)) In terms of the parametrized surface f(r,t)
f
given by
expq(rv(t)),
=
.
0 < r < e
,
we must prove that
=o -5 E
for all
(r,t).
Now a
Fr
of D of
D of of
of of Nr,a't>
_ < cTr cTr'3f>
+
.
The first expression on the right is zero since the curves
r -+ f(r,t) are geodesics.
The second expression is equal to
of D of > = 2 aE a
since IIII
=
dent of
But for
r.
IIv(t) II
1.
=
of of
=
0
Therefore the quantityCTF<,> is indepen-H we have
r = o
f(o,t) = expq(0) = q hence
M(0,t)
Therefore < Fr- 7E > is identically zero, which com-
= 0.
,
pletes the proof.
Now consider any piecewise smooth curve CO:
Each point
w(t)
o < r(t) < e,
[a,bl -+ Uq - (q)
can be expressed uniquely in the form
and
IIv(t) II
LEMMA 10.6.
expq(r(t)v(t))
= 1, v(t) e TMq.
The length
equal to
Ir(b) - r(a)I,
function
r(t)
a
IIdII dt is greater than or
where equality holds only if the
is monotone, and the function v(t) is constant.
with
§10. GEODESICS AND COMPLETENESS
61
Thus the shortest path joining two concentric spherical shells around
is a radial geodesic.
q
PROOF:
Let
so that
expq(rv(t)),
f(r,t)
Then
m(t)
=
f(r(t),t)
afr,(t) +af
dm =
CIE
3E
cTr
Since the two vectors on the right are mutually orthogonal, and since 1,
this gives
II
Ir'(t)12 +
=
II2
where equality holds only if
=
b
S
a
II2 > Ir'(t)12
II
0;
hence only if
dv
=
0.
Thus
b
-Ildt > S Ir'(t) Idt > Ir(b) - r(a) I a
where equality holds only if
v(t) is constant.
is monotone and
r(t)
This completes the proof.
The proof of Theorem 10.4 is now straightforward.
from q
piecewise smooth path m
q'
where
0 < r < e,
IvIi
to a point
expq(rv) E Uq
=
1.
=
Then for any
s > o
tain a segment joining the spherical shell of radius shell of radius segment will be
> r - s;
will be
> r.
If
the path m must cons
and lying between these two shells.
r,
hence letting
m([0,1])
Consider any
s
tend to
does not coincide with
easily obtain a strict inequality.
0
to the spherical
The length of this the length of
y([0,1)),
m
then we
This completes the proof of 10.4.
An important consequence of Theorem 1o.4
is the following.
Suppose that a path co: [o,A] - M, parametrized by arc-length, has length less than or equal to COROLLARY 10.7.
the length of any other path from m(0) to is a geodesic. PROOF:
Consider any segment of
above, and having length 10.4.
< e.
Hence the entire path m DEFINITION.
A geodesic
co
m(A).
Then m
lying within an open set W, as
This segment must be a geodesic by Theorem is a geodesic. y:
[a,b]
-r M will be called minimal
if
62
II.
RIEMANNIAN GEOMETRY
its length is less than or equal to the length of any other piecewise smooth path joining its endpoints.
Theorem 10.4 asserts that any sufficiently small segment of a On the other hand a long geodesic may not be minimal.
geodesic is minimal.
For example we will see shortly that a great circle arc on the unit sphere If such an arc has length greater than
is a geodesic.
n,
it is certainly
not minimal.
In general, minimal geodesics are not unique.
For example two anti-
podal points on a unit sphere are joined by infinitely many minimal geodesics. However, the following assertion is true.
Define the distance
p(p,q)
p,q e M
between two points
to be the
greatest lower bound for the arc-lengths of piecewise smooth paths joining these points.
This clearly makes M into a metric space.
It follows
easily from 10.4 that this metric is compatible with the usual topology of M.
Given a compact so that any two tance less than s are joined by length less than s. Furthermore and depends differentiably on its COROLLARY 10.8. a number
PROOF.
6 > 0
Cover
K by open sets
set K C M there exists points of K with disa unique geodesic of this geodesic is minimal; endpoints.
Wa,
as in 10.3, and let
small enough so that any two points in K with distance less than
in a common Wa.
be
s s
lie
This completes the proof.
Recall that the manifold M is geodesically complete if every geodesic segment can be extended indifinitely.
THEOREM 10.9
(Hopf and Rinow*). If M is geodesically
complete, then any two points can be joined by a minimal geodesic.
PROOF.
Up
Given p,q e M with distance
as in Lemma 10.3. Let
Compare p. 341 of
S C Up
r > 0,
choose a neighborhood
denote a spherical shell of radius
s < a
G. de Rham, Sur la r6ductibilite d'un espace de
Riemann, Commentarii Math. Helvetici, Vol. 26 (1952); as well as H. Hopf and W. Rinow, Ueber den Begriff der-vollstandigen differentialgeometrischen Flache, Commentarii,Vol. 3 (1931), pp. 209-225.
GEODESICS AND COMPLETENESS
§10.
about
is compact, there exists a point
S
Since
p.
p0 on
S
63
expp(sv)
=
for which the distance to
,
1,
=
11V II
We will prove that
is minimized.
q
expp(rv)
=
This implies that the geodesic segment
t
is actually a minimal geodesic from p
to
q.
0 < t < r,
y(t) = expp(tv),
-+
q.
The proof will amount to showing that a point which moves along the geodesic
must get closer and closer
y
to
In fact for each
q.
t e [s,r]
we will prove that
(it)
p(y(t),q)
This identity, for
r-t
=
.
will complete the proof.
t = r,
First we will show that the equality
path from p
to
must pass through
q
p(p,q) = Min
se S
p(p0,q) = r - s.
Therefore
is true.
(it)
Since every
we have
S,
(p(p,s) + p(s,q)) = s + p(po,q) Since
p0 = y(s),
this proves (1s).
denote the supremum of those numbers
t0 e [6,r]
Let
is true.
(18)
Then by continuity the equality
t
for which
is true also.
(it ) 0
If
to < r we will obtain a contradiction.
cal shell of radius point of
S'
about the point
s'
with minimum distance from
p(Y(t0),q) =
Let
y(to); q.
S'
denote a small spheri-
and let
pp e S'
(Compare Diagram 10.)
be a Then
p(s,q)) = 6' + p(pp,q)
hence
p(po,q) = (r - t0) - 6'
(2)
We claim that
po
is equal to
y(to + s').
In fact the triangle
inequality states that p(p,pp') > p(p,q) - p(pp,q) = to + s'
(making use of (2)). po
But a path of length precisely
is obtained by following
a minimal geodesic from
y(t0)
y
from p to
po.
to
y(to),
to + s'
from p
to
and then following
Since this broken geodesic has
minimal length, it follows from Corollary 10.7 that it is an (unbroken)
II.
64
REIMANNIAN GEOMETRY
geodesic, and hence coincides with Thus (1
y(to + s') = pp. 0(y(to +
to+s')
y.
Now the equality (2) becomes ,'),q) = r - (to + s')
This contradicts the definition of
t0;
and completes the proof.
Diagram 10. As a consequence one has the following.
If M is geodesically complete then every bounded subset of M has compact closure. Consequently M is complete as a metric space (i.e., every Cauchy sequence converges). COROLLARY 10.10.
PROOF. expp: of
If
X C M has diameter
TMp -+ M maps the disk of radius
d d
then for any p E X
in TMp
M which (making use of Theorem 10.9) contains
of X
the map
onto a compact subset X.
Hence the closure
is compact.
Conversely, if M is complete as a metric space, then it is not difficult, using Lemma 10.3, to prove that M
is geodesically complete.
For details the reader is referred to Hopf and Rinow.
Henceforth we will
not distinguish between geodesic completeness and metric completeness, but will refer simply to a complete Riemannian manifold.
GEODESICS AND COMPLETENESS
§10.
65
In Euclidean n-space,
FAMILIAR EXAMPLES OF GEODESICS.
with
Rn,
the usual coordinate system x1,...,xn and the usual Riemannian metric
dx®® dx1 +...+ dxn ® dxn we have desic
y,
ri
and the equations for a geo-
0
=
given by t- (x1(t),...,xn(t)
become
d2xi
dt whose solutions are the straight lines. follows:
'
This could also have been seen as
it is easy to show that the formula for arc length
S i=1 (
\
2dt
coincides with the usual definition of aru length as the least upper bound of the lengths of inscribed polygons;
from this definition it is clear that
straight lines have minimal length, and are therefore geodesics. The geodesics on intersections of
PROOF.
Reflection through a plane
with a unique geodesic
C = Sn n E2.
and
E2
I(C')
I(y) = y.
Sn.
is an isometry
I:
Sn -+ Sn
Let x and y be two points of
of minimal length between them.
C'
is an isometry, the curve between
are precisely the great circles, that is, the
Sn with the planes through the center of
whose fixed point set is
I(x) = x
Sn
Then, since
is a geodesic of the same length as Therefore
C' = I(C').
C I
C'
This implies that
C' C C.
Finally, since there is a great circle through any point of
Sn
in
any given direction, these are all the geodesics.
Antipodal points on the sphere have a continium of geodesics of minimal length between them.
All other pairs of points have a unique geo-
desic of minimal length between them, but an infinite family of non-minimal geodesics, depending on how many times the geodesic goes around the sphere and in which direction it starts.
By the same reasoning every meridian line on a surface of revolution is a geodesic.
The geodesics on a right circular cylinder
Z
are the generating
lines, the circles cut by planes perpendicular to the generating lines, and
66
II.
the helices on
Z.
PROOF:
isometry
I:
RIEMANNIAN GEOMETRY
L
If
is a generating line of
Z - L - R2
by rolling
Z
Z
then we can set up an
onto R2:
a1
The geodesics on
Z
are just the images under
in R.2 Two points on
I-1
of the straight lines
Z have infinitely many geodesics between them.
PART III
THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS
§11. The Path Space of a Smooth Manifold.
p and
Let M be a smooth manifold and let sarily distinct) points of
will be meant a map 1)
[0,1]
By a piecewise smooth path from p to q
M.
[0,1] - M such that
w:
there exists a subdivision
0 = to < t1 < ... < tk
is differentiable of class
so that each wl[ti_1,ti] 2)
be two (not neces-
q
=
1
of
C
w(o) = p and w(1) = q.
The set of all piecewise smooth paths from p
by 0(M;p,q),
or briefly by 0(M)
Later (in §16)
0
or
in M will be denoted
q
to
11.
will be given the structure of a topological We will think of
space, but for the moment this will not be necessary.
To start the
"infinite dimensional manifold."
as being something like an
a
analogy we make the following definition.
By the tangent space
of
a
at a path w will be meant the vector
space consisting of all piecewise smooth vector fields W along w
which W(0) = 0
and W(1) = 0.
for
The notation TOW will be used for this
vector space.
If F
is a real valued function on
F*:
it is natural to ask what
0
TQw - TRF(w)
,
the induced map on the tangent space, should mean.
When F
which is smooth in the usual sense, on a smooth manifold F*:
TMp - T RKp)
u - a(u)
in
M,
as follows.
Given X E TMp
which is defined for a(0)
=
p,
=
M,
we can define
choose a smooth path
-e < u < e
moo)
is a function
X
,
so that
68
CALCULUS OF VARIATIONS
III.
Then F*(X)
d(F(a(u))I
is equal to
Tu
multiplied by the basis vector
u=o'
d Uf )F (P) E TRF(p) In order to carry out an analogqus construction for
F:
a
R,
-
the following concept is needed.
m
A variation of
DEFINITION.
(keeping endpoints fixed) is a
function
(-e,e) -. 0,
a: e > o,
for some
of
such that m
1)
&(o)
2)
there is a subdivision
=
0 = to < t1 < ... < tk =
1
so that the map
[0,11
(-e,e) x [0,11 -+ M
a: defined by
is
a(u,t) = &(u)(t)
C°°
on each strip
(-e,e) x [ti-1,ti
Q = 0(M;p,q),
note that:
i = 1,...,k.
Since each 3)
belongs to
&(u)
a(u,o) = p,
a(u,1) = q
We will use either
a
for all u e (-e,e)
a
or
to refer to the variation.
generally if, in the above definition,
hood U of tion of
in Rn,
then a
(or
(-e,e)
More
is replaced by a neighbor-
is called an n-parameter varia-
a)
m.
Now vector"
0
.
a
cTu (o) d.
may be considered as a "smooth path" in e Tom
Its "velocity
11.
is defined to be the vector field W along m
given
by
Wt
Clearly W
E T11m.
=
m(o)t =(o,t)
This vector field W is also called the variation vec-
tor field associated with the variation
a.
Given any W E Tom note that there exists a variation (-e,e) - 11
which satisfies the conditions
a(o) = to,
mo(o)
=
W.
In fact one can set &(u) (t)
=
expm(t)(u Wt)
.
By analogy with the definition given above, if
F
is a real valued
function on
we attempt to define
a,
F* : as follows.
Given W e Tow
d F(n)'
=
mo(o)
w,
d(F())lu=o
equal to
F*(W)
-+ TRF(m)
TO
choose a variation
a(o)
and set
69
THE PATH SPACE
§11.
CLU
=
a:
(-E,e) - 0
with
W
multiplied by the tangent vector
Of course without hypothesis on F
there is no guarantee that
this derivative will exist, or will be independent of the choice of
a.
We will not investigate what conditions F must satisfy in order for F* We have indicated how F*
to have these properties.
might be defined only
to motivate the following. DEFINITION. F:
A path w
0 -+ R if and only if
dF(
u
is a critical path for a function is zero for every variation
I
a
of
u=o w.
EXAMPLE.
derivatives
(
If
F
takes on its minimum at a path mo, are all defined, then clearly
mo
and if the
is a critical path.
CALCULUS OF VARIATIONS
III.
70
The Energy of a Path.
§12.
The length of a vec-
Suppose now that M is a Riemannian manifold. i
v E TMp will be denoted by
for
energy of
from a to
co
o < a < b < 1)
(where
b
For
< v,v >2.
_
IlvOO
define the
a e a
as
=SI1-df
Ea(m)
a
We will write E
for E.
This can be compared with the arc-length from
a
to
given by
b
b b
La ces)
=
S
IIdt
II
a
Applying Schwarz's inequality
as follows.
b
b
b
S fgdt)l2 < \ S f 2dt') ( .S g2dt)l a
a with
f(t)
=
1
g(t) =
and
a
we see that
IIdo) II
< (b - a)Ea
(Lb
where equality holds if and only if the parameter
t
,
is constant; that is if and only if
g
is proportional to arc-length.
Now suppose that there exists a minimal geodesic to
q = m(1).
y
from p = m(o)
Then E(y)
=
L ( - / )
L ( w )
.
L(m)2
can hold only if
m
geodesic, possibly reparametrized.
(Compare §10.7.)
On the other hand
Here the equality L(y)2
the equality
L(w)2
to arc-length along
a minimal geodesic.
=
w.
=
E(m)
can hold only if the parameter is proportional
This proves that
E(y) < E(m)
unless
In other words:
Let M be a complete Riemannian manifold p,q e M have distance d. Then the energy
LEMMA. 12.1.
and let
is also a minimal
function E:
a(M;p,q) - R
a
is also
TEE ENERGY OF A PATH
§12.
takes on its minimum
geodesics from p
precisely on the set of minimal
d2
to
q.
Let
are critical paths for the
m e 9
We will now see which paths energy function
E.
(-e,e) - 2
a:
be a variation of
=
=
and let Wt =
m,
Tu-
(o,t)
Furthermore, let:
be the associated variation vector field.
Vt
71
m
velocity vector of
UE DcT
At
otv
- =
Vt. - Vt_
=
=
discontinuity in the velocity vector at
0
where otV
Of course
=
0
m
acceleration vector of
.
for all but a finite number of values of
THEOREM 12.2 (First variation formula). dE(a(u)) is equal to u=o
2 -
t,
t.
The derivative
1
1
PROOF:
- S <Wt,At> dt
<Wt,AtV>
t
0
According to Lemma 8.3, we have
as as D as as Tu <-cf,ZT > = 2 < Tu Z' Therefore
dE(a(u))
_
d
TU-
J(
as, as > dt
0
0
D as
By Lemma 8.7 we can substitute Choose
each strip
(-e,e) x [ti_1,ti1.
as as -ate
D as 3u 3T in this last formula.
for
3ETu
0 = t0 < t1 <...< tk =
[ti_1,til, as follows.
< D -3 as,- as > dt
2)
a
so that
1
Then we can
is differentiable on
"integrate by parts" on
The identity
D as ,as
as D as
>+
-3y
>
implies that ti
S
3E 3u ,3T > dt
< aa, aa
Tu3t>
=
ti-1
(ti ,1
)a dt , 2T)
ti-1
CALCULUS OF VARIATIONS
III.
72
i = 1,...,k;
Adding up the corresponding formulas for
that u = o
for
t = 0 or 1,
this gives
k-1 1
2 Setting
as pti < U, L i=1
dE(a(u))
u
u = 0,
and using the fact
as
1
]-
as D as < y.-, ZE ZE > dt 0
we now obtain the required formula
2 dE a (o)
< W,AtV > - S < W,A > dt
t
0
This completes the proof.
Intuitively, the first term in the expression for da(0) shows that varying the path w
in the direction of decreasing
"kink," tends to
E; see Diagram 11.
decrease
The second term shows that varying the curve in the direction of its
() tends to reduce
acceleration vector
Recall that the path w e 0 w of
is
w
C`°
is called a geodesic
on the whole interval [0,11,
is identically zero along COROLLARY 12.3.
function E
E.
if and only if
and the acceleration vector -(
w.
The path w
if and only if
co
is a critical point for the is a geodesic.
THE ENERGY OF A PATH
§12.
PROOF:
Clearly a geodesic is a critical point.
There is a variation of
critical point. f(t)
73
m with W(t) = f(t)A(t)
is positive except that it vanishes at the
2 mo(o)
_
m be a
Let
ti.
where
Then
- f(t) < A(t),A(t) > dt. 0
This is zero if and only if
A(t) = 0
for all
t.
Hence each nI[ti,ti+1]
is a geodesic.
Now pick a variation such that W(ti) = Ati V.
2 (0) _ m
< AtiV,AtiV > .
is differentiable of class
Then
If this is zero then all AtV C1,
even at the points
ti.
from the uniqueness theorem for differential equations that everywhere:
thus
co
is an unbroken geodesic.
are
0,
and
Now it follows m
is
C°°
CALCULUS OF VARIATIONS
III.
74
The Hessian of the Energy Function at a Critical Path.
§13.
Continuing with the analogy developed in the preceding section, we now wish to define a bilinear functional
E**: when
-R
TOY x TOy
is a critical point of the function
y
i.e., a geodesic.
E,
bilinear functional will be called the Hessian of If
point
p,
f
E
at
This
y.
is a real valued function on a manifold M with critical
then the Hessian f**:
can be defined as follows. (u1,u2)
a(u1,u2)
values in
M,
TM p x TMp -+ R
Given X1,X2 E TMp
choose a smooth map
defined on a neighborhood of
(0,0) in R2,
with
so that
a(o,o)
p,
=
=
X1(0,0)
X2(0,0)
x1
=
x2
Then 2f(a(u1,u2)) f**(X1,X2)
=
au1 au2
This suggests defining E**
as follows.
(0,0)
Given vector fields
W1,W2
e Tsar
choose a 2-parameter variation
a: where
U
is a neighborhood of
a(o,o,t)
=
y(t),
Ux[0,11
-M
(0,0) in R2,
(o,o,t)
so that
W1(t),
=
au (0,0,t)
(Compare §11.)
=
W2(t)
2
1
Then the Hessian E**(W1,W2)
will be defined to be the
second partial derivative 2E(a(ut,u2))I
auI au2
where
a(u1,u2) e 0
denotes the path
(0 , 0)
a(u1,u2)(t)
=
a(u1,u2,t)
.
This
second derivative will be written briefly as o u2(0,0) 1
2
The following theorem is needed to prove that
E**
is well defined.
THE HESSIAN OF E
§13.
75
THEOREM 13.1 (Second variation formula).
Let
be a 2-parameter variation of the geodesic variation vector fields
Wi
=
x,(0,0) e Tay, 1
Then the second derivative
2
&:
U -+sa
with
y
i = 1,2
a"E
of the energy
(0,0)
u3 you
function is equal to DW
-
D 2W
<W2(t),AtUU1> -S
t
I
< W21
+ R(V,W1)V > dt
0
where V = d
denotes the velocity vector field and where
ofd _(t+) -(t ) DW
denotes the jump in
-dy at one of its finitely many points of discontinuity in the open unit interval.
PROOF:
According to 12.2 we have 1
1
aE
2 3u2
=
-
as as as D as At < cTu2,t > - S
t
0
Therefore
1
f
a2E u2 u1
=
-
D
t
as
> -
3T as
D
C
ac
< cTu1
D as
D
as
2P1
t
as
> dt
< cTu1 Tu2 ) 3T 'R
as At 'R >
D D as
37
0
0
Let us evaluate this expression for
(u1,u2) = (0,0).
Since
y = &(0,0) is
an unbroken geodesic, we have as
At 3T
=
0,
D as
0
WE ZT
e
so that the first and third terms are zero.
Rearranging the second term, we obtain
(13.2)
1
2
6E
0 0)
-
W2 At aE W1
< W2, 1 -RV > dt
)
0
In order to interchange the two operators
and
D
3
,
we need to
1
bring in the curvature formula,
1
D
V - A 1V = R( 4,Tu- )V 1
=
R(V,W1)V
dt.
76
CALCULUS OF VARIATIONS
III.
V
Together with the identity
=
1
(13.3)
D _3T
=
3U T_
1
_CTU
this yields
W1
1
DST
V
+ R(V,W1)V
=
d
ff
Substituting this expression into (13.2) this completes the proof of 13.1. 2 )E
The expression E**(W1,W2) =
COROLLARY 13.1+.
-2(0,0)
is a well defined symmetric and bilinear function of W and W2. PROOF:
The second variation formula shows that
depends only on the variation vector fields
This formula also shows that
is well defined.
E**(W1,W2)
and W2,
W1
ua u 7-0, 0) 2 1
so that
E**
is bilinear.
The symmetry property E**(W1,W2)
E**(W2, W1)
=
is not at all obvious from the second variation formula; but does follow immediately from the symmetry property REMARK 13.5. E**
=
aEu2 u1
The diagonal terms
uc
E**(W,W)
6 E 7 1
of the bilinear pairing
can be described in terms of a 1-parameter variation of
E**(W,W)
where
(-e,e) -+s1
a:
In fact
a(o)
denotes any variation of
equal to W.
field T- (o)
d E
=
y.
with variation vector
y
To prove this identity it is only necessary to
introduce the two parameter variation
(u1,u2)
=
a(u1 + u2)
and to note that
u=ci' t
a9
da
a2E °
u
= d2
d2E ° a
As an application of this remark, we have the following.
LEMMA 13.6.
If y is a minimal geodesic from p to q then the bilinear pairing E** is positive semi-definite. Hence the index X of E** is zero.
PROOF:
d2E(a(
,
The inequality
evaluated at
E(a(u)) > E(y)
u = 0, is > 0.
Hence
=
E(a(o))
implies that
E**(W,W) > 0
for all W.
JACOBI FIELDS
§14.
The Null Space of
Jacobi Fields:
§14.
A vector field
along a geodesic
J
77
E**
is called a Jacobi field
y
if it satisfies the Jacobi differential equation 2
D + R(V,J)V
=
o
dt
where V
=
This is a linear, second order differential equation.
Ur .
[It can be put in a more familiar form by choosing orthonormal parallel vector fields
along
P1,...,Pn
Then setting
y.
J(t) = E fi(t)Pi(t),
the
equation becomes 2 i da
+
L a'(t)fj(t)
o,
=
i = 1,..., n;
j=1
where
a3 = < R(V,Pj)V,Pi > .]
Thus the Jacobi equation has 2n linearly
independent solutions, each of which can be defined throughout solutions are all
A
C°°-differentiable.
given Jacobi field J
y.
The
is com-
pletely determined by its initial conditions: J(o), uE(o) e TM 'Y(0)
Let
with a
and
p = y(a)
q = y(b)
be two points on the geodesic
y,
b.
and
p
DEFINITION.
J
non-zero Jacobi field
p
The multiplicity of
q
along and
q
are conjugate* along y
y
which vanishes for
if there exists a t = a
and
t = b.
as conjugate points is equal to the dimen-
sion of the vector space consisting of all such Jacobi fields.
Now let
y
be a geodesic in
S2
= S0(M;p,q).
Recall that the null-
space of the Hessian
E**:
TOY x TOy--. R
is the vector space consisting of those
*
W1 e TOy
such that
E**(W1,W2) = 0
If y has self-intersections then this definition becomes ambiguous. One should rather say that the parameter values a and b are conjugate with respect to y.
CALCULUS OF VARIATIONS
III.
78
for all W2.
The nullity E**
null space.
v
is degenerate
is equal to the dimension of this
E**
of
if
v > 0.
THEOREM 14.1.
A vector field W1 e TOy belongs to the null space of E** if and only if W1 is a Jacobi field. Hence E** is degenerate if and only if the end points p and q are conjugate along y. The nullity of E** is equal to the multiplicity of p and q as conjugate points.
(Compare the proof of 12.3.)
PROOF:
p
vanishes at
and
then J
q,
J. is a Jacobi field which
If
certainly belongs to
The second
TOO.
variation formula (§13.1) states that Cl
t Hence
0
belongs to the null space.
J
Conversely, suppose that W1 Choose a subdivision W1l[ti_1,ti1
belongs to the null space
o = t0 < t1 <...< tk =
is smooth for each
Let
i.
of
1
E**.
so that
[0,11
[0,11
f:
function which vanishes for the parameter values
of
be a smooth
-+ [0,11
t0,t1,...,tk and is
positive otherwise; and let D
2W
2 + R(VW1)V)t
W2(t) = f(t)( dt
Then 2E**(W1,W2)
f(t)t1
0 +
=
2
2W
1
-
E
o
Since this is zero, it follows that W1j[ti-1,ti1 each
dt
+ R(V,W1)VII
is a Jacobi field for
i.
DW1
I
Now let W2 e TOOy
be a field such that W2(ti)
=
At i
i = 1,2,...,k-1.
--
for
Then k-1
DW1
Ati
- 2E**(W1,WI)
2
.-
1
0 dt = 0
+
0
i=1
DW1
Hence
-cTE
has no jumps.
But a solution W1
of the Jacobi equation is DW1
completely determined b by the vectors W 1(ti) lows that the
k Jacobi fields W11[ti_1,ti1,
to give a Jacobi field
W1
which is
and
Ur-(ti).
i = 1,...,k,
Thus it folfit together
C`°-differentiable throughout the
JACOBI FIELDS
§14.
79
This completes the proof of 14.1.
entire unit interval.
It follows that the nullity
is always finite.
E**
of
v
For
there are only finitely many linearly independent Jacobi fields along Actually the nullity
REMARK 14.2.
satisfies
v
precisely
it is clear that
n,
This will imply that
V = 7 denotes the velocity vector field.
DJ
V+ t DV
1
(IT
=
but not for
t = 0,
In fact let
v < n.
has dimension
We will construct one
v < n.
example of a Jacobi field which vanishes for t = 1.
Since
0 < v < n.
t = 0
the space of Jacobi fields which vanish for
y.
Jt = tVt
where
Then V
2
Uy since R
= 0
hence
= o),
(Since
dt7
=
R(V,J)V = tR(V,V)V Thus
is skew symmetric in the first two variables.
satisfies the Jacobi equation.
J
Furthermore
0.
J 0 = 0,
Since
J1 # 0,
this
completes the proof.
Suppose that M is "flat" in the sense that the curva-
EXAMPLE 1.
ture tensor is identically zero. dt2 = 0.
J(t) =) fi(t)Pi(t)
Setting
this becomes
Then the Jacobi equation becomes
2 i ddf7 = 0.
where
Pi
are parallel,
Evidently a Jacobi field along
y
can have
Thus there are no conjugate points, and E**
at most one zero.
is
non-degenerate. EXAMPLE 2.
unit sphere
Sn,
and let
Then we will see that
and
p
Suppose that
be a great circle arc from p to
y
and
p
are antipodal points on the
q
q
are conjugate with multiplicity
Thus in this example the nullity
n-1.
largest possible value.
q.
v
of
E**
takes its
The proof will depend on the following
discussion.
Let
a
be a 1-parameter variation of
the endpoints fixed, such that each
x [0,11
a: be a
by
COO
map such that
&(u)(t)
=
a(u,t)]
a(o,t)
&(u)
=
y(t),
is a geodesic.
y,
not necessarily keeping
is a geodesic.
That is, let
-+ M and such that each
a(u)
[given
80
LEMMA 14.3.
III.
CALCULUS OF VARIATIONS
a
is such a variation of
If
y
through
geodesics, then the variation vector field W(t) _(O,t) is a Jacobi field along
a is a variation of
If
PROOF:
is identically zero. 0
°
through geodesics, then
y
D as
dT3£
Hence
D as D D D as Tu3-E3y - 3TTu 3T D2
(Compare §13.3.)
Y.
as
as
R( as as \ as
+
ZE
CTU
a(X ) as
+ R\.t'Tu
at
Therefore the variation vector field
Y
is a Jacobi
field.
Thus one way of obtaining Jacobi fields is to move geodesics around.
Now let us return to the example of two antipodal points on a unit n-sphere.
Rotating the sphere, keeping p
vector field along the geodesic and
q.
and
q
fixed, the variation
will be a Jacobi field vanishing at p
y
Rotating in n-1 different directions one obtains n-1 linearly
independent Jacobi fields.
multiplicity
n-1.
Thus
p
and
q
are conjugate along
y
with
§14.
JACOBI FIELDS
81
Every Jacobi field along a geodesic y: [0,11-+ M through geodesics. y
LEMMA 14.4.
may be obtained by a variation of
Choose a neighborhood U
PROOF: of
of
so that any two points
y(o)
U are joined by a unique minimal geodesic which depends differentiably
on the endpoints.
Suppose that
construct a Jacobi field W along values at
t = 0
and
Similarly choose
b:
(-E,E)
We will first
0 < t < s.
with arbitrarily prescribed
yl[o,s]
Choose a curve
t = s.
and so that mo(o)
a(o) = y(o)
for
y(t) e U
(-e,E)
a:
is any prescribed vector in
-+ U with b(0)
y(s)
so that
-+ U
TM y(0).
and T(o)
arbitrary.
Now define the variation a: (-E,E) x [o,s]
by letting to
b(u).
-+ M
M be the unique minimal geodesic from
a(u):
Then the formula
t -+(o,t) (TU
a(u)
defines a Jacobi field with the
given end conditions.
Any Jacobi field along
7(y)
yl[o,s]
can be obtained in this way:
denotes the vector space of all Jacobi fields W along
formula W
(W(0), W(s))
dimension
2n
then the
defines a linear map
f: 7(y) We have just shown that f
y,
If
-+ TM
is determined by its values at
f
TM
7(s)
Since both vector spaces have the same
is onto.
it follows that
Y(0) X
is an isomorphism.
y(o)
and
y(s).
I.e., a Jacobi field
(More generally a Jacobi
field is determined by its values at any two non-conjugate points.)
There-
fore the above construction yields all possible Jacobi fields along yl[o,s1.
The restriction of If
u
a(u)
to the interval
10,81
is not essential.
is sufficiently small then, using the compactness of
[0,11,
can be extended to a geodesic defined over the entire unit interval
a(u) [0,11.
This yields a variation through geodesics: a&:
(-E',E') X [0,11
-+ M
with any given Jacobi field as variation vector.
REMARK 14.5.
This argument shows that in any such neighborhood
the Jacobi fields along a geodesic segment in U are uniquely determined
U
82
III.
CALCULUS OF VARIATIONS
by their values at the endpoints of the geodesic.
REMARK 14.6. (-S,s)
of
The proof shows also, that there is a neighborhood
so that if
0
t e (-s,s)
y(t)
is not conjugate to
We will see in §15.2 that the set of conjugate points to
y(o)
along
y(o)
along the entire geodesic
y.
then
y
has no cluster points.
The index
§15.
THE INDEX THEOREM
§15.
The Index Theorem.
83
of the Hessian
%
E**:
TOy x TOy-+ R
is defined to be the maximum dimension of a subspace of
Tay
on which E**
We will prove the following.
is negative definite.
THEOREM 15.1 (Morse).
The index X of E** is equal y(t), with 0 < t < 1, such that y(t) is conjugate to y(o) along y; each such conjugate point being counted with its multiplicity. This index is always finite*. to the number of points
.
As an immediate consequence one has: COROLLARY 15.2.
A geodesic segment y: [0,11 - M can contain only finitely many points which are conjugate to y(o) along y.
In order to prove 15.1 we will first make an estimate for splitting the vector space
one of which E**
into two mutually orthogonal subspaces, on
is contained in an open set
y(t)
o = to < t1
..< tk =
so that each segment
that each
Let
such that any two
y[ti_i,til
yl[ti_1,tiI
Choose a subdivision
of the unit interval which is sufficiently fine
1
lies within such an open set
TOy(to,tl,t2,...,tk) C TOy y
and so
be the vector space consisting of
such that
is a Jacobi field along
1)
WI[ti_i,til
2)
W vanishes at the endpoints
T11y(t0,tl,...,tk)
U;
is minimal.
all vector fields W along
yl[ti_1,tiI
for each
i;
t = 0, t = 1.
is a finite dimensional vector space consisting of
broken Jacobi fields along *
U
U are joined by a unique minimal geodesic which depends differ-
entiably on the endpoints. (See §10.)
Thus
by
is positive definite.
Each point points of
Tay
)
y.
For generalization of this result see: W. Ambrose, The index theorem in Riemannian geometry, Annals of Mathematics, Vol. 73 (1961), pp. 49-86.
CALCULUS OF VARIATIONS
III.
84
Let
be the vector space consisting of all vector fields
T' C Tsar
for which W(t0) = 0,
W E TOY
W(t1) = 0,
W(t2) = 0,..., W(tk) = o.
LEMMA 15.3. The vector space Tsar splits as the direct sum TOy(to,t,,...,tk) ® T'. These two subspaces are mutually perpendicular with respect to the inner product
restricted to
E**
Furthermore,
E**.
is positive
T'
definite.
Given any vector field W E TOY
PROOF:
"broken Jacobi field" in TOy(t0,t1,...,tk) It follows from §14.5 that
i = 0,1,...,k.
Clearly W - W1 and
belongs to
generate
T'
If
denote the unique
W1
exists and is unique.
W1
Thus the two subspaces,
T'.
for
Tsay(to,t1,...,tk)
and have only the zero vector field in common.
Tsar,
belongs to
W1
let
such that W1(ti) = W(ti)
and W2
Tsay(to)tl,...,tk)
belongs to
T',
then the second variation formula (13.1) takes the form
<W2(t),pt
-liE**(Wj
1
LW -1
<W2,o > dt
> -
t
0
=
0
Thus the two subspaces are mutually perpendicular with respect to E**
For any W E TO y
can be interpreted as the
the Hessian E**(W,W)
2
d E 2 a (0);
second derivative y
where
7,
with variation vector field mo(o)
W belongs to points
for
y(ti)
=
Proof that
&
for W e T'.
y(o) to
y(t1)
E(a(u)) > E(y)
to
Proof that
E**(W,W) > 0
were equal to
a(u) e 0
to ... to
is a piece-
y(1).
But
=
This proves that E(a(o))
0.
for W E T',
.
u = 0,
W
0.
must be
> 0.
Suppose that
Then W would lie in the null space of
In fact for any W1 e T0y(t0,t1,...,tk) 0.
Each
y(t2)
Therefore the second derivative, evaluated at
=
is chosen so as to leave the
is a minimal geodesic, and therefore has smaller energy
yl[ti_1,ti]
E**(W1,W)
If
In other words we may assume that
than any other path between its endpoints.
E**(W,W)
(Compare 13.5.)
W.
i = 0,1,...,k.
E**(W,W) > 0
wise smooth path from each
fixed.
is any variation of
(-s,e)
equal to
then we may assume that
T'
y(to),y(t1),...,y(tk)
a(u)(ti)
&:
For any W2 e T'
we have already seen that
the inequality
E**.
THE INDEX THEOREM
§15.
0 < E**(W + c W2, W + c W2)
for all values of null space. T'
implies that
c
But the null space of
85
2c E**(W2,W) + C
=
E**(W2,W) = 0. E**
2
E**(W2,W2)
Thus W lies in the
consists of Jacobi fields.
Since
contains no Jacobi fields other than zero, this implies that W = 0.
Thus the quadratic form E**
is positive definite on
This
T'.
completes the proof of 15.3.
An immediate consequence is the following:
LEMMA 15.4.
The index (or the nullity) of E** is equal to the index (or nullity) of E** restricted to the space TOy(t0,t1,...,tk) of broken Jacobi fields. In particular (since To (t0,tl,...,tk) is a finite dimensional vector 7 space) the index X is always finite. The proof is straightforward. Let Thus
denote the restriction of
yT
[O,T] -+ M is a geodesic from
yT:
the index of the Hessian Thus
X(1)
( Eo )**
to the interval
y
y(o) to
y(T).
Let
[O,T].
denote
?(T)
which is associated with this geodesic.
is the index which we are actually trying to compute.
First
note that:
ASSERTION (1).
X(T)
For if
then there exists a
T < T'
vector fields along Hessian
( Eo )**
yT
is a monotone function of
which vanish at
X(T)
y(o)
y(T)
dimensional space
and
y(T)
and
y(T').
Thus we obtain a
yT'
X(T)
P of
such that the
is negative definite on this vector space.
field in '' extends to a vector field along between
T.
Each vector
which vanishes identically dimensional vector space
Eoi
of fields along
yT,
on which
)**
is negative definite.
Hence
%(T) <
ASSERTION (2).
For if hence
X(T) = 0
T
T.(-T)
= 0
for small values of
is sufficiently small then
yT
T.
is a minimal geodesic,
by Lemma 13.6.
Now let us examine the discontinuities of the function note that
X(T)
is continuous from the left:
ASSERTION (3).
A.(T-E)
=
X(T).
X(T).
For all sufficiently small
e > 0
we have
First
86
III.
CALCULUS OF VARIATIONS x(1) can be interpreted as
According to 15.3 the number
PROOF.
the index of a quadratic form on a finite dimensional vector space T0y(to,t1,...,tk). say
We may assume that the subdivision is. chosen so that
ti < T < ti+1.
Then the index
of a corresponding quadratic form
broken Jacobi fields along using the subdivision
can be interpreted as the index
X(T)
on a corresponding vector space of
HT
This vector space is to be constructed
yT.
0 < t1 < t2 <... < ti < T
of
Since a
[0,T1.
broken Jacobi field is uniquely determined by its values at the break points y(ti),
this vector space is isomorphic to the direct sum
E
=
TMy(t1) ® TMy(t2) ® ... ® TMy(ti)
ratic form H. Now X(T).
on E
HT
Evidently the quad-
is independent of T.
Note that this vector space E
varies continuously with
.
T.
of dimension
is negative definite on a subspace V C E sufficiently close to
For all r'
negative definite on V.
it follows that
T
then we also have ?V(T-e) < >,(T) by Assertion 1. Hence ASSERTION (4).
Let
v
e > 0
.l.(T+e) x(t)
( Eo ).
we have
x(T) + V
=
jumps by
a conjugate point of multiplicity
< T
'X(T-e) _ X(T).
be the nullity of the Hessian
Then for all sufficiently small
Thus the function
T' = T - e
But if
X(T') >
Therefore
is
HT,
when the variable
v
passes
t
and is continuous otherwise.
v;
Clearly
this assertion will complete the proof of the index theorem. PROOF that of Assertion 3.
a(T+e) < X(T) + v
Since
some subspace )' C Z ly close to
T,
HT
Let
we see that
dim E = ni
of dimension
it follows that
.
fields along
yT,
X(T+e) > X(T) + v.
is positive definite on
For all
T'
sufficient-
is positive definite on n".
X(T') < dim E - dime" PROOF that
H.
ni - X(T) - v.
HT,
E be as in the proof
and
l.(T) + v
Let W1,-..,wx(T)
be
X(T)
vanishing at the endpoints, such that the matrix ( Ep )+ (Wi,Wj)
Hence
1
vector
THE INDEX THEOREM
§15.
is negative definite.
fields along
Let
J1,...,J,
be
87
linearly independent Jacobi
v
Note that the
also vanishing at the endpoints.
yT,
v
vectors
DJh E TMY(T)
-HE('r)
are linearly independent.
along
X1, ... ,X,
Hence it is possible to choose
is equal to the v x v over
DJh
\\
dt Extend the vector fields Wi
identity matrix.
by setting these fields equal to
yT+E
vector fields
YT+E' vanishing at the endpoints of rT + E, so that
(
Jh
v
T < t < T + E.
for
0
and
Using the second variation formula we see easily that Toe )**
Eo
Now let
Jh' WO
=
o
Eo+EI**( Jh' Xk)
=
2shk
be a small number, and consider the
c
W1,...,Wx(T),
along
YT+E'
c_1
J1
- c X1,..., c
on which the quadratic form
)(T) + v
definite.
In fact the matrix of
vector fields
Jv - c Xv
is negative
c A
-4I+ c 2
c At
and B
( E"'),,
(Eo+E)** with respect to this basis is
ET )**( WI,W
A
X(T) + v _1
We claim that these vector fields span a vector space of
dimension
where
(Kronecker delta).
are fixed matrices.
If
c
B
/
is sufficiently small, this
compound matrix is certainly negative definite.
This proves Assertion (4).
The index theorem 15.1 clearly follows from the Assertions (2),(3),
and (4) .
88
CALCULUS OF VARIATIONS
III.
§16.
A Finite Dimensional Approximation to
0c
Let M be a connected Riemannian manifold and let two (not necessarily distinct) points of piecewise
C°°
paths from p
to
and
a = 0(M;p,q)
The set
M.
p
can be topologized as follows.
q
be
q
of
Let
denote the topological metric on M coming from its Riemann metric. with arc-lengths
w, W' e n
s(t),
1
Max
0< t< 1 \
2 .ds'\ dt]
C r
p(a(t), W'(t)) +
/
L
Given
respectively,define the distance
s'(t)
to be
d(W,W')
p
J0
\
i
z
1
(The last term is added on so that the energy function 2
Eb(W)
=
dt
( d
a
will be a continuous function from induces the required topology on
Given and let
c > 0
Int Sac
0c
let
to the real numbers.)
This metric
D.
E-1([O,cl) C n
denote the closed subset
denote the open
is the energy function).
a
subset
E-1([o,c))
E = E1: n - R
(where nc
We will study the topology of
by construct-
ing a finite dimensional approximation to it.
Choose some subdivision val. W:
Let
Sa(to,t1,...,tk)
0 = to< t1 <...< tk =
be the subspace of
a
I
of the unit inter-
consisting of paths
(0,11 - M such that 1) a(o) = p and a)(1) =q 2)
ml[ti-1,ti)
is a geodesic for each
i = 1,...,k.
Finally we define the subspaces n(to,t1,...,tk)c
=
nc n n(to,t1,...,tk)
Int n(to,t1,...,tk)c
=
(Int 0c) n n(to,...,tk)
Let M be a complete Riemannian manifold; and let c be a fixed positive number such that Dc / 0. Then for all sufficiently fine subdivisions (to,tl,...,tk) of [0,11 the set Int n(to,tl,...,tk)c can be given the LEMMA 1"6.1.
structure of a smooth finite dimensional manifold in a natural way.
AN APPROXIMATION TO
§16.
PROOF:
Let
89
sac
denote the ball
S
{x E M : o(x,p) < %r-c1
Note that every path w E sac
lies within this subset
from the inequality L2 < E < c
.
Since
M
S
e > 0
so that whenever
exists
is complete,
unique geodesic from x
depends differentiably on x
and
Choose the subdivision difference
x, y E S
to y of length
there
there is a
p(x,y) < s
and so that this geodesic
< e;
y.
(t0,t1,...,tk)
is less than
ti - ti-1
This follows
Hence by 10.8
is a compact set. and
S C M.
e2/c.
of
[0,11
so that each
Then for each broken geodesic
w E o(t0,t1,...,tk)c we have ti
ti
2
Lti-1 cu)
=
(ti - ti_1) ( Eti-1 cu) < (ti
(E w)
< (ti - ti_1)c < e2 mj[ti_1,ti1
Thus the geodesic
is uniquely and differentiably determined by
the two end points.
The broken geodesic
w
is uniquely determined by the (k-1)-tuple
0)(t1), w(t2),...,w(tk_1) E M X M x...x M.
Evidently this correspondence w
--F (cu(t1),...,aa(tk-1))
defines a homeomorphism between
Int 0(t0,t1,...,tk)c
subset of the (k-1)-fold product M x...x M.
and a certain open
Taking over the differentiable
structure from this product, this completes the proof of 16.1.
To shorten the notation, let us denote this manifold Int 0(t0,t1,...,tk)c
of broken geodesics by B.
Let
E': B -'R denote the restriction to B
of the energy function E : 0 - it.
9o
III.
CALCULUS OF VARIATIONS
R is smooth. This function E': B Furthermore, for each a < c the set Ba = (E')-1[o,a] is compact, and is a deformation retract* of the corresponding set 0a. The critical points of E' are precisely the same as the critical points of E in Int 0c: namely the unbroken geodesics from p to q THEOREM 16.2.
of length less than '. The index or the nullity] of the Hessian E'** at each such critical point y E** at
is equal to the index [ or the nullity ] of
provides a faithful model
Thus the finite dimensional manifold B for the infinite dimensional path space
y.
As an immediate conse-
Int no.
quence we have the following basic result.
Let M be a complete Riemannian manifold p,q e M be two points which are not conjugate along any geodesic of length < %ra. Then as has the homotopy type of a finite CW-complex, with one cell of dimension X for each geodesic in 0a at which E** has index X. THEOREM 16.3.
and let
(In particular it is asserted that
na
contains only finitely many
geodesics.)
This follows from 16.2 together with
PROOF.
PROOF of 16.2. on the
Since the broken geodesic
§3.5.
w e B
depends smoothly
(k-1)-tuple
w(t1),w(t2),...,w(tk-1) E M x...x M
it is clear that the energy El(w) tuple.
also depends smoothly on this
In fact we have the explicit formula
k p(w(ti_1),a(ti))2/(ti _ ti_1)
E'(w) i=1
Similarly B
itself is a deformation retract of
Int sac.
(k-1)-
the set Ba
a < c
For
91
is homeomorphic to the set of all (k-1)-
(p1,...Pk_1) e S X S x...X S
tuples
0c
AN APPROXIMATION TO
§16.
such that
k p(pi_1,pi)2/(ti - ti_1) < a i=1
po = p,
(Here it is to be understood that
pk = q.)
As a closed subset
of a compact set, this is certainly compact.
A retraction
Int 0c , B
r:
is defined as follows.
from
< e
r(e)
such that each r(w)I[ti_1,ti]
denote the unique broken geodesic in B a geodesic of length
Let
to
w(ti_1)
w(ti).
is
The inequality
p(p,w(t))2
< (L e)2 < E w < c
implies that
Hence the inequality
w[o,1] C S.
t.
2
p(w(t1_1),w(ti))
w) <
< (ti - ti_1)( Et'
.
c
=
e2
i-
implies that
can be so defined.
r(w)
Clearly E(r(a)) < E(e) < c.
This retraction
r
fits into a 1-
parameter family of maps
Int 0c -+ Int sa c
ru : For
as follows.
t1-1 < U < ti
let
ru(e)I[o,ti_1I
=
r(e)I[o,ti-1]
ru(w)I[ti_1 u]
=
minimal geodesic from
,
e(ti_1) to e(u)
,
ru(w)I[u,1]
Then
B
.
is the identity map of
ro
fied that that
eI[u,1]
=
ru(w)
r1
= r.
It is easily veri-
is continuous as a function of both variables.
is a deformation retract of Since
and
Int 0c,
E(ru(w)) < E(w)
mation retract of
This proves
Int Sac.
it is clear that each Ba
is also a defor-
saa.
Every geodesic is also a broken geodesic, so it is clear that every "critical point" of
E
in
Int sac
automatically lies in the submanifold
Using the first variation formula (§12.2)
points of
E'
are precisely the unbroken geodesics.
Consider the tangent space Y.
it is clear that the critical
TBy
to the manifold B
This will be identified with the space
T0y(to)tt,...,tk)
at a geodesic
of broken
B.
III.
92
Jacobi fields along
as described in §15.
y,
justified as follows.
CALCULUS OF VARIATIONS This identification can be
Let B
be any variation of
y
variation vector field
through broken geodesics. (o,t)
along
y
Then the corresponding
is clearly a broken Jacobi field.
(Compare §14.3)
Now the statement that the index (or the nullity) of is equal to the index (or nullity) of quence of Lemma 15.4.
REMARK.
E,*
at
y
E**
at
is an immediate conse-
This completes the proof of 16.2.
As one consequence of this theorem we obtain an alternative
proof of the existence of a minimal geodesic joining two given points of a complete manifold.
ponding set function y
E':
y
For if
oa(p,q)
is non-vacuous, then the corres-
Ba will be compact and non-vacuous.
Hence the continuous
Ba _ R will take on its minimum at some point
will be the required minimal geodesic.
p,q
y E Ba.
This
THE FULL PATH SPACE
§17.
§17.
93
The Topology of the Full Path Space.
Let M be a Riemannian manifold with Riemann metric p
be the induced topological metric.
sarily distinct) points of
Let
p
and
q
g,
and let
be two (not neces-
M.
In homotopy theory one studies the space
0*
of all continuous
paths w: from
p
to
-M
[0,11
in the compact open topology.
q,
This topology can also be
described as that induced by the metric
M x
plw(t),w'(t)
t
On the other hand we have been studying the space
paths from p
to
q
0
of piecewise
with the metric
i
1
dc°,w') Since
> d*
d
=
d*(°,
)
+
r L
C°°
S 0 (dT
-
')2 dt I
2
J
the natural map
is continuous.
THEOREM 17.1.
lence between
This natural map 0 and f*.
[Added June 1968.
i
is a homotopy equiva-
The following proof is based on suggestions by
W. B. Houston, Jr., who has pointed out that my original proof of 17.1 was incorrect. S2*
The original proof made use of an alleged homotopy inverse
O which in fact was not even continuous.]
PROOF: We will use the fact that every point of M has an open neighborhood N which is "geodesically convex" in the sense that any two points of N are joined by a unique minimal geodesic which lies completely within N and depends differentiably on the endpoints. to J. H. C. Whitehead.
(This result is due
See for example Bishop and Crittenden, "Geometry of
94
CALCULUS OF VARIATIONS
III.
manifolds," p. 246; Helgason, "Differential geometry and symmetric spaces," p. 53; or Hicks, "Notes on differential geometry," p. 134.)
Choose a covering of
M by such geodesically convex open sets Na
Subdividing the interval [0,1] into 2k equal subintervals [(j-l)/2k,j/2k], let Qk denote the set of all continuous paths
w
from
[(j-l)/2 k,j/2k] should be contained in one of the sets Clearly each C1
p
to
q, and clearly C2
p
to
q
which sat-
the image under m of each subinterval
isfy the following condition:
is an open subset of the space
Na of the covering.
S2* of all paths
from
is the union of the sequence of open subsets SLR
C C2 C 0
C
Similarly the corresponding sets
"k = i-1(C1 ) are open subsets of 0 with union equal to
0.
We will first show that the natural map
"k -3 'k
(iIC<)
is a homotopy equivalence.
For each w E OO let h(w) E S2k be the broken
geodesic which coincides with w
for the parameter values
t = j/2k,
j = 0,1,2,...,2k, and which is a minimal geodesic within each intermediate interval [(j-1)/2k, j/2k).
This construction defines a function
h
:
0 -* 0k
and it is not difficult to verify that
h
is continuous.
Just as in the proof of 16.2 on page 91, it can be verified that the composition (ilCak) ° h
composition h e (iI0k)
is homotopic to the identity map of Ck and that the is homotopic to the identity map of cc.
This proves
that ilQlc is a homotopy equivalence.
To conclude the proof of 17.1 we appeal to the Appendix. Using Ex-
ample 1 on page 149 note that the space 0 is the homotopy direct limit of
the sequence of subsets S.
Similarly note that 4* is the homotopy direct
limit of the sequence of subsets Sc. that i
:
0-* S3*
Therefore, Theorem A (page 150) shows
is a homotopy equivalence.
This completes the proof.
THE FULL PATH SPACE
§17.
It is known that the space
95
has the homotopy type of a
st*
OW-
(See Milnor, On spaces having the homotopy type of a CW-complex,
complex.
COROLLARY 17.2.
0
Therefore
pp. 272-280.)
Trans. Amer. Math. Soc., Vol. 9o (1959),
has the homotopy type of a CW-
complex.
This statement can be sharpened as follows. THEOREM 17.3.
(Fundamental theorem of Morse Theory.)
Let M be a complete Riemannian manifold, and let p,q E M be two points which are not conjugate along Then 0(M;p,q) (or 0*(M;p,q)) has the any geodesic. homotopy type of a countable CW-complex which contains
one cell of dimension to
of index
q
for each geodesic from p
%
X.
The proof is analogous to that of 3.5. a0 < a1 < a2 < ...
energy function
Choose a sequence
of real numbers which are not critical values of the
E,
so that each interval
one critical value.
Consider the sequence 0ao
C
,al
C
(a1_1,ai)
,a2 C
...
contains precisely
;
a
where we may assume that
with 3.3 and 3.7 that each
0
ai
has the homotopy type of
finite number of cells attached:
in
K0 C K1 C K2 C
I K0
of homotopy equivalences.
Since
0
stal
f
C
C K1
Letting
;tae
C
...
C K2
C
...
0 - K be the direct limit mapping,
f:
induces isomorphisms of homotopy groups in all dimen-
is known to have the homotopy type of a OW-complex (17.2)
completes the proof. EXAMPLE.
f
is a homotopy equivalence. This
[For a different proof, not using 17.2, see p. 149.1 The path space of the sphere
are two non-conjugate points on p'
%
1
1
it follows from Whitehead's theorem that
where
with a
CW-complexes with cells of the required
of
...
,a0 C
sions.
together
one %-cell for each geodesic of index
description, and a sequence
it is clear that
9
ai-1
Now, just as in the proof of 3.5, one constructs a se-
E-1(a1_1,ai).
quence
It follows from 16.2
is vacuous.
o
st
Suppose that p and q
That is, suppose that
Sn.
denotes the antipode of
Sn.
p.
a / p,p'
Then there are denumerably many
CALCULUS OF VARIATIONS
III.
96
geodesics
from p
yo'y1' 2'
short great circle arc from p let
pq'p'q;
circle are
to to
q,
as follows. let
q;
k denotes the number of times that
subscript interior of
denote the
yo
and so on.
pgp'q'pq;
denote the arc
y2
Let
denote the long great
y1
p
or
The
occurs in the
p'
yk.
The index
x(yk) =
µ1 +...+ µk
p
plicity
Therefore we have:
n-1.
p'
since each
k(n-1),
in the interior is conjugate to
of the points
or
is equal to
p with multi-
has the homotopy The loop space st(Sf) CW-complex with one cell each in the dimensions
COROLLARY 17.4.
type of a 0, n-1, 2(n-1), 3(n-1),...
For n > 2 from this information.
the homology of Since
Q(Sn)
can be computed immediately
Q(Sn)
has non-trivial homology in infinite-
ly many dimensions, we can conclude:
COROLLARY 17.5. for
n > 2.
M
Let
have the homotopy type of
Then any two non-conjugate points of
Sn,
M
are
joined by infinitely many geodesics.
This follows since the homotopy type of 0(M))
depends only on the homotopy type of
geodesic in 3(n-1),
0(M)
and so on.
with index
0,
M.
o*(M)
(and hence of
There must be at least one
at least one with index
n-1, 2(n-1),
§17.
REMARK.
THE FULL PATH SPACE
More generally if M
is any complete manifold which is
not contractible then any two non-conjugate points of infinitely many geodesics.
97
M are joined by
Compare p. 484 of J. P. Serre,
Homologie
singuliere des espaces fibres, Annals of Math. 54 (1951), pp. 425-505.
As another application of 17.4, one can give a proof of the Freudenthal suspension theorem.
(Compare
§22.3.)
CALCULUS OF VARIATIONS
III.
98
Existence of Non-Conjugate Points.
§.18.
Theorem 17.3 gives a good description of the space viding that the points geodesic.
and
p
O(M;p,q)
pro-
are not conjugate to each other along any
q
This section will justify this result by showing that such non-
conjugate points always exist.
Recall that a smooth map
N - M between manifolds of the same
f:
dimension is critical at a point x e N
f.: of tangent spaces is not
1-1.
if the induced map
TNx - TMf(x) We will apply this definition to the ex-
ponential map
TMp
exp = expp:
(We will assume that M
M
.
is complete, so that exp
is everywhere defined;
although this assumption could easily be eliminated.)
THEOREM 18.1.
is critical at
v(0) = v
and mo(o) = X.
Let
a
Then the map
Therefore the vector field W given by yv.
Obviously W(O) = 0.
W(1) = .(exp v(u))I
=
v
be a path in TMp
u-. v(u)
defined by
is a variation through geodesics of the geodesic
field along
Then
v e TMp.
the tangent space at
for some non-zero X E T(TMP)v,
considered as a manifold.
if and only if the
v.
is critical at
exp
Suppose that
exp v
to
mapping
PROOF: 0
from p
yv
exp
is conjugate to p along
exp v
The point
the geodesic
t
to
exp*(X) TMp,
such that
a(u,t) = exp tv(u)
given by t-* exp tv.
yv
-(exp tv(u))u-o
is a Jacob'
We also have
exp dvu)(0)
=
exp*X = 0.
u=o
But this field is not identically zero since
dT(o)
`
TU_ 3£
(exp tv(u))I(0,0)
So there is a non-trivial Jacobi field along vanishing at these points; hence
p and
=
7v
v(u)Iu=o
from p
to
o
exp v,
exp v are conjugate along
7v
§18.
NON-CONJUGATE POINTS is non-singular at
Now suppose that exp*
ent vectors X1,...,Xn in T(TMP)v.
and -emu (0)
with vi(o) = v Then
choose paths
In T)Mp dvi(u)
linearly independent.
a1,...,an,
W1,...,Wn along
Since
which vanish at
at both p
p,
u - v1(u),...,u -' vn(u)
constructed as above, provide p.
Since the
Wi(1) = exp.(Xi)
Wi
are
can vanish at
yv
vanishes
This completes the proof.
Let
is not conjugate to
theorem (§6.1).
n Jacobi fields
clearly no non-trivial Jacobi field along
and exp v.
PROOF.
are
n is the dimension of the space of Jacobi fields along yV,
COROLLARY 18.2.
p
n independ-
exp*(X1),..., exp.(Xn)
independent, no non-trivial linear combination of the exp v.
Choose
v.
Xi
=
vanishing at
yv,
Then
99
p e M. q
Then for almost all
q E M,
along any geodesic.
This follows immediately from 18.1 together with Sard's
III.
100
§19.
CALCULUS OF VARIATIONS
Some Relations Between Topology and Curvature.
This section will describe the behavior of geodesics in a manifold
with
"negative curvature" or with "positive curvature."
LEMMA 19.1.
for
Suppose that < R(A,B)A,B > < 0
in the tangent space Then no two points of
every pair of vectors A,B TMp and for every p E M.
M are conjugate along any geodesic. PROOF.
let
J
Let
y
V;
be a geodesic with velocity vector field
be a Jacobi field along
and
Then
y.
D 2 J + R(V,J)V
=
0
dt2 so that
DJ J>
< R(V,J)V,J > > 0.
,
dt
Therefore 2
J J> _ < 7.7 , J> +
J
vanishes both at
DJ, J > also vanishes at throughout the interval
0
J
is identically zero.
REMARK.
> 0
and at
and
to,
to > 0,
then the function
and hence must vanish identically
This implies that =
dt(o)
0,
This completes the proof.
If A and B are orthogonal unit vectors at
quantity K R(A,B)A,B > B.
0
[o,t01. J(o)
A and
2
/ 0.
If
so that
CTE
is monotonically increasing, and strictly
Thus the function < CTE , J >
so if d
DJ
p
then the
is called the sectional curvature determined by
It is equal to the Gaussian curvature of the surface (u1,u2)
expp(u1A + u2B)
spanned by the geodesics through p with velocity vectors in the subspace
spanned by A and
p. 101.)
B.
(See for example, Laugwitz "Differential-Geometrie,"
TOPOLOGY AND CURVATURE
§19.
101
[Intuitively the curvature of a manifold can be described in terms of "optics" within the manifold as follows.
Suppose that we think of the
geodesics as being the paths of light rays.
Consider an observer at
looking in the direction of the unit vector U towards a point A small line segment at
q with length
sponding to the unit vector W e TMP,
p
q = exp(rU).
pointed in a direction corre-
L,
would appear to the observer as a
line segment of length
L(1 + - < R(U,W)U,W > + (terms involving higher powers of
r))
Thus if sectional curvatures are negative then any object appears shorter than it really is.
A small sphere of radius
an ellipsoid with principal radii where
W
K1)K2,...,Kn
e(1 +
-K1
at
q
would appear to be
+ ...), ..., e(1 +
r2 67Kn + ...)
denote the eigenvalues of the linear transformation
Any small object of volume
- R(U,W)U.
e
r2
would appear to have volume
v
2
v(1 + -(K1 + K2 +...+ Kn) + (higher terms)) to the "Ricci curvature"
K(U,U),
where
K1 +...+ Kn
is equal
as defined later in this section.]
Here are some familiar examples of complete manifolds with curvature
< 0: (1)
The Euclidean space with curvature
(2)
The paraboloid
(3)
The hyperboloid of rotation x2 + y2 - z2 ture
(4)
0.
z = x2 - y2, with curvature < 0. = 1,
with curva-
< 0.
The helicoid x cos z + y sin z = 0, with curvature
(REMARK.
< 0.
In all of these examples the curvature takes values arbi-
trarily close to o.
Of. N. V. Efimov, Impossibility of a complete surface
in 3-space whose Gaussian curvature has a negative upper bound, Soviet Math., Vol. 4 (1963), pp. 843-846.)
A famous example of a manifold with everywhere negative sectional curvature is the pseudo-sphere
z
=
-
1
- x2 - y2
+
with the Riemann metric induced from R3.
sech 1 Jx2 + y2, z > 0 Here the Gaussian curvature has
the constant value -1.
No geodesic on this surface has conjugate points although two geodesics may intersect in more than one point.
The pseudo-sphere gives a
III.
102
non-Euclidean < it
radians.
CALCULUS OF VARIATIONS
geometry, in which the sum of the angles of any triangle is This manifold is not complete.
In fact a theorem of Hilbert
states that no complete surface of constant negative curvature can be imbedded in R3.
(See Blaschke, "Differential Geometric I," 3rd edn., §96;
or Efimov, ibid.)
However, there do exist Riemannian manifolds of constant negative curvature which are complete.
(See for example Laugwitz, "Differential
and Riemannian geometry," §12.6.2.) Such a manifold can even be compact; for example, a surface of genus > 2. (Compare Hilbert and Cohn-Vossen, "Geometry and the imagination," p. 259.) THEOREM 19.2 (Cartan*).
Suppose that M is a simply connected, complete Riemannian manifold, and that the sectional curvature KR(A,B)A,B > is everywhere < 0.
M
Then any two points of desic.
Furthermore,
M
are joined by a unique geois diffeomorphic to the
Euclidean space Rn. PROOF:
Since there are no conjugate points, it follows from the
index theorem that every geodesic from p Theorem 17.3 asserts that the path space of a 0-dimensional
to
q
0(M;p,q)
has index
X = 0.
has the homotopy type
CW-complex, with one vertex for each geodesic.
The hypothesis that M is simply connected implies that is connected.
Thus
Q(M;p,q)
Since a connected 0-dimensional CW-complex must consist of
a single point, it follows that there is precisely one geodesic from p q. *
to
See E. Cartan, "Lecons sur la Geometrie des Espaces de Riemann," Paris,
1926 and 1951.
§19.
Therefore, the exponential map
is a global diffeomorphism.
expp
see that
- M is one-one and
is non-critical everywhere;
is locally a diffeomorphism.
expp
so that
TMp
expp:
expp
But it follows from 18.1 that
onto.
103
TOPOLOGY AND CURVATURE
Combining these two facts, we This completes the proof of
19.2.
is not simply connected; but is
More generally, suppose that M complete and has sectional curvature flat torus
S1 X S1,
negative curvature.)
< 0.
(For example M might be a
or a compact surface of genus
> 2
with constant
Then Theorem 19.2 applies to the, universal covering ti
space
M
of
For it is clear that M inherits a Riemannian metric
M.
from M which is geodesically complete, and has sectional curvature
paths from p
q
to
it follows that each homotopy class of
p,q E M,
Given two points
< 0.
contains precisely one geodesic.
The fact that M is contractible puts strong restrictions on the topology of
M.
For example:
COROLLARY 19.3. < 0
then
If M is complete with
the homotopy groups
ai(M) are zero for contains no element of finite order other than the identity.
i > 1;
and
PROOF:
Clearly
a1(M)
M is
=
j(M)
=
contractible the cohomology group
11k(M)
can be identified with the co-
homology group 202 of that
ai(M)
of the group
Hk(a1(M))
S. T. Hu "Homotopy Theory," a1(M)
for
0
a1(M).
(See for example pp. 200-
M
Hk(G)
of M we have =
Hk(M)
Now suppose
Academic Press, 1959.)
contains a non-trivial finite cyclic subgroup
suitable covering space
Since
i > 1.
=
0
a1(M) = G; for
G.
Then for a
hence
k > n
.
But the cohomology groups of a finite cyclic group are non-trivial in arbitrarily high dimensions.
This gives a contradiction; and completes the
proof. Now we will consider manifolds with "positive curvature."
Instead
of considering the sectional curvature, one can obtain sharper results in this case by considering the Ricci tensor (sometines called the "mean curvature tensor").
CALCULUS OF VARIATIONS
III.
io4
fold M
of a Riemannian mani-
p
The Ricci tensor at a point
DEFINITION.
is a bilinear pairing TMp
K:
K(U1,U2)
Let
defined as follows.
R
x TMp
be the trace of the linear transforma-
tion
W from TMp
to
R(U1,W)U2
(In classical terminology the tensor K is obtained
TMp.
from R by contraction.)
It follows easily from §9.3 that K is symmetric:
K(U1,U2) = K(U2,U1).
The Ricci tensor is related to sectional curvature as follows.
U1,U2,...,Un be an orthonormal basis for the tangent space
tures < R(Un,Ui)Un,Ui >
for
By definition K(Un,Un)
PROOF:
( K R(Un,Ui)Un,Uj > )
.
zero, we obtain a sum of
TMp.
is equal to the sum of the sectional curva-
K(Un,Un)
ASSERTION.
Let
i = 1,2,...,n-1.
is equal to the trace of the matrix
Since the n-th diagonal term of this matrix is n-1 sectional curvatures, as asserted.
THEOREM 19.4 (Myers*).
Suppose that the Ricci curvature
K satisfies K(U,U) > (n-1)/r2
for every unit vector U at every point of M; where r Then every geodesic on M of length > Ar contains conjugate points; and hence is not
is a positive constant. minimal. PROOF:
Let
y:
[0,11 - M be a geodesic of length
P1,...,Pn along
parallel vector fields
y
point, and hence are orthonormal everywhere along Pn
points along
y,
L.
Choose
which are orthonormal at one y.
We may assume that
so that
DP
V Let Wi(t)
_
(sin
=
=
at) Pi(t).
L Pn ,
and
i
=
0
Then
See S. B. Myers, Riemann manifolds with positive mean curvature, Math. Journal, Vol. 8 (1941), pp. 4o1-404.
Duke
TOPOLOGY AND CURVATURE
§19.
D2W. dt2
1
2E**i,Wi)
- $ < Wi,
_
105
R(V,Wi)V > dt
+
0
1
(sin at)2
S
(a2 -
L2 < R(Pn,PI)Pn,Pi > ) dt.
0
we obtain
i = 1,...,n-1
Summing for
1 7-
(sin
n t ) 2 ((n-1
2
) ic
-
L2 K(P n, P n)> dt
0
>
(n-1)/r2
and L > ar
E**(Wi,Wi) < 0
for some
i.
Now if Hence
< o. y
K(Pn,Pn)
then this expression is
This implies that the index of
is positive, and hence, by the Index Theorem, that
contains conju-
y
gate points.
It follows also that
dE(a(u))
u = 0.
In fact if
is a variation with variation vector field Wi
a
for
is not a minimal geodesic.
y
Hence
d2E(a(u)) 2 du
= 0,
CTU_
< o
for small values of
E(a(u)) < E(y)
then
u # 0.
This com-
pletes the proof.
If M is a sphere of radius
EXAMPLE.
curvature is equal to (n-1)/r2.
Hence
1/r2.
takes the constant value
It follows from 19.4 that every geodesic of length >
tains conjugate points:
COROLLARY 19.5.
If
M
compact, with diameter PROOF.
If
Then the length of
p,q e M y
ar con-
a best possible result.
is complete, and
for all unit vectors
(n-1) /r2 > o
< ar.
K(U,U)
then every sectional
r
K(U,U) >
then M
is
< ar.
let
must be
U,
y
< ar.
be a minimal geodesic from p
to
q.
Therefore, all points have distance
Since closed bounded sets in a complete manifold are compact, it
follows that M itself is compact.
M
of
a1(M)
is
This corollary applies also to the universal covering space M.
Since M is compact, it follows that the fundamental group
finite.
This assertion can be sharpened as follows.
106
CALCULUS OF VARIATIONS
III.
If M
THEOREM 19.6.
is a compact manifold, and if the
Ricci tensor K of M is everywhere positive definite, 11(M;p,q) then the path space has the homotopy type of a CW-complex having only finitely many cells in each dimension.
Since the space consisting of all unit vectors
PROOF.
is compact, it follows that the continuous function a minimum, which we can denote by
of length > ar
y E 0(M;p,q)
(n-1)/r2 > 0.
has index
i = 1,2,...,k
E**(Xi,Xi )
=
dimensional subspace of
(
i-1
i
ki
for
0
TOy
j;
from p
p
which vanishes
E**(Xi,Xi) < 0.
span a k -
is negative definite. q
are not conjugate along any
Then according to § 16.3 there are only finitely many geodesics to
q
of length < ksr.
desics with index
REMARK.
< k.
Hence there are only finitely many geo-
Together with §17.3, this completes the proof.
I do not know whether or not this theorem remains true if
M is allowed to be complete, but non-compact.
The present proof certainly
breaks down since, on a manifold such as the paraboloid curvature
y
so that X1,...,Xk
and
Then a
In fact for each
>v > k.
on which E**
Now suppose that the points geodesic.
of length > kirr.
y
and such that
),
,
takes on
Then every geodesic
one can construct a vector field Xi along
outside of the interval Clearly
has index
y
K(U,U) > 0
X > 1.
More generally consider a geodesic similar argument shows that
on M
U
K(U,U)
z = x2 + y2,
the
will not be bounded away from zero.
It would be interesting to know which manifolds can carry a metric so that all sectional curvatures are positive.
provided by the product
Sm X Sk
An instructive example is
of two spheres; with m,k > 2.
manifold the Ricci tensor is everywhere positive definite.
For this
However, the
sectional curvatures in certain directions (corresponding to flat tori S1 X S1 C Sm X Sk)
are zero'.
It is not known whether or not
be remetrized so that all sectional curvatures are positive. partial result is known:
can
The following
If such a new metric exists, then it can not be
invariant under the involution from a theorem of Synge.
Sm x Sk
(x,y) -+ (-x,-y)
of
Sm x Sk.
This follows
(See J. L. Synge, On the connectivity of spaces
§19.
of positive curvature,
TOPOLOGY AND CURVATURE
107
Quarterly Journal of Mathematics (Oxford), Vol. 7
(1936), pp. 316-320.
For other theorems relating topology and curvature, the following sources are useful.
K. Yano and S. Bochner, "Curvature and Betti Numbers,"
Annals
Studies, No 32, Princeton, 1953. S. S. Chern, manifold,
On curvature and characteristic classes of a Riemann
Abh. Math. Sem., Hamburg, Vol. 20 (1955), pp. 117-126.
M. Berger, Sur certaines varietes Riemanniennes a courbure positive, Comptes Rendus Acad. Sci., Paris, Vol. 247 (1958), pp. 1165-1168. S. I. Goldberg, "Curvature and Homology," Academic Press, 1962.
109
PART IV.
APPLICATIONS TO LIE GROUPS AND SYMMETRIC SPACES
§20.
Symmetric Spaces.
A symmetric space is a connected Riemannian manifold M
for each p e M
there is an isometry
and reverses geodesics through then
LEMMA 20.1 p = y(o)
M which leaves
M
(assuming Iqlp
PROOF:
Let
y
y(t)
preserves parallel vector fields along
y'(t)
y(t + c).
=
IgIp(y(t))
=
Then
is a geodesic and
y'
Iq(y(-t))
y
Iq(y'(-t - c)) =
=
y(t + 2c).
=
If the vector field V is parallel along
=
y(0) = p
q = y(c).
Therefore
parallel (since Ip*V(t)
fixed
p
is a geodesic and
y
be a geodesic in M, and let Then IgIp(y(t)) = y(t + 2c) and y(t + 2c) are defined). More-
Let and
over,
y'(t + c)
:
= y(-t).
Ip(y(t))
y'(o) = q.
I
p i.e., if
p,
such that,
Ip
-V(-t).
is an isometry) and Therefore
COROLLARY 20.2. Since
M
y
then
Ip*V(o) = -V(o);
Iq* Ip*(V(t))
=
Ip*(V)
is
therefore
V(t + 2c).
is complete.
20.1 shows that geodesics can be indefinitely extended.
COROLLARY 20.3.
1P
is unique.
Since any point is joined to COROLLARY 20.4.
fields along field along
y
p by a geodesic.
U,V and W are parallel vector then R(U,V)W is also a parallel If
y.
PROOF. If X denotes a fourth parallel vector field along y, note that the quantity < R(U,V)W,X > is constant along y. In fact,
110
APPLICATIONS
IV.
q = y(c),
given p = y(o), p
carries
to
consider the isometry
Then
q.
= < R(T*UU,T*VV)T. WW,T. Xp>
by 20.1.
is an isometry, this quantity is equal to
T
Since
< R(UJ,VV)WW,Xp>
.
Thus
It clearly follows that
vector field X.
T = Iy(c/2)Ip which
is constant for every parallel
R(U,V)W
is parallel.
Manifolds with the property of 20.4 are called locally symmetric. (A classical theorem, due to Cartan states that a complete, simply connected, locally symmetric manifold is actually symmetric.)
In any locally symmetric manifold the Jacobi differential equations
y: R - M be a geodesic in a local-
Let
have simple explicit solutions.
be the velocity vector at
Let V = mo(o)
ly symmetric manifold.
p = y(o).
Define a linear transformation KV:
by
KV(W)
=
R(V,W)V.
Let
TMp -+ TM__ TM
el,...,en
denote the eigenvalues of
Kv.
The conjugate points to p along y are the points y(ak/e) where k is any non-zero integer, and ei is any positive eigenvalue of KV. The multiplicity of y(t) as a conjugate point is THEOREM 20.5.
equal to the number of tiple of PROOF:
ei
such that
is a mul-
t
n/ e .
First observe that
KV(W),W'>
KV is self-adjoint:
<W,Kv(W')
=
This follows immediately from the symmetry relation
KR(V',W')V,W
R(V,W)V',W' > =
Therefore we may choose an orthonormal basis Kv.(Ui)
where
e1, ... ,en
along
y
KV
=
are the eigenvalues.
by parallel translation.
U1, ...,Un
eiUi
Mp
so that
,
Extend the
Then since
for
M
Ui
to vector fields
is locally symmetric,
should not be confused with the Ricci tensor of §19.
SYMMETRIC SPACES
520.
111
the condition
R(V,Ui)V remains true everywhere along
y.
eiUi
=
Any vector field W along
y
may be
expressed uniquely as W(t)
w1(t)U1(t) +...+ wn(t)Un(t)
=
.
2
Then the Jacobi equation
+ KV(W)
=
0
takes the form
Tt d2w dt2 Ui +
L Since the
Ui
the system of
eiwiUi
=
0.
are everywhere linearly independent this is equivalent to
n equations d2wi
et
+ eiw1
We are interested in solutions that vanish at wi(t)
=
Then the zeros of If
ci sin (f t),
wi(t)
t = 0.
If
for some constant
are at the multiples of
then wi(t) = cit
ei = o
0
and if
ci.
then
wi(t) = ci sinh (41
for some constant
vanishes only at
This completes the proof of 20.5.
t = 0.
then
t = n/qre-i
ei < 0 ci.
ei > 0
Thus if
ei < 0,
wi(t)
112
APPLICATIONS
IV.
Lie Groups as Symmetric Spaces.
§21.
In this section we consider a Lie group G
with a Riemannian metric
which is invariant both under left translations
G -* G,
LT: and right translation, certainly exists. as follows:
RT(a)
aT.
=
If
=
G
Ta is commutative such a metric
is compact then such a metric can be constructed
G
If
LT(a)
Let <> be any Riemannian metric on
the Haar measure on
Then
G.
new inner product <<, >>
<< V,W >>
=
G,
and Let
denote
.t
Define a
is right and left invariant.
.t
G by
on
< La*RT*(V), La*RT*(W) > du(a) du(T)
S GxG
Then
<<,>>
is left and right invariant.
LEMMA 21.1
If G is a Lie group with a left and right invariant metric, then G is a symmetric space. The reflection IT in any point T E G is given by the formula IT(a) = Ta-1 T.
PROOF: Ie:
By hypothesis
and RT
LT
Ie*:
TGe
TGe
Ie
shows that
Ie*:
TGa --) TGa-1
reverses the tangent space at
Finally, defining shows that each
IT
=
reverses the tangent space of
an isometry on this tangent space.
Princeton, 1946.)
so is certainly
R1IeLa_1
is an isometry for any
a E G.
Since
it reverses geodesics through
e,
Ta-1T,
=
the identity
Ie
e.
I. = RTIeRT1
is an isonetry which reverses geodesics through of
G
is a C°°
e.
T.
homomorphism of R into
It is well known that a 1-parameter subgroup of
its tangent vector at
e;
Now the identity
=
IT(a)
A 1-parameter subgroup G.
Define a map
G by
G
Ie(a)
Then
are isometries.
G
is determined by
(Compare Chevalley, "Theory of Lie Groups,"
LIE GROUPS
§21.
113
LEMMA 21.2. The geodesics y in G with y(O) = e are precisely the one-parameter subgroups of G.
the map
Iy(t)Ie
takes
y(t)y(u)y(t)
=
for any integer
n.
so
for some
y(u)
into
gent vector of Hence
and
at
e.
y(nt) = y(t)n t" = n"t
and
y(t' + t") = y(t)n
then
n"
+n"
=
is a hor..omorphism.
1-parameter subgroup.
such that the tangent vector of
e y
n'
G be a
y: R
geodesic through
y
Iv(t)Ie(a) = y(t)a y(t)
t' = nit
is rational so that
t'/t"
If
Now
y(u + 2t).
By Lemma 20.1
= e.
y(o)
By induction it follows that
y(u + 2t).
By continuity
Now let
We have just seen that
Let at
y'
y'
e
be the is the tan-
is a 1-parameter sub-
7
This completes the proof.
y' = y.
A vector field X and only if
be a geodesic with
and some integers
t
y(t')y(t").
group.
y: R - G
Let
PROOF:
on a Lie group
(La)*(Xb) = Xa,b
are left invariant then
[X,Y]
for every
is called left invariant
G a
and
in
b
If X and Y
G.
The Lie algebra
is also.
if
g
of
is the
G
vector space of all left invariant vector fields, made into an algebra by the bracket g
[
1.
is actually a Lie algebra because the Jacobi identity [[x,Y],Z] + [[Y,Z],x] + [[Z,X],Y]
=
0
holds for all (not necessarily left invariant) vector fields X,Y and
THEOREM 21.3.
Let
G be a Lie group with a left and
right invariant Riemannian metric.
If
X,Y,Z
are left invariant vector fields on G
then:
<[X,YI,Z>
R(X,Y)Z = - [[X,Y],ZI
c)
< R(X,Y)Z,W >
PROOF:
and W
<X,[Y,ZI>
a) b)
=
Z.
< [X,YI,[Z,W1 >
As in §8 we will use the notation X F Y for the covariant
derivative of Y in the direction X.
For any left invariant X
the iden-
tity
x FX = 0 is satisfied, since the integral curves of X
are left translates of
parameter subgroups, and therefore are geodesics.
1-
IV.
114
APPLICATIONS
Therefore
(X + Y) F (X + Y)
=
(X F X) + (X F Y)
+
(Y F X) + (Y F Y)
is zero; hence
X F Y+ Y F X
=
0.
X FY -Y FX
=
[X,Y]
On the other hand
Adding these two equations we obtain:
by §8.5. d)
2X F Y
[X,Y]
=
Now recall the identity
Y (X,Z> = (See §8.4.) constant.
+
<X,Y F Z > .
The left side of this equation is zero, since <X,Z > is in this equation we obtain
Substituting formula (d) 0
=
< [Y,X]
,
Z> +
Finally, using the skew commutativity of
<X, [Y, Z] > [Y,X],
.
we obtain the required
formula
<[X,Y],Z>
(a) By definition,
R(X,Y)Z
- X F (Y F Z) Substituting formula (d), -
<X,[Y,Z]>
_
is equal to
Y F (X F Z)
+
[X,Y] F Z.
+
this becomes
-[X,[Y,Z]] + -[Y,[X,Z]] + 2[[X,Y],Z]
Using the Jacobi identity, this yields the required formula
R(X,Y)Z
(b)
The formula
=
7[[X,Y],Z]
(c) follows from (a) and (b)
.
.
It follows that the tri-linear function X,Y,Z -+ is skewsymmetric in all three variables. Thus one obtains a left invariant differential 3-form on G, representing an element of the de Rham cohomology group H3(G). In this way Cartan was able to prove that H3(G) # 0 if G is a non-abelian compact connected Lie group. (See E. Cartan, "La Topologie des
Espaces Representatives des Groupes de Lie," Paris, Hermann, 1936.)
LIE GROUPS
§21.
115
The sectional curvature < R(X,Y)X,Y K [X,Y],[X,Y] > is always > 0. Equality holds if and only if [X,Y] = 0. COROLLARY 21.4.
Recall that the center
the set of X E
such that
g
of a Lie algebra
c
[X,Y] = 0
is defined to be
g
for all Y E
g.
If G has a left and right invariant COROLLARY 21.5. metric, and if the Lie algebra g has trivial center, then G is compact, with finite fundamental group.
Let
This follows from Meyer's theorem (§19).
PROOF:
unit vector in
X1
and extend to a orthonormal basis X1,...,Xn.
g
be any The Ricci
curvature
K(X1,X1)
_
2. (R(X1,Xi)X1,Xi i=1
must be strictly positive, since K(X1,X1)
[X1,Xi)
/
for some
0
Furthermore
i.
is bounded away from zero, since the unit sphere in
Therefore, by Corollary 19.5, the manifold
G
g
is compact.
is compact.
This result can be sharpened slightly as follows.
A simply connected Lie group G with left
COROLLARY 21.6.
and right invariant metric splits as a Cartesian product
G' is compact and Rk denotes the additive Furthermore, the Lie
G' x Rk where
Lie group of some Euclidean space. algebra of
has trivial -center.
G'
Conversely it is clear that any such product
G' x Rk possesses a
left and right invariant metric. Let
PROOF.
g'
_
c
be the center of the Lie algebra
(X E g : <X, C> =
be the orthogonal complement of
if X,Y E g'
and
C E
0
c.
for all
Then
g'
[X,Y] E g'.
Lie algebras.
is compact by 21.5 and
For
is a Lie sub-algebra.
then
c
It follows that
Hence G
and let
CE c)
[X,Y],C > = <X, 1Y,C1 > = hence
g
g
o;
splits as a direct sum
splits as a Cartesian product G"
G' x G";
g' ®
C
where
of G'
is simply connected and abelian, hence isomorphic
116
APPLICATIONS
TV.
to some Rk. (See Chevalley, "Theory of Lie Groups.")
This completes the
proof. THEOREM 21.7 (Bott). Let G be a compact, simply conThen the loop space 0(G) has the nected Lie group.
homotopy type of a CW-complex with no odd dimensional cells, and with only finitely many X-cells for each even value of
X.
Thus the x-th homology groups of free abelian of finite rank for REMARK 1.
an example, if
G
.
odd, and is
a. even.
This CW-complex will always be infinite dimensional. is the group
that the homology group REMARK 2.
is zero for
01(G)
As
of unit quaternions, then we have seen
S3
is infinite cyclic fqr all even values of i.
Hi01(S3)
This theorem remains true even for a non-compact group.
In fact any connected Lie group contains a compact subgroup as deformation retract.
(See K. Iwasawa, On some types of topological groups, Annals of
Mathematics 50 (1949), Theorem 6.) PROOF of 21.7.
Choose two points
conjugate along any geodesic.
p
and
Q(G;p,q)
By Theorem 17.3,
type of a CW-complex with one cell of dimension
p
to
each
of index
q
X.
in G which are not
q
X
for each geodesic from
By §19.4 there are only finitely many
Thus it only remains to prove that the index
X.
has the homotopy
X
%-cells for
of a geodesic is
always even.
Consider a geodesic
y
V
=
According to §20.5
p with velocity vector
starting at mo(o)
e TGp = g p
the conjugate points of
.
on
y
eigenvalues of the linear transformation KV :
TGp - TGp
,
defined by KV(W)
=
R(V,W)V
=
7[[V,w],V]
Defining the adjoint homomorphism
Ad V:
S1 9
are determined by the
117
LIE GROUPS
§21.
by Ad V(W) _ [V,W]
we have
KV = - - (Ad V)
o
is skew-symmetric; that is
The linear transformation Ad V
Ad V(W),W' >
_
(Ad V)
-
< W,Ad V(W') >
This follows immediately from the identity 21.3a. an orthonormal basis for
(4
.
Therefore we can choose
so that the matrix of
takes the form
Ad V
a2
0
-a2
It follows that the composite linear transformation
(Ad V)o(Ad V)
has
matrix
2
2
a2
Therefore the non-zero eigenvalues of
KV
-
--(Ad V)2
are positive, and
occur in pairs.
It follows from 20.5 that the conjugate points of
p
y
also
In other words every conjugate point has even multiplicity.
occur in pairs.
Together with the Index Theorem, this implies that the index geodesic from
along
p
to
q
is even.
This completes the proof.
?
of any
118
IV.
APPLICATIONS
Whole Manifolds of Minimal Geodesics.
§22.
So far we have used a path space
o(M;p,q) based on two points
p,q c M which are in "general position."
However, Bott has pointed out
that very useful results can be obtained by considering pairs special position. let
As an example let
p,q be antipodal points.
desics from p
to
q.
M be the unit sphere
p,q Sn+1,
in some
and
Then there are infinitely many minimal geo-
In fact the space
saa
of minimal geodesics forms
a smooth manifold of dimension n which can be identified with the equator Sn C Sn+1.
We will see that this space of minimal geodesics provides a
fairly good approximation to the entire loop space
0(Sn+1)
Let M be a complete Riemannian manifold, and let points with distance
p,q c M be two
p(p,q) =-,r-d.
THEOREM 22.1.
If the space Std of minimal geodesics from p to q is a topological manifold, and if every non-minimal geodesic from p to q has index > Xo, then the relative homotopy group ai(o,Std) is zero for 0 < i < X0.
It follows that the inclusion homomorphism
ai(rid is an isomorphism for group
ai(Sc)
i < Xo - 2.
is isomorphic to
- ai(a) But it is well known that the homotopy
ai+1(M)
for all values of
i.
(Compare
S. T. Hu, "Homotopy Theory," Academic Press, 1959, p. 111; together with §17.1.)
119
MANIFOLDS OF MINIMAL GEODESICS
§22.
Thus we obtain: COROLLARY 22.2.
ei(od) With the same hypotheses, for 0 < i < Xo - 2.
is
a1+1(M)
isomorphic to
Let us apply this corollary to the case of two antipodal points on the (n+1)-sphere.
Evidently the hypotheses are satisfied with
1`0 = 2n. Sn+1;
For any non-minimal geodesic must wind one and a half times around and contain two conjugate points, each of multiplicity
n, in its interior.
This proves the following. (The Freudenthal suspension theorem.) COROLLARY 22.3. is isomorphic to The homotopy group ,ri(Sn)
"i+1(Sn+1)
for
i < 2n-2.
also implies that the homology groups of the loop
Theorem 22.1 space
0
are isomorphic to those of
fact follows from
in dimensions
S;d
< %0 - 2.
This
22.1 together with the relative Hurewicz theorem. Compare also
for example Hu, p. 306.
(See
J. H. C. Whitehead, Combinatorial
homotopy I, Theorem 2.)
The rest of §22 will be devoted to the proof of Theorem 22.1.
The
proof will be based on the following lemma, which asserts that the condition
"all critical points have index > %o"
remains true when a function is
jiggled slightly.
Let K be a compact subset of the Euclidean space Rn; a neighborhood of
let U be
and let
K;
U
f:
R
be a smooth function such that all critical points of
f
in K have index
)o.
LEMMA 22.4.
If
is "close" to _
3xi
la
of
I
'3i
<
R is any smooth function which
U
g:
f,
in the sense that
E
a2g xT
uniformly throughout
_
K,
for some sufficiently small constant
then all critical points of (Note that the application, points.)
f
a2f
8x c)x
i
s,
g in K have index > Xo.
is allowed to have degenerate critical points.
In
g will be a nearby function without degenerate critical
120
IV.
APPLICATIONS
The first derivatives of
PROOF of 22.4.
are roughly described
g
by the single real valued function
kg(x) on
U;
=
>o
mo1 f
which vanishes precisely at the critical points of
derivatives of
can be roughly described by n
g
continuous functions
U - R
eg,..., eg: as follows.
The second
g.
Let e11(x) < e2(x) <... < eg(x) 9 9 2
denote the
x
of
g
n eigenvalues of the matrix
has index > ?
(
)
x
.
if and only if the number
The continuity of the functions
eg
Thus a critical point is negative.
eg(x)
follows from the fact that the
X-th eigenvalue of a symmetric matrix depends continuously on the matrix*. This can be proved, for example, using the fact that the roots of a complex
n vary continuously with the coefficient of the poly-
polynomial of degree
(Rouche's theorem.)
nomial.
Let
denote the larger of the two numbers
mg(x)
Similarly let
mf(x)
and -ef (x).
The hypothesis that all critical points of
index > X0
kg(x)
and -e g W.
denote the larger of the corresponding numbers
implies that
-ef (x) > 0
whenever
f
kf(x) = 0.
kf(x)
in K have In other words
for all x E K.
mf(x) > 0 Let
s > 0
denote the minimum of
is so close to
f
(*)
kg(x) - kf(x)
for all x E K. critical point of
mf
on
K.
Now suppose that
g
that
Then mg(x) g
I
< e,
leg (x)
- of (x)I
will be positive for
< e
x E K;
hence every
in K will have index > X0.
This statement can be sharpened as follows. matrices.
Consider two nxn symmetric If corresponding entries of the two matrices differ by at most
e, then corresponding eigenvalues differ by at most ne. This can be proved using Courant's minimax definition of the X-th eigenvalue. (See §1 of Courant, Uber die Abhangigkeit der Schwingungszahlen einer Membran...,
Nachrichten, Kbniglichen Gesellschaft der Wissenschaften zu Gottingen, Phys. Klasse 1919, pp. 255-264.)
Math.
MANIFOLDS OF MINIMAL GEODESICS
§22.
121
To complete the proof of 22.4, it is only necessary to show that the inequalities (*) will be satisfied providing that
g`zi - 4i l for sufficiently small
a.
2g <E
and
I
2
xcc1 x
6x
x1c
<
E
This follows by a uniform continuity argument
which will be left to the reader (or by the footnote above ). for real valued
We will next prove an analogue of Theorem 22.1 functions on a manifold. Let
M -+ R be a smooth real valued function with minimum
f:
such that each Me = f-1[o,c]
o,
is compact.
If the set M0 of minimal points is a manifold, and if every critical point in M - M0 has index > Xo,
LEMMA 22.5.
ar(M,M°) = 0
then
PROOF:
U C M.
0 < r < X0.
for
First observe that
is a retract of some neighborhood
Mo
In fact Hanner has proved that any manifold M0
neighborhood retract.
is an absolute
(See Theorem 3.3 of 0. Hanner, Some theorems on
absolute neighborhood retracts, Arkiv for Matematik, Vol.
1
(1950), pp.
Replacing U by a smaller neighborhood if necessary, we may
389-408.)
assume that each point of U is joined to the corresponding point of by a unique minimal geodesic. Let
Thus U can be deformed into Mo
denote the unit cube of dimension
Ir
r < Xp,
M0
within
M.
and let
h: (Ir,Ir) _ (M,MO)
h is homotopic to a map
We must show that
be any map.
h'
with
h'(Ir) C Mo. Let
mum of
c
be the maximum of
on the set M - U.
f
since each subset
Mc - U
f
on
h(Ir).
(The function
f
Let
35 > 0
be the mini-
has a minimum on M - U
is compact.)
Now choose a smooth function g:
which approximates is possible by §6.8.
f
Mc+2s
R
closely, but has no degenerate critical points.
To be more precise the approximation should be so
close that: (1)
Jf(x) - g(x)l < s
for all x E
Mc+2s.
and
This
122
APPLICATIONS
IV.
pact set
at each critical point which lies in the com-
g
The index of
(2)
f-1[8,c+28]
> x0.
is
g which approximates
It follows from Lemma 22.4 that any
f
sufficiently closely, the first and second derivatives also being approximated, will satisfy (2).
In fact the compact set
covered by finitely many compact set
f-'[8,0+281
each of which lies in a coordi-
Ki,
Lemma 22.4 can then be applied to each Ki.
nate neighborhood.
The proof of 22.5 now proceeds as follows.
The function
Ience the manifold
points are non-degenerate, with index > ).0.
g
has the homotopy type of g 1(-oo,2s]
(--,c+bl
g is
and all critical
smooth on the compact region g 1[25,c+s] C f 1[s,c+2sl,
-1
can be
with cells of dimension
attached.
> X0
Now consider the map h:
Since
r < X0
Mc,MO
Ir jr
C g 1(-oo,c+s],MO
h is homotopic within
it follows that
g1(-oo,c+s],M0
to
a map Ir,Ir -y g-1 (--22812M 0
h':
But this last pair is contained in MO h":
within
M.
It follows that
and U
(U,MO);
.
can be deformed into
is homotopic within
h'
to a map
(M,MO)
This completes the proof of 22.5.
Ir,Ir - MO,M0.
The original theorem, 22.1,
now can be proved as follows.
Clearly
it is sufficient to prove that
si(Int 0c,fd) for arbitrarily large values of a smooth manifold sad
c.
=
o
As in §16 the space
Int ac(to,t1,...,tk)
Int 0c
as deformation retract.
contains
The space
of minimal geodesics is contained in this smooth manifold.
The energy function Int sac(t0,t1,...,tk),
difficulty is that
when restricted to
ranges over the interval
[o,oo).
F.
be any diffeomorphism.
--' R,
almost satisfies the hypothesis of
E(m)
the required interval
E: a
To correct this, let
[d,c) - [o,o)
22.5.
d < E < c,
The only instead of
§22.
MANIFOLDS OF MINIMAL GEODESICS
Then F ° E:
Int oc(tc,t1,...,tk)
satisfies the hypothesis of 22.5.
R
Hence
si(Int cc(tp,...,tk),12d) = ni(Int fc,0d) is zero for
i < Xc.
This completes the proof.
123
124
APPLICATIONS
IV.
§23.
The Bott Periodicity Theorem for the Unitary Group.
First a review of well known facts concerning the unitary group. of complex numbers, with the usual Her-
Let C n be the space of n-tuples mitian inner product.
of all linear transformations product.
is defined to be the group
The unitary group U(n) S.
Cn -Cn which preserve this inner is the
Equivalently, using the matrix representation, U(n)
group of all n x n
complex matrices
denotes the conjugate transpose of
S
such that
S S*
=
I;
where
S*
S.
For any n x n complex matrix A the exponential of A is defined by the convergent power series expansion
= I+A+ 1 A2+1A3 + ...
exp A
The following properties are easily verified: (1)
exp (A*)
(2)
If A and B
=
exp (TAT-1)
(exp A)*;
T(exp A)T-1.
=
commute then In particular:
_ (exp A)(exp B).
exp (A + B) (3)
(exp A)(exp -A)
(4)
The function exp maps a neighborhood of
= I 0
in the space of
n x n matrices diffeomorphically onto a neighborhood of
If A is skew-Hermitian (that is if A + A* lows from (1) and (3) that exp A
is unitary.
=
0),
Conversely if
then it fol-
exp A
unitary, and A belongs to a sufficiently small neighborhood of it follows from (1), (3), and (4) that A + A*
0.
=
I.
is
then
0,
From these facts one
easily proves that: (5)
(6)
U(n)
is a smooth submanifold
the tangent space
TU(n)I
of the space of
n x n matrices;
can be identified with the space of
n x n skew-Hermitian matrices.
Therefore the Lie algebra
g
the space of skew-Hermitian matrices.
of U(n)
can also be identified with
For any tangent vector at
uniquely to a left invariant vector field on U(n).
I
extends
Computation shows that
the bracket product of left invariant vector fields corresponds to the product
[A,B] = AB - BA
of matrices.
THE BOTT PERIODICITY THEOREM
§23.
Since U(n) Riemannian metric.
125
is compact, it possesses a left and right invariant Note that the function
TU(n) I - U(n)
exp:
defined by exponentiation of matrices coincides with the function exp defined (as in §10) by following geodesics on the resulting Riemannian manifold.
In fact for each skew-Hermitian matrix A
the correspondence
t - exp(t A) U(n) (by Assertion (2) above);
defines a 1-parameter subgroup of
and
hence defines a geodesic.
A specific Riemannian metric on U(n) Given matrices
can be defined as follows.
let denote the real part of the complex
A,B e g
number trace (AB*)
Aij$ij
=
.
1,3
Clearly this inner product is positive definite on This inner product on
Riemannian metric on U(n).
g
.
determines a unique left invariant
g
To verify that the resulting metric is also
right invariant, we must check that it is invariant under the adjoint
action of U(n)
on
g.
DEFINITION of the adjoint action.
Each
S E U(n)
determines an
inner automorphism S-1
X-SX of the group U(n).
U(n).
Ad(S).
(LSRS-1 )X
The induced linear mapping
TU(n)I --. TU(n)I
(LSRS-1)*: is called
=
Thus
Ad(s)
is an automorphism of the Lie algebra of
Using Assertion (1) above we obtain the explicit formula
Ad(S)A
=
SAS 1
,
for A E g, S E U(n). The inner product Ad(S).
In fact if
Al = Ad(S)A,
A1B1*
=
is invariant under each such automorphism B1
= Ad(S)B
SAS-1(SBS 1)*
then the identity SAB*S-1
=
126
1V.
APPLICATIONS
implies that trace (A1B1*)
=
trace (SAB*S 1)
trace (AB*)
=
;
and hence that
KA1,B1 >
< A,B
=
It follows that the corresponding left invariant metric on U(n)
is also
right invariant.
Given A c g T E U(n)
so that
we know by ordinary matrix theory
TAT 1
that there exists
is in diagonal form
ia2 TAT-1
where the
=
Also, given any
ails are real.
there is
S E U(n),
a
T E U(n)
such that
TST-1
=
ian e
where again the
ails are real.
Thus we see directly that
exp:
g
-iJ(n)
is onto.
One may treat the special unitary group SU(n) SU(n)
is defined as the subgroup of
terminant
1.
in the same way.
consisting of matrices of de-
U(n)
If exp is regarded as the ordinary exponential map of
matrices, it is easy to show, using the diagonal form, that det (exp A)
Using this equation, one may show that
the set of all matrices A
etrace A
=
g'
the Lie algebra of SU(n)
,
such that A + A*
=
and
0
trace A = 0.
In order to apply Morse theory to the topology of U(n) we begin by considering the set of all geodesics in U(n)
In other words, we look for all A E TU(n)1 Suppose T E U(n)
A
is such a matrix;
be such that exp TAT-1
TAT-1
=
=
g
is
from
such that
and SU(n), I
to
-I.
exp A = -I.
if it is not already in diagonal form, let is in diagonal form.
T(exp
A)T-1
=
T(-I)T-1
Then =
-I
THE BOTT PERIODICITY THEOREM
§23.
127
so that we may as well assume that A is already in diagonal form
/ ia1
/ eis
1
ian e
if and only if A has the form
exp A = -I
so that
/ k1 in k2in.
I
for some odd integers
k1,...,kn.
Since the length of the geodesic is it
ki
JAI k'1
=
tJ
r AA*,
equals
and in that case, the length is
+ 1,
to Cn
as a linear map of Cn
v E Cn such that
Av = inv;
Av = -inv.
Eigen(-in),
to
t =
is
I
Now, regarding
observe that A is completely
the space of all v E Cn
splits as the orthogonal sum Eigen(in)
is then completely determined by Eigen(in),
which is an arbitrary subspace of Cn. from
n f-n.
the vector space consisting of all
and Eigen(-in),
Since Cn
the matrix A
desics in U(n)
t = 0
the length of the geodesic determined by A
determined by specifying Eigen(in),
such that
from
Thus A determines a minimal geodesic if and only if each
+...+ kn.
such an A
t -+ exp to
to
-I
Thus the space of all minimal geo-
may be identified with the space of all
sub-vector -spaces of C Unfortunately, this space is rather inconvenient to use since it has components of varying dimensions.
replacing U(n)
by SU(n)
above considerations remain valid.
a1 +...+ a2m
=
0
This difficulty may be removed by
and setting n = 2m.
with ai = + n
In this case, all the
But the additional condition that restricts Eigen(in)
to being an arbi-
trary m dimensional sub-vector-space of C. 2m This proves the following:
1
128
IV.
APPLICATIONS
The space of minimal geodesics from I to -I in the special unitary group SU(2m) is homeomorphic to the complex Grassmann manifold Gm(C2m), consisting of all m dimensional vector subspaces of LEMMA 23.1.
C2m.
We will prove the following result at the end of this section. Every non-minimal geodesic from has index > 2m+2.
LEMMA 23.2.
in SU(2m)
to
I
-I
Combining these two lemmas with §22 we obtain:
THEOREM 23.3 (Bott). The inclusion map Gm(CM) _ induces isomorphisms of homotopy groups ci(SU(2m); 1,-I) in dimensions < 2m. Hence tti Gm(CSm)
- tti+1SU(2m)
for i<2m. On the other hand using standard methods of homotopy theory one obtains somewhat different isomorphisms.
LEMMA 23.4.
The group for
tti-t U(m)
U(m) = I(
for
i < 2m.
i-1
i < 2m;
ttiGm(C2m)
is isomorphic to
Furthermore,
U(m+1) = I(
i-1
U(m+2) = ..
and
ttj U(m) = ttj SU(m)
for j X 1. PROOF.
First note that for each m there exists a fibration
U(m) - U(m+1)-wS2m+1 From the homotopy exact sequence
-F tti g 2m+1
xi-1 U(m) -if,-, U(m+1) -1(i-1
...
S 2m+1
of this fibration we see that tti-1 U(m) -
tti_1 U(m+t)
for
i < 2m.
(Compare Steenrod, "The Topology of Fibre Bundles," Princeton, 1951, p. 35 and p. 9o.)
It follows that the inclusion homomorphisms
tti-1 U(m) are all isomorphisms for
-+
5i_1 U(m+t) i < 2m.
-+
tti_1 U(m+2)
-- ...
These mutually isomorphic groups are
called the
129
THE BOTT PERIODICITY THEOREM
§23.
(i-1)-st stable homotopy group of the unitary group.
be denoted briefly by
They will
ai-1 U.
The same exact sequence shows that, for
the homomorphism
i = 2m+1,
n2m U is onto.
n2M U(m) - a2m U(m+1)
The complex Stiefel manifold is defined to be the coset space U(2m)/U(m).
From the exact sequence of the fibration
-'U(2m) -U(2m)/U(m)
U(m) we see that
ni(U(2m)/ U(m))
=
for
0
i < 2m.
The complex Grassmann manifold
the coset space U(2m)/ U(m) x U(m).
Gm(C2m)
can be identified with
(Compare Steenrod §7.)
From the exact
sequence of the fibration
U(m)
--F U(2m) / U(m)
.
Gm(C2m)
we see now that niGm(C2m) for
ai_l U(m)
i < 2m.
Finally, from the exact sequence of the fibration SU(m)
-+ U(m)
-+ S1
we see that
= ai U(m)
ni SU(m)
for
j / 1.
This
completes the proof of Lemma 23.4.
Combining Lemma 23.4 with Theorem 23.3 we see that
= AiGm(C2m)
ni-1 U = ai-1 U(m) for
1
< i < 2m.
= ni+1 SU(2m)
= ni+1 U
Thus we obtain:
PERIODICITY THEOREM.
ni+1 U for
ni-1 U
i > 1.
To evaluate these groups it is now sufficient to observe that U(1) is a circle;
so that
n0 U n1 U
As a check, since SU(2)
=
no U(1)
=
=
n1 U(1)
= Z
0
(infinite cyclic).
is a 3-sphere, we have:
n2 U
=
n3 U
=
n2 SU(2) n3 SU(2)
=
0
=Z
Thus we have proved the following result.
130
APPLICATIONS
IV.
THEOREM 23.5 (Bott). The stable homotopy groups rti U of the unitary groups are periodic with period 2. In fact the groups n0 U = a2 U = n4 U
are zero, and the groups
3U =55U =...
n1U are infinite cyclic.
The rest of §23 will be concerned with the proof of Lemma 23.2. must compute the index of any non-minimal geodesic from
SU(n),
where
n
is even.
matrix A e g'
n x n
to
-I
on
Recall that the Lie algebra T(SU(n))I
=
g'
consists of all
I
We
skew-Hermitian matrices with trace zero.
corresponds to a geodesic from
I
to
A given
if and only if
-I
the eigenvalues of A have the form isk1,...,iakn where
k1,...,kn are
odd integers with sum zero.
We must find the conjugate points to
t
-h exp (tA)
I
along the geodesic
.
According to Theorem 20.5 these will be determined by the positive eigenvalues of the linear transformation g1
K : A
-. g'
where KA(W)
R(A,W)A
=
=
-- [[A,W),A]
.
(Compare §21.7.)
We may assume that A is the diagonal matrix inks
iskn
with
k1 > k2 > ... > kn.
If W
=
(wjQ)
then a short computation shows
that [A,W1
=
(ia(kj
(A,[A,W])
=
(-a2(kj - kp)2 wjQ)
-
kQ)wjf)
,
hence ,
THE BOTT PERIODICITY THEOREM
§23.
131
and it
KA(W)
Now we find a basis for 1)
For each
(kj - kp) 2 wj f)
consisting of eigenvectors of
g'
the matrix
j < P
with
Ej,
+1
as follows:
KAY
in the
(j4)-th
(ej)-th place and zeros elsewhere, is in g'
-1 in the
place,
2
and is an eigenvector corresponding to the eigenvalue 2
"(kj - kf)2 2)
Similarly for each (jfl -th place and
the matrix
j < P
in the
+i
E,',
with
+i
in the
(.j)-th place is an eigenvector,
also with eigenvalue !(k. - kk)2 3)
Each diagonal matrix in
is an eigenvector with eigenvalue o.
g'
KA
Thus the non-zero eigenvalues of
with
kj > kk.
are the numbers
(kj - k1) 2
Each such eigenvalue is to be counted twice.
Now consider the geodesic
y(t)
Each eigenvalue
exp tA.
=
gives rise to a series of conjugate points along
e = 1 (kj - kj)2 > 0
y
corresponding to the values
t (See §20.5.)
a/*, 2r[/,Te-,
=
3a/Ve-,
Substituting in the formula for t
The number of such values of k . - kR j 2 equal to - 1.
t
e,
this gives
, k -k
kj -2 k , k -4 k
-
...
6
in the open interval
Now let us apply the Index Theorem.
For each
, (0,1)
j,P
is evidently
with
k. > kQ
2
we obtain two copies of the eigenvalue
(kj - kR)2,
and hence a contri-
bution of
-kp
2(l to the index.
Adding over all
X
2
j,P
_
kj > kR for the index of the geodesic
As an example, if
y
- 1)
this gives the formula
(kj-kR-2)
Y.
is a minimal geodesic, then all of the
kj
132
APPLICATIONS
TV.
are equal to
+
1
.
Hence
as was to be expected.
x = 0,
Let
Now consider a non-minimal geodesic. CASE 1.
At least
m+1
case at least one of the positive
ki
are (say) negative.
ki's
of the
n = 2m.
must be
> 3,
In this
and we have
m+1
- (-1) - 2)
CASE 2.
all are
m
of the
Then one is
+ 1.
ki
> 3
are positive and m are negative but not and one is
(3 - (-1) - 2) + 1
.
< -3
so that
m-1
m-1
>
2(m+1)
=
(1
- (-3) - 2) + (3 - (-3) - 2)
1
4m > 2(m+1)
Thus in either case we have
X > 2m+2.
This proves Lemma 23.2,
and therefore completes the proof of the Theorem 23.3.
§24.
§24.
133
THE ORTHOGONAL GROUP
The Periodicity Theorem for the Orthogonal Group.
This section will carry out an analogous study of the iterated loop However the treatment is rather sketchy, and
space of the orthogonal group.
The point of view in this section was suggested
many details are left out.
by the paper Clifford modules by M. Atiyah, R. Bott, and A. Shapiro,
which relates the periodicity theorem with the structure of certain Clifford algebras. (See Topology, Vol. 3, Supplement 1 (1964), pp. 3-38.)
Consider the vector space Rn with the usual inner product. consists of all linear maps
orthogonal group 0(n)
T
Rn
:
which preserve this inner product.
real n x n matrices
The
T
Rn
Alternatively 0(n) T T*
such that
=
I.
consists of all
This group 0(n)
considered as a smooth subgroup of the unitary group U(n);
can be
and therefore
inherits a right and left invariant Riemannian metric.
Now suppose that DEFINITION.
mation
J
:
is even.
A complex structure
Rn
the identity
n
J
on Rn is a linear transfor-
belonging to the orthogonal group, which satisfies
J2 = -I.
The space consisting of all such complex structures
on Rn will be denoted by St1(n). We will see presently (Lemma 24.4) that R1(n)
is a smooth sub-
manifold of the orthogonal group 0(n). REMARK.
of 0(n) J1.
Given some fixed
J1 E,Q1(n)
let U(n/2)
be the subgroup
consisting of all orthogonal transformations which commute with
Then fl1(n)
can be identified with the quotient space 0(n)/U(n/2).
The space of minimal geodesdcs from I to -I is homeomorphic to the space n1(n) of complex on 0(n) structures on Rn. LEMMA 24.1.
PROOF:
The space 0(n)
orthogonal matrices. the space of
n x n
can be identified with the group of
Its tangent space g= T O(n)I skew-symmetric matrices.
n x n
can be identified with
Any geodesic
y
with
134
APPLICATIONS
IV.
can be written uniquely as
y(o) = I
y(t)
exp (et A)
=
for some A E g. Let
A
Since
n = 2m.
is skew-symmetric, there exists an element
so that
T E 0(n)
1
-a1 0
o a2
-a2 0 TAT-1 =
o am -am o
with a1,a2,...,am > 0.
A short computation shows that
T(exp it
A)T-1
is
equal to cos na1
sin aa1
0
0
-sin oat
cos rta1
0
0
0
is equal to
Thus exp(aA)
0
cos na2
sin aa2
0
-sin na2
cos na2
-I
if and only if
The inner product < A,A> Therefore the geodesic and only if If
a1,a2,...,a, are odd integers. 2(a? + a2 +...+am).
is easily seen to be
y(t) = exp(nt A)
from
I
to
-I
is minimal if
a1 = a2 = ... = am = 1. y
is minimal then 0
2
1
-1 0 A2
=
0
1
-1 0
T-1
T = -I
hence A is a complex structure. Conversely, let
J
be any complex structure.
nal we have
J J*
=
I
Since
J
is orthogo-
§24.
where
denotes the transpose of
J*
this implies that
J J = -I
THE ORTHOGONAL GROUP
J
/oa T-1
TJ
that
a1 = ... = ah1 = 1;
Thus
is skew-symmetric.
J
Hence
1
1
-a 1 0 =
a1,a2,...,am > 0
for some
Together with the identity
J.
= -J.
1 35
and some
Now the identity
T.
and hence that
exp itJ
This completes the
-I.
=
implies
J2 = -I
proof.
Any non-minimal geodesic from
LEMMA 24.2.
in O(2m)
has index
t
-+ exp(at A)
to
-I
Suppose that the geodesic has
The proof is similar to that of 23.2. the form
I
> 2m-2.
with
A =
where
a1 > a2 > ... > am > 0 are odd integers. Computation shows that the
non-zero eigenvalues of the linear transformation KA = - -147 (Ad A) 2 1)
for each
i < j
the number
2)
for each
i < j
with ai
(ai + aj)2/ 4,
the number
# aj
Each of these eigenvalues is to be counted twice.
I
i<j
(a
+ aj - 2) +
For a minimal geodesic we have >v
= 0,
as expected.
I
ai>aj
and (ai - aj)2/ 4.
This leads to the formula
(ai - aj - 2)
a1 = a2 = ... = am =
For a non-minimal geodesic we have
(3+1-2) + 0
=
are
1
so that
a1 > 3;
so that
2m - 2.
This completes the proof. Now let us apply Theorem 22.1.
The two lemmas above, together with
136
APPLICATIONS
TV.
is a manifold imply the following.
the statement that SL1(n)
THEOREM 24.3 (Bott).
The inclusion map 111(n) -+ o 0(n)
induces isomorphisms of homotopy groups in dimensions < n-4.
Hence
Ai a, (n) = ni+1 0(n) for
i < n-4.
Now we will iterate this procedure, studying the space of geodesics
from J
to
in f1(n);
-J
Assume that
and so on.
n is divisible by a
high power of 2. Let commute *,
J1,...,Jk-1
be fixed complex structures on Rn which anti-
in the sense that
iris for J
r
s.
+
Jsir
=
0
Suppose that there exists at least one other complex structure
which anti-commutes with J1,...,Jk-1' DEFINITION.
Let SLk(n)
denote the set of all complex structures J
on Rn which anti-commute with the fixed structures
J1,...,Jk-1'
Thus we have
SLk(n) C SLk-1 (n) C ... C SL1 (n) C 0(n) Clearly each SLk(n)
is a compact set.
natural to define n0(n) LEMMA 24.4.
To complete the definition it is
to be 0(n)
Each SLk(n)
is a smooth, totally geodesic**
submanifold of 0(n). JR
to
-JR
0
The space of minimal geodesics from in SLR(n) is homeomorphic to SLR+1(n), for
It follows that each component of SLk(n)
For the isometric reflection of 0(n) cally carry ilk(n)
is a symmetric space.
in a point of SLk(n) will automati-
to itself.
These structures make Rn into a module over a suitable Clifford algebra. However, the Clifford algebras will be suppressed in the following presentation.
** A submanifold of a Riemannian manifold is called totally geodesic if each geodesic in the submanifold is also a geodesic in larger manifold.
THE ORTHOGONAL GROUP
§24.
Any point in 0(n)
PROOF of 24.4.
J
close to the identity can be
where A is a "small," skew-
expressed uniquely in the form exp A,
close to the complex structure
Hence any point in 0(n)
symmetric matrix.
where again A is small and
J exp A;
can be expressed uniquely as
137
skew.
is a complex structure if and only if A
J exp A
ASSERTION 1.
anti-commutes with J.
If A
PROOF: I
Therefore
=
anti-commutes with J,
then J-1A J
J-1
(exp A)J exp A
exp(J-1A J) exp A
(J exp A)2
Conversely if
-I.
=
=
(J exp A)2
=
= -A
hence
-I
then the
above computation shows that exp(J-1A J) exp A
=
I
Since A is small, this implies that J-1A J
=
-A
so that A anti-commutes with J. J exp A anti-commutes with the complex structures
ASSERTION 2. J1,...,Jk-1
if and only if A commutes with J1,...,Jk-1'
The proof is similar and straightforward. Note that Assertions I and 2 both put linear conditions on
Thus a neighborhood of
J
in Qk(n)
consists of all points
A.
J exp A where
A ranges over all small matrices in a linear subspace of the Lie algebra This clearly implies that SLk(n)
g.
is a totally geodesic submanifold of
0(n). Now choose a specific point Jk E SLl(n),
exists a complex structure
and assume that there
which anti-commutes with J1,...,Jk.
J
Setting
J = JkA we see easily that A is also a complex structure which anticommutes with Jk.
However,
A comutes with J1,...,Jk-1.
Hence the
formula
t -Jk exp(st A) defines a geodesic from Jk to
-Jk in Sbk(n).
Since this geodesic is
minimal in 0(n), it is certainly minimal in SLk(n).
IV.
138
Conversely, let
ak(n).
Setting y(t)
y
=
APPLICATIONS
-Jk in
be any minimal geodesic from Jk to
A is
it follows from 24.1 that
Jk exp(st A),
a complex structure, and from Assertions 1,2 that A commutes with
belongs to fk+1(n).
This completes the proof of 24.4.
The point JkA enk+1(n)
REMARK.
geodesic
It follows easily that JA
and anti-commutes with Jk.
J1,...,Jk-1
which corresponds to a given
has a very simple interpretation:
y
it is the midpoint
y(z)
of the geodesic.
In order to pass to a stable situation, note that Ilk(n)
imbedded in nk(n+n')
as follows.
Choose fixed anti-commuting complex
structures structure
determines a complex
J e Slk(n)
J ® Jk on Rn ®Rn'
can be
which anti-commutes with Ja E) J.1,
for
a = 1,...,k-1.
Let 12k
DEFINITION.
spaces ak(n),
denote the direct limit as
with the direct limit topology.
The space 0 =110
n--' w
(I.e., the fine topology.)
is called the infinite orthogonal group.
It is not difficult to see that the inclusions 11k+1(n) give rise, in the limit, to inclusions 11k+1 THEOREM 24.5.
nllk
of the
For each k > 0
is a homotopy equivalence.
nh 0 = nh-1 111
-+ nil
- n ak(n)
k '
this limit map 1k+1 Thus we have isomorphisms
ah-2112
= ... -
"1 ah_l
The proof will be given presently.
Next we will give individual descriptions of the manifolds for
k = 0,1,2,...,8.
n0(n)
is the orthogonal group.
a1(n)
is the set of all complex structures on Rn
Given a fixed complex structure space
llk(n)
CnJ2 n2(n)
J1
we may think of Rn as being a vector
over the complex numbers.
can be described as the set of "quaternionic structures" on
the complex vector space C2. n/ Given a fixed J. e 112(n)
we may think of
139
§24. THE ORTHOGONAL GROUP
Cn/2
over the quaternions H.
as being a vector space He/4
be the group of isometries of this vector space onto itself.
Sp(n/4)
Let
Then SL 2(n)
can be identified with the quotient space U(n/2)/ Sp(n/4). Before going further it will be convenient to set
n = 16r.
LEMMA 24.6 - (3).
The space l3(16r) can be identified with the quaternionic Grassmann manifold consisting of
H4r
all quaternionic subspaces of Any complex structure
PROOF:
of H4r = R16r lows.
Let
into two mutually orthogonal subspaces
Note that
J1J2J3J1J2J3
determines a splitting
J3 E a3(16r)
and V2
as fol-
is an orthogonal transformation with square
J1J2J3
equal to
V1
Hence the eigenvalues of
+ I.
V1 C R16r be the subspace on which J1J2J3
J1J2J3
equals
+ I;
are
+ 1.
and let
V.
be the orthogonal subspace on which it equals -I. Then clearly R16r = V1 ® V2. Since J1J2J3 commutes with J1 and J2 it is clear
that both V1
and V2
are closed under the action of
Conversely, given the splitting H4r
=
V1 (D V2
we can define
orthogonal quaternionic subspaces,
J1
and
J2.
into mutually
J3 E a3(16r)
by the
identities J3IV1
J3IV2
=
=
-J1J2lV1
J1J21V2
This proves Lemma 24.6 -(3).
The space a3(16r) varying dimension.
of largest dimension:
spaces of H4r.
is awkward in that it contains components of
It is convenient to restrict attention to the component namely the space of 2r-dimensional quaternionic sub-
Henceforth, we will assume that
this way, so that
J3
has been chosen in
dimes V1 = dimes V2 = 2r.
can be identified The space n4(16r) to with the set of all quaternionic isometries from V1 V2. Thus Sl4(16r) is diffeomorphic to the symplectic group Sp(2r). LEMMA 24.6 - (4).
PROOF:
Given J4 E n4(16r)
commutes with J1J2J3
.
Hence
J3J4
note that the product maps
V1
J3J4
to V2 (and V2
antito
V1).
140
APPLICATIONS
IV.
Since
commutes with J1
J3J4
and
J3J4fV1
T
V1
:
V1
:
- V2
Conversely, given any such isomorphism
is a quaternionic isomorphism. -. V2
we see that
J2
it is easily seen that
is uniquely determined by the
J4
identities: J4IV1
=
J31 T
J4IV2
=
-T-1 J3
This proves 24.6 - (4).
The space Sl5(16r) can be identified with the set of all vector spaces W C V1 such that (1) W is closed under J1 (i.e., W is a complex vector space) and (2) V1 splits as the orthogonal sum W ® J2 W. LEMMA 24.6 - (5).
Given J5 ea5(16r)
PROOF:
commutes with J1J2J3
note that the transformation J1J4J5
and has square
itself; and determines a splitting of
Let W C V1
spaces.
Since
J2
Thus
+ I. V1
into
V1
into two mutually orthogonal sub-
be the subspace on which J1J4J5
anti-commutes with J1J4J5,
maps
J1J4J5
coincides with
it follows that
precisely the orthogonal subspace, on which J1J4J5
J2W C V1
equals
-I.
+ I.
is
Clearly
J1W = W.
Conversely, given the subspace W, it is not difficult to show that J5
is uniquely determined. REMARK.
morphisms of
V1
If U(2r) C Sp(2r)
denotes the group of quaternionic auto-
keeping W fixed, then the quotient space
Sp(2r)/ U(2r)
can be identified with a5(16r).
LEMMA 24.6 - (6).
The space Sl6(16r)
can be identified
with the set of all real subspaces X C W such that W splits as the orthogonal sum X ® J1X. PROOF.
Given J6 e a6(16r)
commutes both with J1J2J3 itself.
Since
note that the transformation J2J4J6
and with J1J4J5.
(J2J4J6)2 = I,
it follows that
Hence
J2J4J6
J2J4J6
maps W into
determines a
THE ORTHOGONAL GROUP
§24.
141
Let
splitting of W into two mutually orthogonal subspaces. equals
subspace on which J2J4J6 subspace on which it equals
+I.
X C W be the
Then J1X will be the orthogonal
-I.
Conversely, given X C W, it is not hard to see that
J6
is unique-
ly determined.
denotes the group of complex automor-
If O(2r) C U(2r)
REMARK.
phisms of W keeping X
fixed, then the quotient space U(2r)/ O(2r)
can
be identified with a 6(16r).
can be identified LEMMA 24.6 - (7). The space Sl7(16r) with the real Grassmann manifold consisting of all real
subspaces of X = R2r.
PROOF:
Given J7,
anti-commuting with J1, ... ,J6
J1J6J7
commutes with J1J2J3,
square
+I.
Thus
J1J6J7
orthogonal subspaces:
J1J6J7
equals
X1
with J1J4J5,
and with J2J4J6;
determines a splitting of X (where
J1J6J7
equals
Conversely, given X1 C X
-I).
note that
+I)
and has
into two mutually
and X2
(where
it can be shown that
J7
is uniquely determined. This space 117(16r), like J1 3(16r), has components of varying dimension.
Again we will restrict attention to the component of largest dimen-
sion, by assuming that dim X1 = dim X2 = r. Thus we obtain: ASSERTION.
The largest component of a7(16r) is diffeomorphic to
the Grassmann manifold consisting of r-dimensional subspaces of
R2r.
LEMMA 24.6 - (8).
The space fl8(16r) can be identified with the set of all real isometries from X1 to X2.
PROOF.
If
J8 e j8(16r)
commutes with J1J2J3, J1J4J5, Hence
J7J8
determines
maps J8
X1
then the orthogonal transformation and
J2J4J6;
isomorphically onto
J7`T8
but anti-commutes with J,J6J7. X2.
Clearly this isomorphism
uniquely.
Thus we see that d18(16r)
is diffeomorphic to the orthogonal
IV.
142
APPLICATIONS
group* 0(r).
Let us consider this diffeomorphism Sl8(16r) - 0(r), the limit as
r
It follows that Sl8
-+ -.
and pass to
is homeomorphic to the infinite
Combining this fact with Theorem 24.5, we obtain the
orthogonal group O. following.
THEOREM 24.7 (Bott).
The infinite orthogonal group 0 has
the same homotopy type as its own 8-th loop space.
ai 0 is isomorphic to
the homotopy group If
Sp =114
Hence "i+8 0 for i > 0.
denotes the infinite symplectic group, then the above
argument also shows that 0 has the homotopy type of the 4-fold loop space
and that
0000 Sp, 0000 O.
Sp
has the homotopy type of the 4-fold loop space
The actual homotopy groups can be tabulated as follows.
i modulo 8
ni Sp
ni O
Z2 Z2
0
2
0
0
3
Z
Z
4
0
5
0
Z2 Z2
0 1
0
6
0
0
7
Z
Z
The verification that these groups are correct will be left to the reader. (Note that
Sp(1)
is a 3-sphere, and that
S0(3)
is a projective 3-space.)
The remainder of this section will be concerned with the proof of Theorem 24.5.
It is first necessary to prove an algebraic lemma.
Consider a Euclidean vector space V with anti-commuting complex structures
*
J1,...,Jk.
For k > 8
it can be shown that dlk(16r)
In fact any additional complex structures
is diffeomorphic to Slk_8(r).
J9,J10,...,Jk
on R16r
give
rise to anti-commuting complex structures J8J9, J8J10, J8J111...,J81k X1;
and hence to an element of
fl k_8(r).
will be sufficient to stop with k = B.
on
However, for our purposes it
§24.
THE ORTHOGONAL GROUP
V is a minimal
DEFINITION.
143
(J1,...,Jk)-space if no proper, non-
trivial subspace is closed under the action of
and
J1,...,
Jk.
Two such
minimal vector spaces are isomorphic if there is an isometry between them which commutes with the action of
LEMMA 24.8
J1,...,Jk.
For k 3 (mod 4), any vector spaces are isomorphic.
(Bott and Shapiro).
two minimal
(J1,...,Jk)
The proof of 24.8 follows that of 24.6.
For
k = 0,1, or 2
a
minimal space is just a 1-dimensional vector space over the re als, the complex numbers or the quaternions.
For
k = 3
a minimal space is still a 1-dimensional vector space
over the quaternions. J3
is equal to
Clearly any two such are isomorphic.
However, there are two possibilities, according as
+J1J2
This gives two non-isomorphic minimal
-J1J2.
or
spaces, both with dimension equal to
For
k = 4
J3J4 mapping H to k = 5,6
For
complex structures spaces.
k = 7
For
4.
H and
Call these
a minimal space must be isomorphic to
The dimension is equal to
H'.
H'.
H ® H',
we obtain the same minimal vector space J5,J6
with
8.
H ® H'.
merely determine preferred complex or real sub-
we again obtain the same space, but there are two
possibilities, according as
J7
is equal to
+J1J2J3J4J5J6
or to
-J1J2J3J4J5J6.
Thus in this case there are two non-isomorphic minimal
vector spaces;
call these
For
with J7J8
k = 8
k > 8
periodically.
L and
L'.
a minimal vector space must be isomorphic to L ® L',
mapping L onto
For
The
L'.
The dimension is equal to 16.
it can be shown that the situation repeats more or less
However, the cases
k < 8 will suffice for our purposes.
Let mk denote the dimension of a minimal
(J1,...,Jk)-vector space.
From the above discussion we see that:
m0 = 1, m1 = 2, m2 = m3 = 4, m4 = m5 = m6 = m7 = 8, m8 = 16. For
k > 8
it can be shown that mk = 16mk-8.
REMARK.
These numbers mk are closely connected with the problem
of constructing linearly independent vector fields on spheres. example that
J1,...,Jk
Suppose for
are anti-commuting complex structures on a vector
APPLICATIONS
IV.
144
space V
of dimension
for each unit vector
u E V the
dicular to each other and
vector fields on an
Here
rmk.
to
can be any positive integer.
r
k vectors
u.
Thus we obtain k
fields on a
(16r-1)-sphere; and so on.
are perpenindependent
3
vector
(8r-1)-sphere;
7 vector fields on an
(4r-1)-sphere;
(Compare
linearly
For example we obtain
(rmk 1)-sphere.
fields on a
and Radon.
J1u, J2u,..., Jku
Then
8 vector
These results are due to Hurwitz
B. Eckmann, Gruppentheoretischer Beweis des Satzes von
Hurwitz-Radon..., Commentarii Math. Hely. Vol. 15 (1943), pp. 358-366.)
J.
F. Adams has recently proved that these estimates are best possible.
k V 2 (mod 4).
PROOF of Theorem 24.5 for
minimal geodesics from J at
of 11k(n)
Let
T
to
-J
in 11k(n).
1)
A
is skew
2)
A
anti-commutes with J
3)
A
commutes with J1,".,Jk-1'
denote the vector space of all such matrices
corresponds to a geodesic
t - J exp (at.4) from J
its eigenvalues are all odd multiples of
Each such A E T KA:
Recall that the tangent space
J A where
consists of all matrices
J
We must study non-
T - T.
can compute
A.
to
A given A E T -J
if and only if
i.
determines a self-adjoint transformation
is a totally geodesic submanifold of 0(n),
Since a k(n)
we
KA by the formula KAB
just as before.
=
-1 [A,[A,B1]
_
(-A2B + 2ABA
-
BA2)/4
,
We must construct some non-zero eigenvalues of
KA
so as
to obtain a lower bound for the index of the corresponding geodesic
t - J exp(irt A)
.
Split the vector space Rn as a direct sum
M1 ® M2 ® ... ® Ms
of
mutually orthogonal subspaces which are closed and minimal under the action of
J1,...,Jk-1, J
and
A.
all equal, except for sign.*
Then the eigenvalues of
A
on Mh must be
For otherwise Mh would split as a sum of
*
We are dealing with the complex eigenvalues of a real, skew-symmetric transformation. Hence these eigenvalues are pure imaginary; and occur in conjugate pairs.
THE ORTHOGONAL GROUP
§24.
A;
eigenvalues of
AIMh;
where
Now note that
is a complex structure on Mh which
J' = ah1JAIMh;
be zero for
(J1,...,Jk-1,J,J')k +
Since
3 (mod 4)
1
are mutually isomorphic.
M1,M2,...,Ms
with h
h,j
we can construct an eigenvector
j
B: Rn _Rn of the linear transformation BIM,
Mh is
Thus
Hence the dimension of Mh is mk+1
For each pair
+ iah be the two
are odd, positive integers.
a1,...,as
anti-commutes with J1,...,Jk-1, and J.
we see that
Let
and hence would not be minimal.
eigenspaces of
minima]..
145
T - T as follows. Let
KA:
Let BIMh be an isometry from Mh to Mj
f / h,j.
which satisfies the conditions
BJa
=
J0B
BJ
=
-JB
and
BJ'
dicates that we have changed the sign of exists by
Finally let
24.8.
B
We claim that eigenvalue
B
J.
BIMh.
Since
T.
w E Mj BIMh
com-
It follows easily that
also commutes with J1,...,Jk-1
BIMj
where the bar in-
Such an isomorphism
It is also clear that
and anti-commutes with
Thus
Mj;
for v E Mh,
is skew-symmetric.
the negative adjoint J.
J
to Mj.
be the negative adjoint of
Kv,- Bw>
=
mutes with J1,...,Jk-1
commutes with
BIMj
on
.
B belongs to the vector space
Proof that
it is clear that
+J'B
=
is an isomorphism from Mh
BIMh
In other words
a = 1,...,k-1;
for
and anti-
B E T.
KA
is an eigenvector of For example if
(ah + aj)2/4. (KAB)v
v e Mh
(-A2B + 2ABA
=
-
corresponding to the then
BA2)v
2ajBahv + Bahv) _
7 -IT
(aj + ah) 2 By
and a similar computation applies for Now let us count.
by
v E Mj.
The number of minimal spaces Mh C Rn is given
For at least one of these the integer
s = n/mk+1'
For otherwise we would have a minimal geodesic. (always for
=
> 3.
This proves the following
2 (mod 4)):
k
ASSERTION.
> (3+1)2/4
ah must be
4.
KA has at least The integer
s-1 eigenvalues which are
s = n/mk+1
tends to infinity with
n.
146
APPLICATIONS
IV.
Now consider the geodesic of
t
e2
Each eigenvalue
-+ J exp(at A).
gives rise to conjugate points along this geodesic for
KA
Thus if
by 20.5.
t = e-1, 2e-1, 3e-1,...
then one obtains at
e2 > 4
Applying the index theorem, this proves
least one interior conjugate point. the following.
The index of a non-minimal geodesic from J
ASSERTION.
to
-J
in
is > n/mk+1- 1.
Slk(n)
It follows that the inclusion map
ak+1(n) -+ o nk(n) induces isomorphisms of homotopy groups in dimensions number tends to infinity with
as n -+ co,
it follows that the inclusion map
i
:
and
But it can be shown
a Dk have the homotopy type of a CW-complex.
fore, by Whitehead's theorem, it follows that
nilk(n)
Thus
ing subspace fk+1(n)
has an infinite cyclic fundamental
has infinitely many components, while the approximathas only finitely many.
To describe the fundamental group f
as follows.
Let
k 0 2 (mod 4).
The difficulty in this case may
k e 2 (mod 4).
be ascribed to the fact. that dlk(n) group.
There-
is a homotopy equivalence.
i
This completes the proof of 24.5 providing that PROOF of 24.5 for
This
-i a S).k induces
S?,k+1
isomorphisms of homotopy groups in all dimensions. that both SZk+1
< n/mk+1 - 3.
Therefore, passing to the direct limit
n.
J1,...,Jk-1
dlk(n)
:
we construct a map
allk(n)
-o S1 C C
be the fixed anti-commuting complex struc-
ture on R. Make Rn into an
(n/2)-dimensional complex vector space by
defining iv = J1 J2 ... Jk-1v
for that
v e Rn;
where
i2 = -1,
i = 47-1
and that
E C.
The condition
J1,J21...IJk-1
k
2 (mod 4)
commute with
i.
Choose a base point J e llk(n) . For any J' e Slk(n) composition
J-1J'
commutes with
i.
Thus
J-1J'
guarantees
note that the
is a unitary complex
linear transformation, and has a well defined complex determinant which will
be denoted by
f(J').
THE ORTHOGONAL GROUP
§24.
147
Now consider a geodesic J exp(sctA)
t
from J
to
-J
Since A
in 1Zk(n).
commutes with
i = J1J2 ... Jk-1
also as a
(compare Assertion 2 in the proof of 24.4) we may think of A
In fact A
complex linear transformation.
trace of A
is a pure imaginary number.
is skew-Hermitian;
Now
f(J exp(stA)) = determinant (exp(stA)) = Thus
f
ent trace A
maps the given geodesic into a closed loop on
pletely determined by the trace of
S1
which is com-
It follows that this trace is in-
A.
n(11k(n);J,-J).
variant under homotopy of the geodesic within the path space The index
is closed under the action of h,
J1,...,Jk-1,J,
and
iah.
A;
and is minimal.
J1,...,Jk-1,
Therefore the trace of A
Now for each h X j
can have only
For otherwise Mh would split into eigenspaces.
Thus AIMh coincides with ahJlJ2 ... Jk-1IMh. the action of
where each Mh
M1 ® ... ® Mr
the complex linear transformation AIMh
one eigenvalue, say
As
of this geodesic can be estimated as follows.
%
before split Rn into an orthogonal sum
Thus for each
hence the
and
Since Mh is minimal under
its complex dimension is
J;
is equal to
mk/2.
i(a1+...+ar)mk/2.
an eigenvector B
of the linear transforma-
tion
B -+ KAB
=
(-A2B + 2ABA - BA2) /4
can be constructed much as before.
Since
Mh and
Mj
are
(J1,...,Jk-1,J)-
minimal it follows from 24.8 that there exists an isometry
BIMh :
and anti-commutes with J.
which commutes with J1,...,Jk-1 the negative adjoint of
BIMh;
Mh-> Mj
and let
BIM1
be zero for
Let
BIMj
e / h,j.
Then
an easy computation shows that KAB
Thus for each ah > aj Since
=
(ah - aj) 2B/4
we obtain an eigenvalue
(ah - aj)2/4
each such eigenvalue makes a contribution of
towards the index
x,
we obtain the inequality
(ah- aj - 2)
2),>
ah > aj
for
(ah - aj)/2 -
KA. 1
be
148
APPLICATIONS
TV.
Now let us restrict attention to some fixed component of
That is let us look only at matrices A c
Q ak(n).
trace A = icmk/2
such that
where
is some constant integer.
Thus the integers
al,...ar
1)
a, a a2 = ... = ar
2)
al +...+ ar = c,
3)
Max lahi > 3
satisfy (mod 2), (since
exp(aA) = -I),
and
(for a non-minimal geodesic).
h Suppose for example that some the positive
ah
and
-q
ah is equal to
the sum of the negative
p-q = hence
2p > r + c.
ah.
Thus
,
Now
ah>a3 hence
p+q>r
c,
(ah - a - 2) >
2X >
Let p be the sum of
-3.
4X > 2p > r + c;
where
follows that the component of
ah>o
=
p
tends to infinity with
r = n/mk C 11k(n)
(ah - (-3) - 3)
n.
It
is approximated up to higher and
higher dimensions by the corresponding component of Sk+l(n),
as
n - C .
Passing to the direct limit, we obtain a homotopy equivalence on each component.
This completes the proof of 24.5.
149
APPENDIX. THE HOMOTOPY TYPE OF A MONOTONE UNION
The object of this appendix will be to give an alternative version for the final step in the proof of Theorem 17.3 (the fundamental theorem of Morse theory).
Given the subsets
C saa2 C
...
and given the information that each
o = o(M;p,q),
space
a1
oao C
of the path a o i has the
homotopy type of a certain CW-complex, we wish to prove that the union
0
also has the homotopy type of a certain CW-complex. More generally consider a topological space X0 C X1 C X2 C ...
X
X
and a sequence
To what extent is the homotopy type of
of subspaces.
Xi?
determined by the homotopy types of the
It is convenient to consider the infinite union X£ = X0 x[0,1] v X 1 X (1,2] u X2 X(2,31 u ...
This is to be topologized as a subset of X x R.
We will say that
DEFINITION. the sequence
(Xi)
EXAMPLE 1.
X£ - X,
:
Suppose that each point of X X
and that
Xi,
p
defined by
is a homotopy equivalence.
p(x,T) = x,
some
is the homotopy direct limit of
X
if the projection map
is paracompact.
lies in the interior of
Then using a partition of unity one
can construct a map
f : X- R f(x) > i+1
so that
pondence
for
x - (x,f(x))
x
Xi,
f(x) > 0
for all
x.
Now the corres-
maps X homeomorphically onto a subset of
is clearly a deformation retract.
and X
and
Therefore
p
X.
which
is a homotopy equivalence;
is a homotopy direct limit.
EXAMPLE 2.
with union
X.
Let X be a CW-complex, and let the Xi be subcomplexes
Since
p : X£ - X induces isomorphisms of homotopy groups
in all dimensions, it follows from Whitehead's theorem that X direct limit.
is a homotopy
APPENDIX
150
The unit interval
EXAMPLE 3.
[0,1]
limit of the sequence of closed subsets
[o]
is not the homotopy direct v [1/i,1].
The main result of this appendix is the following.
Suppose that X is the homotopy direct and Y is the homotopy direct limit Y be a map which carries each Let f: X {Yi). into Yi by a homotopy equivalence. Then f
THEOREM A.
limit of of
[X.1
Xi itself is a homotopy equivalence.
Assuming Theorem A, the alternative proof of Theorem 17.3 can be given as follows.
Recall that we bad constructed a commutative diagram saa,
S2ao C
Ko
of homotopy equivalences.
C K1
Since
1a
C saa2
C ...
C K2 C ... a = U o i
and K = U Ki
are homotopy
direct limits (compare Examples I and 2 above), it follows that the limit
mapping 0- K is also a homotopy equivalence.
PROOF of Theorem A.
Define
f£ : XX -* YY
It is clearly sufficient to prove that
CASE 1.
is a homotopy equivalence.
f£
is homotopic to the identity.
fi
:
Xi-.. Yi
We must prove
f£ is a homotopy equivalence.
REMARK. that
f£(x,t) = (f(x),t).
Suppose that Xi = Yi and that each map
(obtained by restricting f) that
by
Under these conditions it would be natural to conjecture
f. must actually be homotopic to the identity.
However counter-
examples can be given.
For each n
let
Xn
Xn
be a one-parameter family of mappings, with hn Define the homotopy h
: Xz - X&
fn,
hn = identity.
HOMOTOPY OF A MONOTONE UNION
151
as follows (where it is always to be understood that
and
0 < t < 1,
n = 0,1,2,...). (hu(x),n+2t)
hu(x,n+t)
(h(3-4t)u (x),n+1) 0,,n+
for
0 < t < z
for
z < t < -T
for
3
< t < 1
(x),n+1)
Taking u = 0 to
fE.
this defines a map
The mapping
XE
: XE
h1
h0 : XE
which is clearly homotopic
X£
on the other hand has the following
properties: h1(x,n+t)
=
(x,n+2t)
for
0 < t < z
h1(x,n+t)
e
71, +1
for
2 < t < 1
We will show that any such map homotopy inverse
h1
x[n+11
can be defined by the formula
g : XE - XE
(x,n+2t)
(
In fact a
is a homotopy equivalence.
0 < t <
-
g(x,n+t) = j
l h (x,n 2- t)
2
1
-
This is well defined since h1(x,n+2) = h1(x,n+1) = (x,n+1)
Proof that the composition h1g of
XE.
.
is homotopic to the identity map
Note that
h1g(x,n+t) =
Define a homotopy Hu : XE
(x,n+4t)
0 < t < 4
h1(x,n+2t)
4 < t < 2
h1 (X,ri 2 - t)
2
XE
as follows.
h1g(x,n+t)
Hu(x,n+t) =
and h1(x,n+1-u)
For
0 < u < z
let
for
0 < t < (1-u)/2
for
z+u < t < 1
for
(1-u)/2 < t < z + U.
APPENDIX
152
This is well defined since h1g(x,n+(1-u)/2)
Now
h1g and
is equal to
Ho
h1g(x,n+2+u)
=
h1(x,n+1-u).
is given by
Hiz
0 < t < 4
1(x,n+4t) Hi(x,n+t) z
=
=
< t < 1
(x,n+1)
Clearly this is homotopic to the identity. is homotopic to the identity; and a completely analogous
h1g
Thus
argument shows that
is homotopic to the identity.
gh1
This completes the
proof in Case 1.
Now let
CASE 2.
and Y be arbitrary.
X
gn : Yn -+ Xn be a homotopy inverse to
fn.
gri
Yn
For each n
let
Note that the diagram
Xn
tin
Iin gYl+ 1
Xn+1
Yn+1
(where
in
and
in
denote inclusion maps)
is homotopy commutative.
In
fact
1ngn ti gnifn+tingn Choose a specific homotopy and
define
G : Y£
X£
hu
gn+tjnfngn - gn+1jn
=
Yn--' Xn+1
:
with ho = ingn,
h,
gn+1Jn;
by the formula (gn(y),n+2t)
0 < t <
G(y,n+t)
(h2t-1(y),n+1)
We will show that the composition Gf£
Let X£ denote the subset of
T < n.
(Thus
sition
Gf£
identity.
X£ = X0 x[o,11 carries
X£ -+ X.
is a homotopy equivalence.
consisting of all pairs
...
with
(x,T)
Xn_1 x [n-1 n] u Xn x[n].)
The compo-
X2 into itself by a mapping which is homotopic to the
In fact X£
mapping GfE
XE
:
< t < 1
contains Xn x[n]
restricted to Xn x[n]
hence is homotopic to the identity.
as deformation retract; and the
can be identified with
gnfn,
and
Thus we can apply Case 1 to the sequence
HOMOTOPY OF A MONOTONE UNION (X£n}
and conclude that This proves that
argument shows that f£
Gf£ f.
is a homotopy equivalence. has a left homotopy inverse.
f£G : Y£ - Y.
has a right homotopy inverse.
equivalence (compare page 22)
153
This proves that
f£
is a homotopy
and completes the proof of Theorem A.
Suppose that X
is the homotopy direct has the homotopy type of a OW-complex, then X itself has the homotopy
COROLLARY.
limit of
(Xi).
If
A similar
is a homotopy equivalence, so that
each Xi
type of a OW-complex. The proof is not difficult.
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