F. ,lBrisque
81
Gromov's almost flat manifolds by Peter BUSER and Hermann KARCHER*
*
This work was done under the program
Mathematik
(SFB 40) at Bonn
by the Swiss National the I. H. E. S.
Science
University. Foundation
Sonderforschungsbereich The first
and the 'second
societe mathematique -."..,~--------.,---
author
Theoretische
was also supported by an invitation
de france
to
PREFACE
This
expose
almost
gives
a detailed
flat manifolds.
to rewrite aSSumes
a proof.
- we hope
the full proof
geometry
since
trolled
chapters
and global
levels.
These
curvature
develops
metric
methods;
§ 2 contains
are the heart
hope
to the almost
for discussions
with
in 1977 on the present
was born, form.
and at the I.H.E.S. After
that our readers
(countably) profit
way:
nonlinear
group
averaging
group
technique
§ 7
which
and 5.1 is a
§ 1 contains
earlier
theorem
results
and a guide
is in 1.5.
M. Gromov
at the I.H.E.S.
§ 3 after which in 1980 which
many
geometry;
groups.
flat manifolds
of the theorem
the following
in the fundamental
in § 3 - § 5, while
The statement
We are grateful
its final
is given
con-
at several
in Riemannian
discrete
we
curvature
occurs
§ 8 explains
of nilpotent
argu-
Riemannian
us to write
accessible
estimates
different
Secondly,
local
constructions
of Lie groups;
treatment
rather
unconventional
between
persuaded
for
effort
[ 1 ]
to qualitative
constructions
commutator
pertaining
to its proof.
Arbeitstagung
controlled
several
less background.
interplay
geometric
theorem
so much
publication
rather
introduction
of the Gromov-Margulis
proper
and example~
with
and hopefully
properties
new form of Malcev's
The proof
familiar
considerations
in a selfcontained
§ 6 treats
original
the characteristic
analysis
different
Gromov's
requires
an ideal
pinching
for spending
in completing
our presentation
consider
script
is very
and has no difficulties
ments
of M. Gromov's
two reasons
One is that
that the reader
fields
proof
We have
discussions
from the synthesis
and the
the idea of this manu-
helped between
to get § 5.1 in the two of us we
of two different
styles
and
temperaments.
Finally
our thanks
the manuscript for publication.
go to Mrs.
and to Arthur
M. Barr6n L. Besse
for carefully
who
suggested
typing
- and retyping
contacting
Asterisque
_
~1e
1.
of contents
The theorem, Earlier
earlier
results,
examples
5
results 5
Examples
7 The theorem 9 Comments
about
the proof 1J
2.
Products Short
of short
loops
geodesic
loops
and their
holonomy
15
motions
and their product 15
The fundamental
ITl (M)
group
19 Holonomy 22 Commutator
estimates 23
Lower
volume
bounds 28
Number
of generators
ITl (M)
for
30 3.
Loops with Short
small
rotational
parts
33
bases 34
Relative
denseness
of loops with
small
rotation 36
Small
rotational
parts 38
Nilpotency
properties 42
Short
4.
loops modulo
The embedding of finite
of
index
almost
rp
translational
into a nilpotent
ones
45
torsionfree
subgroup
ITl (M)
in
Normal
bases
Almost
translational
49
for lattices
in
subsets
~n of
49 ~n 52
Orbits,
representatives,
The product
projection
in the projection
55
T' 58
5.
The A-normal
basis
The embedding
of
The nilpotent diffeomorphism
for the almost
translational
into the fundamental
Lie group F :
M
N
+ N
set
group
60 63
and the r-equivariant . 71
The Malcev
polynomials
71 The local
diffeomorphisms
The" left invariant Maximal
metric
and their
r-equivariant
and its curvature
agerage
77 80
rank of the average
84
3
TABLE
6.
Curvature
controlled
OF CONTENTS
91
constructions
Curvature
91
Parallel
92
Jacobi
and affine
translation
94
fields
Applications
to geodesic
Comparison
of riemannian
99
constructions and euclidean
parallel 105
translation Aleksandrow's
7.
area estimate
for geodesic
triangles
111 Lie groups Basic
111
notions
Explicit
solutions
for Jacobi
fields, 113
Campbell-Hausdorff-formula
_115
Norms Jacobi
field estimates
Metric
results
Applications
Almost
O(n)
119
case
and to the motions
of
n R
126
flat metrics estimates
Nonlinear
averages
The nonlinear
center
homomorphisms
Averages
122 125
metrics
Remainder
Almost
117
in the biinvariant
to
-Left invariant
8.
106
for the Cfu~pbell-Hausdorff-formula
127 131 131
of mass of compact
of differentiable
groups
138 142
maps
4
1. The theorem,
1.1
Earlier
earlier
results
results,
which
examples
concluded
global
properties
from curvature
assumptions.
(i) The GauE-Bonnet
=
2TI- X(M) shows
JM
that
KdO s2
of positive role
formula , together
and
characteristic,
the topological
are the only compact
- Such proofs
by integral
classification surfaces
of nonpositive
formulas
theorem
curvature
states
is covered
that a complete by
Rn.
of the fundamental
exp : T M + M p p group TIl(M,p)
geodesic
so that,
exp
with
the Riemannian
loop at
p
an extension
compact
convex
[9
balls.
geodesic
sufficiently
precise
(iv) The topological
n
Riemannian
t~
mizing.
R
manifold
imply
has maximal contains
noncompact
by exhibiting
distance
geodesic
closer
of
M.
map.
rank
and
exactly
one
_ § 2 starts
manifold
Mn
an exhaustion
of of
theorem [ 3 [20 n M with sectional
first from
is sharp
to one pole
as at several
the situation n spaces, S
described
A simply
curvature
bounds
pn(C)
p E Mn
argu-
the
<
K ~ 1
a metric
has a unique
antipode
"hemispheres"
up to length though
connected
t
carries
than to the other)
depend,
of Gromov's
- convexity
in this case is
] states:
since
that each p.; then
are at least
- The details
of geodesics
control.
segments
in the proof
- The - global
the behaviour
the result
K ~ 1 . One proves
since
n
J,
sphere
Sn;
(the sets of points balls
Riemannian
is a covering
p
a complete
to
curvature
manifold
to
at maximal
that
submanifold
under
with
no
Cheeger-Gromoll [ 6 ] extended this to nonnegative n M is diffeomorphic to the normal bundle of a _
work because
q E M
play
case
in the proof
is homeomorphic
J
is diffeomorphic
in which
totally
map
in fact,
ments
complete
metrics
ideas.
proved
curvature
curvature,
exponential
of these
(iii) Gromoll-Meyer posit~ve n M with
however
The assumptions
b) that each element
a) that
of surfaces
admitting
follows.
(ii) The Hadamard-Cartan M
with
p2(R)
curvature.
in what
for the Euler
more
of these poles
are diffeomorphic TI distance
complicated
miniinstances
theorem on a curvature controlled comparison between n on M and the corresponding situation on the model
5
to
1I
Ia~~
P. BUSER AND H. KARCHER
(v) The differentiable
sphere
curvature
0.7
assumptions
to a space
of constant
isometric F :
Mn
action
+ sn
maximal
ideas
Gromov's
theorem
Bieberbach's TIl(M,p)
and
finite
order
TIl-action on The main
spaces
but also
rational.
n
M
geometric
r\T M p
1.3 Definition. is bounded
rotational of this
arbitrary
important
and prove
- Corre-
hold [24
TIl(M)
n;
which
leads
Riemannian
of the diameter
flat we usually
mean
t.
~
space
they are
differential
flat Riemannian by rigid in
of
the factor
M
covers
part
A
motions
group
with
manifold
=
r
r\TIl(M)
of the G
deckgroup
of each motion
rotational either
It is the discovery to Gromov's
(deck-
, then: G
parts
A E O(n)
is always
manifold,
TIl(M,p)
of rotational
has all its main
of the theorem angle
however
for the understanding
TpM = ~n
on
is a torus
if the model
J,
of
of rank
< By almost
map
sn
control.
via partial
as the group
fact which
A compact
in terms
resp.
in the sequel.
be a compact
acting
subgroup
As a consequence
proof
J,
the set of all translations
in the deckgroup
or has a maximal
is diffeomorphic
M
results
step is to show that the rotational
x + Ax + a
under
(differentiable)
on
curvature
[23
version
Let
group r
normal
P
the
and are proved
and is obtained
T M
local
] states
(iv) one has to find an
actions
does not occur
J:
[ 4
its fundamental
is a free abelian
construct
similar
symmetric
in spirit
theorem
to
[17
MP
a complete
TIl(M)
is the following
transformations),
has
Sn,
J,
[13
§§ 5.1, 5.2, 5.4.
which
different
that
refined
to formulate
- a method
1.2 Rather
on
theorem
In addition
In principle
by other
complicated
1
for the
with more
spaces.
is replaced
equations
TIl(M)
are behind
(vi) Symmetric
more
dF
2
K
curvature.
equivariant
rank of
sponding
sn
of
2
pinching
angles the identity
of a direct
theorem.
is called
s-flat
if the curvature
as follows:
s ~d
(M)-2
that the manifold
s > 0 . If one multiplies
an s-flat
s-flat.
6
carries
metric
s-flat
metrics
by a constant
for
it remains
THE THEOREM
1.4 Examples of almost flat manifolds.
It is essential to realize that
aI.ost flat manifolds which do not carry flat metrics exist and occur rather naturally. Each nilmanifold
(i)
compact quotient of a nilpotent
(=
Lie group) is
·almost flat (7.7.2). (ii) An illustrative
special case of
(i)
is obtained if on the nilpotent
Lie algebra 0 a .. ) { ( ..... 1J o ·0
the following
:=
A ;
a
ij
~ IR ,
family of scalar products
.I
:==
1 < j
1 ~ i < j
~
n}
is introduced
a~. 2(j-i) 1J q
They give left invariant Riemannian metrics on the corresponding Lie group
1/
N
of upper triangular matrices.
[A,B] Ilq :: 2(n-2) /lA Ilq • liB Ilq
q-independent
and 7.7.1 one derives the following
(!) bound for the curvature tensor
24(n-2)21IA Therefore each compact quotient
~
theorem subgroup
M
group
of finite index in
112 • liB /12q • Ilc 112 q q is almost flat, since obviously q
sufficiently
ITl(M)
r.
r
=
would then contain an abelian
Hence we would have implies, that
N
zm
uniform in
{A EN;
0
<- a 1J.. <
can see the deviation
(1 0 1) 010 001
N
gives an example with the compact fundamental 1}
, a hypercube.
In the 3-dimensional
from the flat situation
(1 0 0) Oil
Er
act on
001
7
in a simple picture: 1 (
z
X)
o 1 Y
001
E
N
N,
itself is abelian
a contradiction.
The integer subgroup of domain
of these metrics:
q
small by choosing
which via the Campbell-Hausdorff-formula (5.1.6),
R
cannot carry a flat metric since by Bieberbach's
(1.2) the fundamental zm
r\N
=
its diameter can be made arbitrarily small. On the other hand
nilpotent
From
by
case one
IIII
~:~!!~
P. BUSER AND H. KARCHER
left translations of the x-axis,
as in the torus
resp.
the y-axis,
( 1 1 0)
case,
i.e. by translation
but
acts as the affine
010
x ->-x + y y->-y z ->-z + 1
map
001
The above
left invariant
Ilad II
in the direction
metrics
are given
1 , independent
of
by
q.
only the
"vertical"
fication
differs
torus
identi-
from the
case.
/T4~ I
Finally, since
the mulpiplication
it deviates
II (1 This notion manifolds,
+
plays
can be called
+
a central
only by quadratic
+
(1
B)
role
"almost
+
A
B)
II q
<
errors:
(n-2)
for the fundamental
translational"
IIA II q • IIBII q .
group
of almost
flat
see 3.5.
with
c : [0,00) ->-M these
N
from translation
A) • (1
(iii) Parabolic manifold
in
ends.
curvature
decreasing
be a simply
on
to
Jrt
them onto
c , then
1{ [ 15 o
connected
complete
Riemannian
2 2 _b ~ K : _a < 0 . Each geodesic
its family
by projecting
asymptotic metrics
H
bounds
determines
horospheres
of geodesics
Let
]
of horospheres.
Jro
with
the perpendicular
one gets a family
:
8
ray
If one identifies
of exponentially
family
THE THEOREM
go • e-
On the other
hand,
for horospheres of
t,
2bt
<
2at
<
go • e-
the GauE equations
show
that their
and the nonexistence
intrinsic
One therefore
gets an almost
II of isometries quotient [8
- e.g. ].
Jet
of
H
if
(iv) The binary
of S-fl~t
dodecahedron 1.3 works
i~
"flatter"
space
of
which
The examples
~
with
o
from any parabolic free) with
group
compact
volume
is naturally
the
83
of
is contained well point.
(in the sense
0(4) 3
8
(s + 0)
metrics
(~)2 ~ 0.15S-flat
by dividing
II\1fo
(fixed point
to an end of a finite
subgroup
equally
to a distinguished
therefore
Jro
by moving
icosahedral
spherical
distance
on
quotient
given
II-equivariant
by
gt
geodesic
with
has as fundamental
in a spherical
the diameter
The quotient
- but still
of
83
by the symmetry
group
a
of radius
curvature
curvature
1 ! An even
can be obtained
dodecahedral
tessel-
.
1.4. (i) are up to finite
quotients
the only
sectional
exp(-exp(exp
(i)
is covered
by a nilmanifold.
The fundamental normal
< ~
subgroup
almost
flat manifolds:
J
M
flow
by the maximal
by that
of constant
of the above
domain
ball
replaced
of 1.3) of nonnegative
1.5 Main Theorem (Almost flat manifolds) [ 1 n Let M be a compact Riemannian manifold with
then
of
1.5.2) .
Definition
lation
independent
2b (b - a) .
flat manifold
acting
obtained
(cf. also
<
II corresponds
A family
or equivalently to
of focal points
are bounded
in fact
<
H
curvatures
subgroup
group
r
ITl (M)
of rank
curvature
2
n »
bounds
,
More precisely:
contains n
9
a torsion
(4.6.5).
free nilpotent
P. BUSER AND
(E)
H. KARCHER
isomorphic
~ 2. G = r\TI (M) has order 1 to a subgroup of O(n) (3.6.4).
The finite
covering
1
The quotient
(iii)
G
(iv)
The simply by
bach
connected
It is not known
to the quotient suitable
of
N
with
nilpotent
of
metric
but a little
In the Bieberbach M
case
at
be described
p
whether
fundamental r\N
Lie group
M
by a uniform
left invariant
theorem;
group
M
to a nilmanifold
and is
group
deckgroup
rand
(5.2).
N
is uniquely
more
G
,and
on
r,
is - as in the flat case discrete
N.
than
group
Therefore stated
of isometries
in 1.5 is true
as the set of loops
at
p
with
rotational
part
~
(Le.
a rotational
• Again,
these
r
is almost
loops
holonomy
rotations
r\TI (l1,p) 1
Remark
at
p
En
same method
for small
1.4
a small
smaller
to the length
of
correction,
(1.2). loops
part
(2.3)
than
0.48
of the loop so that
(ii)). Moreover,
classes
in 1.5 reflects
can yield;
The twodimensional
IKI:
(see example
are, after
proof
to
are in fact much
~
"short"
if one chooses
TIl(M,p) mod
a subgroup
of
r
then
O(n)
n
isomorphic
much better
for larger constants
n
approximately
can be obtained
with
case of 1.5 follows
from the classification
integral
to obtain
1 2TI
the
of surfaces:
(with curvature
normalized
1
<
what
(= 3,4).
to the GauB-Bonnet
IX(M) I
their
(3.6.4).
2. The number
6.4.1
parts
proportional
in the equivalence
the present
Apply
bound
translational
shortest
to
rotational
an upper
the Bieber-
in that direction:
by those
(3.5); they have
for a
r\TI (M,p) is naturally isomorphic to the holonomy 1 the set of translations in TIl(M,p) , can also
flat case the group r is generated n lengths::: 4 (6TI)I (n-l)d(M)) which have
with
- diffeomorphic
1.5 does not imply
In the almost
: 0.48
determined
TIl(M) , (5.1. 7).
1.
Remark
of
is diffeomorphic
(6TI) n (n-l)
d(M)
J
<
o
10
sinh r dr
cosh d(M)
- 1
THE THEOREM
therefore,
if
<
IKld(M)2
arccosh(l+a)
then
x
implies
On the other
~
hand
following
r.
for
1.5.1
are corollaries
The proof
Corollary
Let
~
H
with
acting .
with
(parabolic
curvature
<
ends)
bounds
quotient
Iwasawa
H
normal is
is the restriction so is any uniform
1.6 Comments In pinching
to compare
discrete
theorems
methods
to write
the following
a
M
< 0
M
is
is covered
by a torus.
and the parabolic be given
0
r
of finite
r
In particular
If
group
r
to Gromov
in question
H = G/K
from
index,
group 1.4
<
-1
K
Je o .
of
N
Now
N
(iii).
if
space
< - ~ ,
4 such that the nilpotent
to
IT
and the
is abelian
is a symmetric
bounds
r\N IKI
2
nothing
his goal with a guide
remarks
of
and the part
is 2-nilpotent
N
and
(5.1.6).
curvature model
has to be constructed Ed(M)-2
could
only
assumptions
space
after parts
moreover
information
the universal
arguments
it seems
allowed
(1.1). In the almost
in the proof; so weak
said about
different
the proof;
clear
'''''·~''''''~1'--
are at first
have been
so widely
through
become
suitable to a given
11
,,".
2
_a
can be taken
read.
I
2
curvature
subgroup
prior
inequalities
achieves
~.
of
the proof.
the manifold
earlier
K
is sharp:
G = KAN
flat case the nilmanifold curvature
<
of the isometry
about
3.5
J subgroup
-
has the precise
decomposition
and
on the horosphere]l
r
of
2
_b
4.
rank one, then
helpful
radius
is abelian
[11 2
a nilpotent
of nilpotency 2 -1
estimate
and commutativity)
C TIl(M)
that the last statement
Gromov
1.5 and from the commutator
radius
r
'
compact
-1 < K < -a
with
TI2/4 '"2.47 -flat.
E < En . If the injectivity then
IT contains
degree
from
is
o .
-
(~)1/2d(M) En
Then
plane
a . In particular
in 5.5.
(Injectivity
be E-flat,
1.5.2 Corollary
Note
is given
3
> 2-n
Let
the projective
<
Ixl
that covering.
that it may be
unavoidable of the proof
that some of have
been
P. BUSER AND H. KARCHER
of
(2.3). Geodesic
T M
P
can be multiplied composition
loops
of the corresponding
controlled
by curvature
§ 3. This chapter berbach's
shorter
than the maximal
(2.2.3) in a way which holonomy
motions
what
Consider
Gromov
calls
the following
under
strong
priori
8
rea)
p
First,
8)
in the ball This
A certain
p
(3.1); den)
induction
parts
p ,
and
rea)
and
IIr (a) II denotes
the
of § 3 is:
p » d(M)
some
rp
are trans-
(also a
~ It(a) I ' Ilr(a) II < - p
satisfy
r p with very small rotational are relatively densely distributed
a €
which
(3.2) .
P
translations"
iterated
at
3.5).
T M
in
is used
commutators
set of generators
commutators
a E
loops
of Bie-
tea)
exists
such that the loops
of radius
d-fold
(3.4.2,
translational
set of "almost
trivial
An
parameter
have
there
such that loops
is selected
«<
part
is
motion
. The achievement
assumptions
above)
an adjustable
id € O(n)
from
curvature
bounded
the
of the proof
set of geodesic
:= {a; It(a) I < p, Ilr(a) II < O.48} , where p lational and rotational part of the holonomy
of
of
with
(2.3.1). The error
"imitation
r
distance
rank radius
compatible
and by loop length.
contains
theorem".
is almost
in fact have
r
for
does have
p
is an a priori
over wordlength
to show that those
bound
loops
a rotational
trivial crucial
(3.3) .
<
part
d(n)-fold
with
in
iterated
for the proof.
(3.4) and a self-improvement
(3.5) complete
the
argument. Knowing
this much
free nilpotent group
r
subgroup
of equivalence
the factor
rp
about
group
G
is still
TIl (M,p)
of
classes
= r\TI
mod
estimates
of § 2 are repeatedly
<
P
; but since
S
the loops
(3.6.4) - which
index
is shown
p
torsion
form
in 4.6.5
of
G
a
to be
already.
1
the enormous
elements
r
the finite
p (M,p) - one can see the properties
Note that
§ 4. with the results
far from having
curvature
assumptions used
for loops
of § 3 one should
of a slightly
arise
deformed
12
of lengths
rp
picture lattice
since
in
the definitions up to
p.
as consisting T M . After p
and
of the
tedious
error
THE THEOREM
controls,
an induction
produces
exactly
n
All loops
in
generators
Yl' ... 'Y
for
r p
n
with
the additional
property:
n
some suitable
R
generates
an
as subset
(4.6.2).
be written
€
. (4.5). W~th 1 ~ J ~-1 the set of geodesic loops r C rp R torsionfree group r which contains
(1. €:I';) and ~ (d(M)« R «p)
(abstract)
can uniquely
[Jy.,y.
y111 .. ·y ln
as words
rp
nilpotent
r
R structure
really
of length finite
~
index
parts
R
In a final
yields
of the loops
in
loops
r
R
in
about
different
into
and that all relations (i.e. among
information
represent
subgroup
step one shows
<
classes
; the proof
are fairly
of lengths
the fundamental
homotopy
ITl(M,p)
ITl(M,p)
that the discovered
dense
group:
and
r
All loops
embeds
as a
uses that the translational BR C T M
in the ball
are generated
5d(M)
algebraic
from
(3.6.3)
P
"short"
relations
) (2.2.7).
§ 5. The multiplication
in r is shown to be polynomial in the exponents 11 ln etc. ) (5.1 ) and by extending this polynomial Y1 •. ·Y product from n n to IR one embeds r as a uniform discrete subgroup in a nilpotent Lie
(of z'n
group
N
to which exp
Y
(reproving a finite
Malcev's
covering
M
Only now do we have
is to be compared!
the model
We find
exp-l from balls around the "lattice points" yo p e yp N which are compatible with the two actions of r on
0
into N
(5.2); the local maps
of
§
other
result).
of
are interpolated
8) to a r-equivariant pinching
applied
to
maximal
rank
M
theorems
differentiable
- standard
and to a suitable for
F
and completes
(with the center map
curvature
F
left invariant the proof
13
:
M
control
+ N
local maps
M M
(y E
. Finally
(5.3) on
(5.4).
f)
resp.
of mass
of geodesic
metric
of 1.5
r\N
on
averaging - as in
constructions N
proves
2. Products
of short
geodesic
In this
section
defined
(2.2). It determines
assumptions
(2.2.7).
the easily loops,
the Gromov
of this
product
product
group
controlled
volume
In the almost
flat case the commutator
Usually
of the fundamental
we work with
manifold
M;
curvature
bounds
IKI
S
2.1 The many
group
curvature
sometimes K
S ~ . more
loops
at
geodesic under
of loops
of the holonomy
(2.3, 2.4).
lemma
(2.4) allows
IKI < A2
convenient
important
is compared
curvature
to the
S-flat
be a Riemannian
a point
p EM.
und upper
case
section.
p
The lift of the Riemannian M
proof.
Riemannian
lower
metric
to
T M .
P Let
(2.5.3).
to find nilpotent
for a compact
in the next
with
of the
applications
of Gromov's
to distinguish
The specialization
is
(2.5.2) and the
estimate
point
loops
mild
motions
Immediate
of negative
- the starting
it is more
S
short
manifolds
bounds
becomes
: 0
S • d(M)-2
2.1.1
for compact
motions.
already
are the Margulis
corresponding
subgroups
group
and the commutator
technique"
bound
holonomy
for sufficiently
und commutator
is curvature
"discrete
and their
the fundamental
The product
computable
the error
loops
manifold.
Then we have
Assume
curvature
from 6.4.1
bounds
for the maximal
IKI
<
A2
rank radius
and fix of
exp p
(i)
In a ball metric gives metric
Bp
lifted
of radius from
of
p
<
'IT
A
•
•
A-1
-1
via
T M P Ap sinh Ap • d(v,w)
Length
Any closed
'IT
around 0 E T M we have the Riemannian p exp-1 . If v,w E B , Ivl+lwl < p then 6.4.1 P p comparison between the lifted Riemannian and the euclidean
M
the following
(H)
2.1.2
>
r max
decreasing
curve
c
at
<
Iv-wi
Ap sin Ap • d(v,w)
homotopies. p
of length
15
I!II!I!II!I!I!I
<
<
p
can be lifted
via
to a
P. BUSER AND
curve
c
in
T M . By continuously p
H. KARCHER
replacing
longer
and longer
arcs of
c
decreasing
homotopies
N
C
by geodesic from
c
segments
Ivl < p-d(M) t.+ exp tv
geodesic
2.1.3
<
t
c
<
ray
"natural"
[O,lJ • c(l)
be the closed
1)
d(M))
by joining
with
curve
point
Proposition. vET
p
(Existence Ivl
M
is obtained exp v
from the geodesic
by a shortest
Then its lift and
c
provides
exp w = p . This and
a
(2.1.1)
<
of many
p - d(M)
loops)
, there
is a geodesic
loop
a
[O,lJ + M
with
Or refo~ulated tangents
in terms
of geodesic
Ap sinh Ap • d(M)
of 2.1.4 with
loops
at
p
a:=
.A~ A p •
SLn
are a-dense
d(M)
: The initial
in the ball
Bp-d(M)
around
p Definition.
in a metric such that Among
2.1.5
A discrete
ball
B
d(v,w)
<
the geodesic
fundamental
Let
p.
< d(M)
d(v,w)
<
2.1.4
which
v € Tp M ,
for any
the
p
OETM
length
. In particular,
the endpoint
to the initial Ovw
triangle
For any at
, let (0
(S
geodesic
proves
(fig.) one has
to the geodesic
of radius
p
D
of a metric
p , if for each
space
v E B
is called there
a-dense
is a
wED
P
a loops of 2.1.3
fairly
short
ones
suffice
to generate
the
group:
Proposition. M
subset
be a compact
(Short generators Riemannian
for
TIl(M,p))
(or Finsler)
16
manifold.
For each
n
>
°
the
PRODUCTS
fundamental
group
~ 2d(M)
<
Proof.
Any closed
2d(M)
Thus
c
which
generate
curve
points
is represented
is particularly
there
1. E l
sufficiently
2.2.1
under
2.2.2
Proposition. at
by geodesic of closed
decreasing
of
• These
of length:
of lengths:
homotopies (2.1.2);
max only finitely
are eliminated
of
into arcs of lengths:
segments
curves
+ n < r
and hence
2d(M)
to geodesic also,
many
+ n ,
loops.
in this case,
loops
exist
by now choosing
of loops
homotopies
There
at
rank
p
is called
radius
of
exp
max short homotopy
will be called
is exactly
short,
r
one geodesic
loop in each
with
n
if each of its . Equivalence p classes.
short
homotopy
p.
definition
of "short"
On the other
ments
pointing
2.2.3
Definition. and
and assume
S lal
namely
S
*a
class
of at least would
can be lifted
(Gromov's be geodesic
lsi <
the curve
r
product loops
is the unique
in
If there by the
T M with fixed endp loops are radial seg-
geodesic
p . Denote
S•
a
their
loops) lengths
be the product
followed
by
geodesic
loop in the short
S·
in 2.1.2. them which
a contradiction.
of short
at
. Let
between
of the two geodesic
directions,
max "a
of the curve
homotopy
to a homotopy
hand the lifts
in different
+
one loop is explained
be a short
a .
17
S" .
by
lal,ISI
used in homotopy
Define homotopy
n
d(M)
product
than the maximal
The existence
theory,
points
A homotopy
two, then there
a
p
2d(M)
and Gromov's
short
class
points.
loops
small.
is shorter
Proof.
by geodesic
can be subdivided
p to
length
(2d(M),2d(M)+n]
Definition.
classes
Let
via
easy if
2.2 Short homotopies
curves
at
as a product
are no conjugate
lengths
were
c
are joined
then are deformed
This
is generated
r > 2d(M) (see 2.1.1) then the loops max IT1(M,p) . (See also 2.5.6)
+ n • If
length length
The division
IT1(M,p)
OF GEODESIC LOOPS
P. BUSER AND
2.2.4
one obtains geodesic
(ii)
a geodesic
segments
triangle.
from 2.1.2
of 2.2.1;
every
loop and the Gromov
locally
isometric
bound
M =
lifts
T M as a group p formations.
of fixed point
2.2.5
(Restricted
If
proposition.
a,13,y
are geodesic
length
loops
-1 to T M p P decreasing homotopy by via
important
product
exp
in this case
homotopy
is represented
(see 2.3.1).
is short
by a unique
group
Tp M , exp p : T p M + M
acts on the universal
free isometries,
in
geodesic
in the fundamental
to all of
TIl (M,p)
and
a
K ~ 0 , then every class
is the ususal
metric
covering
13'
and
The natural
homotopy
product
TIl (M,p) . The Riemannian
a
is particularly
If one has a curvature
the sense
13 ~
If one lifts
(i)
Remarks.
H. KARCHER
is a
covering
the so called
deck
trans-
associativity) at
which
p
lal
satisfy
+ 1131 + Iyl
< rmax
then
a ~
*
(3) t: y
(a"¥.
y)
is a multiplicative {p} =: The constant loop -1 is 'the loop obtained from a by reversing
a
a
Proof.
The length
homotopy
a ~ (13
theory
* y)
statement
to
2.2.2
2.2.6 Notations. loops
(shorter
-1
assumptions
are short;
a'
completes
2.1.4, than
~ a
=
unit.
If
to
(a'
< rmax
the parametrization
and
then
1 .
are such that the standard
for example
21al
we have
(3) •
y
to
short
(a'"
homotopies
homotopies
from
from
(3) "If. Y . The uniqueness
the proof.
2.2.4,
rmax)
2.2.5
together
18
and 2.2.7 with
show that
the Gromov
the set of short
product
*
are closely
PRODUCTS
OF GEODESIC LOOPS
related to the fundamental group (for
p
TIl(M,P) • We emphasize this by introducing
< r
max 7fp
{a
1=
We write *-products
a
a geodesic loop at
p
with
lal ~
without brackets:
L: la.1 < r ~ max
(see 2.2.5). We abbreviate
a
-1 -1 f ... fa
[S,al [S,a]
(if k·lal < r max )
(if 21al + 21s1 < r max
is called a 2-fold commutator; m-fold commutators
ductively:
p} .
y
If
(k+l)-fold. Note that in this definition
.
are defined in-
0 an l-fold commutator, then
is a k-fold and
)
I
[y,oJ
is
it is not assumed that the lengths
of the _loops involved are so short that the associative law (2.2.5) can unrestrictedly be used; for example, if in 0 l' Y JI: 0-1 Y-1 we substitute
Y
=
y
S
-1
fat
S
¥
a
we require only that
to be defined and
2.2.7 Proposition. Assume
*
-1
y
and
0
a
and
are short enough for
(TI determines p 2d(M) < P <
S
are short enough for
[y,oJ
to be defined.
TIl(M,p»
r - 2d(M) max W(TI) be the free group of words in the elements of TI ; let N (TI) p -1 pop be the set of words a S y where y (the "short relations", one
Let
needs
lal + lsi ~ p+2d(M»;
W (TIp) which contains
let
N(TI) P
be the smallest normal subgroup in
N (TI) . Then o P :=
is the group presented
by
TI P
There is a natural isomorphism
which can be defined by mapping the word class of the closed curve
1
19
22ii"
w
~(w):= a •••.• a
m
P. BUSER AND
~.
The proposition
loops
and their
that the set
that
if only
TIl(M,p)
4d(M)
TIp can be identified
called
injectivity
s-flat
manifolds,
Gromov's
states
products
theorem
of
H. KARCHER
with
TIp is false
but this
<
is determined
by the short
r . It does not state or imply max a subset of TIl(M,p) . This so
in general;
fact is established
it is true only
for sufficiently
late in the proof
of
(4.6.5). -1
Proof.
Since
elements
----
of
N(TI)
are products
of words
a v a
of the form
p
with
V EN,
curves
a E W , it is clear that
o
M.
in
~
maps
N(TI) to nullhomotopic p clearly a homomorphism and,
A
Therefore
~
is well
defined,
A
because word
of 2.1.5,
wE
to show
surjective.
To prove
W(TI ) the closed curve p w w E N(TI ) by reducing p
c
=
injectivity ~(w)
of
~
assume
is nullhomotop
to the trivial
in
loop with
that
M;
for a
we have
a finite
number
of substitutions
... as ..........
c = ~(w)
The curve a = 0
o
word
<
w
0
< •••
(or vice versa),
y ...
<
1 corresponds
: [a,b] .... M 0
n
= b
of
[a,b]
to the restriction
~(ai)
We first
comes
show that words
which
where
with
a,S,y E
TIp and
a ~ S .
y
a subdivision letter a. E TI of the 1. p to one of the subintervals:
such that each of
c
cl[o. 1.-1 ,0.J 1. correspond
are
to further
mod N(TI ) the same as w. By induction p subdivision point s in one of the intervals
J·
[oi_1,oi
c(s)
p
20
refinements
it suffices
of
to include Join
[a,b] one more
c(s)
by a
PRODUCTS
minimizing resp.
c
I
OF GEODESIC LOOPS
geodesic g to p. Then each of the closed curves g. c -1 is at most as long as a. E TI and therefore [s,a.J • g
short
homo top to a geodesic
la:1 1.
since
Next to
let
=
s
o =
t
a
n = j
o o
< s < t
2
< -
la.1. I
t
2
c
t
o
c
1
By uniform
1 1
<
t
)-
min (rmax - 4d(M) (s.)
such that
-
{a.})
of
t
from
c
=
S
si
and
, injectivity geodesics
lengths
radius g ..
of
is represented
(s) , si_1 j where
n ,
_
M).
of length
as a product
s + ct
1 < -3
1.)-
c
homotopy
subdivisions
the curves
have
)
1.
curve
differentiable
we choose
a. mod N(TI ). 1. P
1.
-
by minimizing
=
1. 1.
(a refinement
< te 1.
i
a~'a:
this proves
max
J
,s
1
* ad. = a
ai
1.
< r
continuity b
a'.' . We have
resp.
1 , be a piecewise
< sL
<
a:1.
loop
+ 2d(M)
t + C (s.) , t. 1 < t < t.
and
ct
=
{p}
la~1 1.
+
i
p
1.
J,.
I [a _
Again
s
join the points
< 2d(M)
of the short
S
to
p.
Then each
curves
j
I
-1 gi-1,j
. [si_1,sJ
each of which
is short homotopic
to a geodesic
B ..
loop
1.)
We also
introduce
short homotopic
the geodesic
to gij+1
loops
• ct(s.) 1.
is the trivial
Bi,j+1
~ Yi-1,j
I
rt
I,. j'
y..1.) , t j+1
J.
IY· .I 1.)-
-1 g ...
21
1.)-
1
2d (M) + -3
Then:
I
Each
<
n
2d(M)
+ -31 n
which
are
of the loops
1.)
loop and we have
or
<
, lB ..
the short
Bi,j+1
relations
Yi-1 , J' mod N(TIp )
c
o
P. BUSER AND
since
only
associativity
W(TI ) p
in
SL,j+l
This
is true
into the affine
To each geodesic
loop
a
and proves
rotational
at
P
T M""
0
the translational
m(a)
We use the distance motion motions
2.3.1 Let
of
a
and
proposition. M
IKI ~
product
(The holonomy
a,S E TI -1 ' then
and is called
.the
a,
0
a(o)
of the geodesic
or of
S;f: a
Riemannian
A2 = so d(M)-2
(e.g.
a
loop and is called
a. group
from
(7.3) to compare
to the composition
the holonomy
of the holonomy
S·
be a complete
p,?
(6.2.2)
o
on the orthogonal
of the Gromov
around
r(a)
at the endpoint m(a)
motion
x + t(a)
or of the loop
a(l)
part of
its holonomy
translation
o
vector
mod N(TI ) P
P
r(a)
t(a)
is the tangent
immediately
T M
P
of the motion
and
this and
w
we associate
is the Levi-Civita
part
with
holonomy
m(a) (x)
r (a)
gives
i
over
S . oJ
m(a)
where
are involved.
Yoj
o, ... ,e
j
32 n
+
induction
•. , Soj+1
for
2.3 The map
2 4d(M)
of length
curves
H. KARCHER
S
*a
map is almost
homomorphic)
manifold
curvature
with
in the almost
is defined
bounds
flat case).
Consider
loops
and
A (i)
(ii)
d(r(S)
It(m(S)
oda),
0
m(a))
r(6*"a))
- t(6 *" all
22
2
<
A
<
A
0
2 0
It(a)1
o
It(6) I
It(a)1
0
It(S)1
(It(a)1 + It(S)I)·
PRODUCTS
Proof.
By definition
the change
of
m
m(S)
under
the one from 2.2.4(i) homotopy
is
we have
S· a
The area
F
(r (6)
and apply
with
This proves
2.3.2
(i),
Remark.
reformulation lation. m(S)
0
m(a)
controlled
The proof
estimates
usually
its applications
follows
shows
that
obtained
<
IKI
then 6.2.1
• F
with 6.7, 6.7.1
similarly
<
A
2
g~ves
•
as
2.3.1
from 6.2.2
can be regarded
of Levi-Civita
result,
since
2.3.1
when
in section
.
as a convenient
and affine
parallel
the composition
and approximates
occuring
importance
we refer
m(S ~ a)
do not allow
commutators
to take
are computed;
trans-
of motions
up to curvature advantage
of
good commutator
2.4.
of an inequality
the reader
It is also the key result
(Commutator
be a Riemannian
be loops
from which
manifold;
p,
Gromov's
and 7.3
through
the following
arguments
start.
Re-
.
estimates) assume
which
curvature
bounds
either
(b)
or
at
can only be explained
to 2.5 to appreciate
from 2.2.6
(a)
a,S
in this
• It(S)1
(ii)
2.4.1
Let
curve
2
and definitions
M
(6.1.1),
a) • x) I
It(a)l
1.25
call notations
Let
S ~ a . We choose
estimates
the geometric
Theorem.
*"
is estimated
The estimates
are derived
2.4 Commutator
theorem.
< ~ A2
pinching
is easily
one has to control
to
It(a) I + It(S) I . From
of the path-dependence
errors.
cancellations
Since
and
S· a
6.2 and 6.7. The longest
• x , r(S
<
It is a typical
from
tensor
of the homotopy
F
m(S • a) • Therefore
length
r(a)
0
=
m(a)
a good homotopy
for the curvature
11:'
0
OF GEODESIC LOOPS
in case
23
(a)
also
satisfy
lal + lsi < TI/3A.
P. BUSER AND
Then the Gromov commutator
(i)
It([S,a]) I
A·
(ii)
is defined
:: Ilr(a) II· sinh Alt(S)
+
and trivially
[S,a]
H. KARCHER
~A2It(a)I'lt(6)1
and satisfies
i
+ lir<s)11• sinh Alt(a) I
sinh A(lt(a)1
+ It(S)I)
also:: 2A(lt(a) I + It(S) I)
d(r([S,a]),
<
[r(S),r(a)])
I
A2 It(a) I • It(S) I + ~ A2 It( [S,aJ) I • (It(a) I + It<S) I) .
(iii)
(If
It([S,a])
K
<
We include
two immediate
2.4.2 Corollary.
corollaries
(Simplification
Choose in definition lal ,lsi ~ (16 A)-l
(i)
by
1 resp. !. 2
.)
reference.
of 2.4.1)
<
1.006
c > 8 . Assume in addition
(1Im(a)11 • It(6) I +
(ii), (iii)
0.21 A( Ilm(a)11• It(S) I +
Ilm([S,a] )11
2.4.3 Remark.
replaced
be
to 2.4.1
, then
It([S,a]) I
<
can
for quicker
7.3.2 the parameter
The error term from 2.4.1
(iii)
5 5 3 resp. 6
then the constants
0
<
- t([m(S) ,m(a)]) I
~
2.03
simplifies
Ilm(6)II· It(a) I)
to
Ilm(6}ll' It(a} I}
Ilm(a)II' Ilm(S)11
In this form the result
24
can easily be compared
with commutator
PRODUCTS
estimates
in the group of motions:
:s
Ir([m(Bl,m(al]ll
We see that the homotopy
is standard
lI[r
:s
2l1r(allio
errors are under effective
to conclude
set of motions converge
OF GEODESIC LOOPS
from these estimates
with rotational
to the identity.
parts
<}
IIr(BllI .
curvature
that iterated
control.
commutators
and bounded" translational
Now almost the same follows
for Gromov
It of a
parts
commutators
of loops:
2.4.4 Corollary. Assume
in addition
an additional
(Convergence
of commutatorsl
to 2.4.2
IIm(alli , IIm
assumption
(il
(iil
on the rotational
0.49,
parts.
which is essentially
Then
IIm([B,a]l II
:s
0.995 min( IIm(alli , IIm
It([B,a]l I
:s
0.493
Proof. For the corollaries
1 r sinh
:s
r < - 1.001
if
simplify
(iil, The other inequalities
2.4.1
(iil
(iiil
2.4.1
f sinh
~
:s
0.084
and
(il which then implies
are restatements.
follow with the same arguments
we lift the closed null-homotopic
curve
J
as in section
a -1 °13-1 ° [B,a °aoB
via
2.3: If
exp -1 to p
~
[(3,0(]. eX· (3
25
~
(il with
r < _ 116 . This gives 2.4.2
2.4.2
and
(It(all + It(8lll
---P. BUSER AND
we obtain
T M
the geodesic
H. KARCHER
pentagon
with
Isl,lal,I[S,a11,
edge lengths
p
shown
ISI,lal geodesics
of length
distance these
triangles.
between
%
:
decreasing
translation
F:
in the figure.
It(a) I + It(S) I
homotopies
The curvature
(6.2.1) along
r ([S,a] )
and
the control
that the longest
curve
along
a).
and 6.6.1
A.
path
S)
which
start
result
gives gives
[S,a J
2.4.1 B(l)
a(o)
(6.1.1)
2.4.1
(i) introduce (resp.
along
a(l)
B
and
(ii)
2.4.1 -1 S·a·S·a
to
into
the distance
1\2
(6.7,6.7.1)
from
and use the
of Levi Civita
IIR II
(dotted)
surfaces"
translations
from
of
triangles
"ruled
to estimate
inserting
to obtain
translation
Rauch's
;
by two
dependence
is used
in our homotopy
a
by parallel
controlled
(Ial + lSi»)
desics obtained
to span
(6.2.2) of affine
2(lal + lSi) . Finally, (resp.
into three
this homotopy
length
this pentagon
from 2.1.2
[r (8) ,r (a) ]
+ I[S,a]l·
(2Ial·ISI
Similarly
We subdivide
(iii) ; note -1 has
two more
geo-
) in directions
(resp. of
B(o)
(6.4.1) gives
A·
d(a:6(1),a(1)
<
IIr(S) lI·sinh
A·
d(§';'a(1),S(l»
<
IIr(a) II· sinh Alt(8)1
Alt(a)1
'
gives
<
d(S(1),a(l»
The sum of these
distances
~A2
It(a)1
• It(8)1 sinh A
is the bound
2.4.1
(i)
(It(a)1 + It(8)I)
for
•
It([S,aJ) I .
2.5 Applications We discuss
examples
same purpose,
namely
and in particular proves Lemma with again
of global
follows
to illustrate
of loops with
from 2.4.4;
to obtain
bounds
which
small
it implies
controlled
of the discrete
estimates.
A simple
rotational
a lower
- 1 : K < 0 • Finally,
curvature
were used by Gromov
the power
of the commutator
the existence
curvature
results,
volume
information
26
bound
the volume about
group
volume
parts;
for the technique
comparison
then the Margulis for compact
comparison
manifolds
is used
the fundamental
group.
PRODUCTS
2.5.1
Proposition.
Assume
curvature
n-
=
o.9 9n
OF GEODESIC LOOPS
(Existence bounds
of small
IKI ~ A2
rotational
and choose
parts)
n,
0
<
n
2
0.49
Put
. /- n/2 . , m = 2 ~nt (2ITn) . Then: For each geodes~c loop a IT 1+n/2 . a 2 8A • (n/2IT) there ex~sts k < m such that
at
p
I I
of length k IIm(a )
II
<
n .
J'roof. First,
with
=
c
(Use
8
7.6.1
in 7.3.2).
(ii)
The area of the natural
choose
homotopy
2
k
m
such that
(2.1.2) from
Ilr(a)k II
a· ..• ·a
to
ak
=
2
n
a
* ...
*a
."..---------- ...
is bounded homotopy
by applying
6.7, 6.7.1
<
F
Now 6.1.1
to the
(k-1)
and 6.2.1
%
2 lal •
(1+2+ ...+(k-1))
2.5.2
the assumption
Corollary. curvature
geodesic
the
loops
(Margulis
Alai,
S
at
Aisl
2
"'2
8 A mlal <
n 2}
we obtain
<
n .
Lemma) IKI < A
bounds a,
.
m
1 (1 + 256)n
Assume
into which
give
<
and using
triangles
can be decomposed:
p
2
Which
If
a,b € IT1(M,p)
are represented
satisfy
<
(n
27
dim M)
,
by
-
P. BUSER AND
subgroup of
such that
k,l ::2(3.6)n
then there are
H. KARCHER
a
k
TIl(M,p) •
Proof. With 2.5.1 choose
k
and
1
2.4.4 one finds a number
d
1 k Ilm(a ) II, Ilm(13 )II : 0.49
such that
From the existence of a shortest nontrivial
and
generate a nilpotent
loop at
p
and the estimate
such that any d-fold commutator
131 is the trivial loop. Now the commutator
(2.2.6)of
a
k
identity
[a,bc]
implies that any d-fold commutator in the group generated by TIl(M,p)
is trivial, since d-fold commutators
be an n-dimensional
<
bounds (i)
K
<
0
compact Riemannian manifold with curvature
and diameter
There exists a point
d(M)
E M
q
such that the injectivity
q
r
q
vol(M)
(ii)
4-(n+3)
>
P
> -
n1 voleS n-l )Pn
,
note that this bound depends only on the dimension,
(iii)
in particular
not
max K For all
r p
~.
radius of
is
eXP
on
of the generators
(Volume bound from below)
2.5.3 Proposition. ~
in
of products can be rewritten
as (much longer) products of at least d-fold commutators ak,bl
Let
l
ak,b
p E M
>
the injectivity
TI voleS n
n-l
)
(
satisfies
radius
c:inhPC!IM)
)
n. sinh d(M) •
vol (Sn)
The last inequality says that in this general situation one has a
similar phenomenon
as for surfaces of constant curvature
injectivity
(for surfaces: a very thin handle) can only occur if the
radius
diameter is large.
28
- 1 : A very small
... PRODUCTS
OF GEODESIC LOOPS
Proof. (ii) follows from (i) since exp ---q given bound is the volume of a ball of radius (iii)
follows
shortest
from
closed
(ii)
by estimating
geodesic
c
on
is expanding P
in
the normal
Musing
6.3.8
(6.4.1) and the
T M •
q exponential
(see [
16
map of the
] for more
details) :
<
vol(~)
length
n-2
(c) • vol (S
d(M)
f
).
sinh
n-2
r'
cosh r
dr
o <
To prove
(i)
of compact
of
a
since
Z
(i) shows
of negative
r
subgroup of
has an axis
If
B
and
that
B(c)
the axis of"
r
of
the elements
of
M
(ii) and the following
Z
Z
~
with
no fixed points.)
c
and
a,
of the deck-
axis
c
geodesic
of the axis).
in particular
Now every
nilpotent
N
and
B
over
the elements
of
results
2.5.4 Margulis
lemma
be loops and
a
By induction
standard
a
Therefore
they commute.
These
then
subgroup
(Preismann):
hence
it follows that
of
a,B
c-
a
then
=
B(c)
c
=
B a B
so that
-1
c
is
B
group
Let
to
map is an isomorphism
(ii) For a,B E r assume that a and commute, i.e. have [B,a] -1 -1 axis c (by (i». Then a(c) = c and B a B a (c) = c imply -1 -1 a(B (c» = B (c) so that B-1(c) is invariant under a , i.e. (uniqueness
c,
is a
and - if restricted
is the axis of
under
a
one invariant
a common
cyclic
from the geometry
Each element
(The restriction
have
is invariant
r
facts
exactly
is discrete
and is therefore
commute
M
a E
since
of
curvature:
covering
The set of all
subgroup
a translational
N
2.2.4
has in the universal
the "axis" cyclic
recall
manifolds
r
group
1 n. n-1 (c) • 2rr • voleS ) • slnh d(M).
length
B
at
have
are combined
for negative q E M a common
with
a cyclic
the iterated N
have
commutator
a common
the commutator
B(c)
=
c
subgroup, sub-
axis.
estimates
to give
curvature.
.
.
whlch
satlsfy
axis,
29
'II"!'!'
generate
a common
hence
Ia,I I B I generate
<
2'4
- (n+3)
a cyclic
= 2p ,
subgroup
of
r.
P. BUSER AND
by 2.5.2 suitable
Indeed,
of the deckgroup c ; clearly To finish
c
take
exp
there
a
a
of
a
a(y(o)
, k(y(O)))
2.5.5
There
for all
<
and
y
T > 0
with
q
have
a common
axis
used
volume
in Gromov's
angle
comparison
basis"
trick
paper.
to produce
cyclic
k E H
geodesic
=
p
p
k
0
M and
covering subgroup
which
which
has
is orthogonal
to
we can have
for some
ko
E
is
161,lk
for geodesic
o i.e.
which
I
>
p,
-
< 2p
since
otherwise 6
there
¢H
and 2.5.4 imply
that
6
6 € H , a contradiction.
gave
2.5.1 via 7.6.1 is frequently
it is combined
with
Toponogow's
(6.4.3) and with the "short
triangles
implied
Hand
(Y{T))) = 2p .
yeT)
a),
argument
curvature
on
a(o)
161 < 2p . Then on the one hand
In the following
theorem
universal
be a geodesic
-
(namely
comparison
choose
> a(y(T),
o The simple
< 2p
lal
such that we have
q = exp
at
at
6 . closed
If
N
geodesic
6.5.1 implies:
therefore
radius
and
subgroup
invariant
be a shortest
many
(2.5.5); on the other hand
of
k
6
a
isometric
finitely
k E H : a(y(T), k(y(T)))
be a loop
because
2p,
is a smallest
Now the injectivity would
For only
a nilpotent
a common
be the maximal
as axis and let
a(o) = yeo).
at
a
to prove.
Her
generate has
axis of
as the locally
(2.1.1, 2.2.4 (ii) ). Let the lift
recalled
(i) let
is nothing
: T M + M p
p
as we have
of 2.5.3
61
k a ,
powers
is also the common
the proof
lal ~ 2p
If
which
H. KARCHER
information
about
the fundamental
group.
2.5.6 Proposition. (i)
Let Then
(ii) Let
~
(Number of generators
be a complete
the fundamental Mn
Riemannian
group
be a compact
and curvature
K > -A
s
TIl(M)) of nonnegative
can be generated
Riemannian 2
for
manifold
manifold
by
with
, then the fundamental
<
2
s
diameter group
in
0
(3 + 2 cosh AD) ~
elements.
30
S
curvature.
1n 2 51 0
elements.
d(M)
< D/2
can be generated
by
PRODUCTS
Proof.
Represent
each element
at
and call
lal
p
OF GEODESIC LOOPS
TIl (M,p)
of
the length
of the homotopy
{a1, .•. ,as}
Pick a short basis
by a shortest
TIl (M,p)
for
(i)
a1
represents
a nontrivial
homotopy
(ii)
If
a1' ... '~
have
been
a homotopy group
class
already
of minimal
generated
>
a.~
to the universal
or
a.J
11 la.~ I , la.1 , la.a.J ~ J to
la.a~11 ~ J
were
.
max{
as follows:
class
of minimal
then
~+l
length.
represents
in the complement
of the sub-
I a.~ I , I a.J I} ,
not chosen
M
covering
a
{a1' ... '~}
by
(t)
(otherwise
loop
class.
chosen,
length
geodesic
of
M
minimally).
and obtain
By Toponogow's
is not smaller
theorem
a triangle 6.4.5
than the corresponding
a.~ , a.J
We lift the loops with
edge
the angle
angle
~
lengths opposite
in a triangle
with
the same edge lengths
In case
Now there
a)
in the euclidean
plane
b)
in the hyperbolic
a)
the inequality
(~)
<
la.~ I
plane
K ~ 0
of curvature
implies
2
+
la·1 J
2
_A2
~ > 600
if
K > _A2 .
since
- 21a.~ I . la·1 J
cos ~ •
are at most
1 +. Si~ (
unit vectors
with pairwise
the endpoints
and on the other
hand
ang}es
of these
T ~) ~ ~
vectors
contained
since the balls of radius n in R are on the one hand
in the ball of radius
31
....
n
2" ~
s~n
around
if
(1 + sin
sin
T~
disjoint
T~)
around
P. BUSER AND H. KARCHER
the origin, vectors.
so that
If one takes
contained
21 ~ ,f,) ~
n.
To get a lower bound sufficient
.f an 1.mprovement 1.
,
for
¢
small.
as in case
a)
cosh A cos ¢
in case
generators
arbitrarily
gives
<
hence
1 + sin
-2 1
cosh AD 1 + cosh
2¢ <
} ¢)~
,f, ~
b)
which
recall
(~)
I a.1.I •
1fD
3 + 2 cosh
AD .
32
of
then these
are
gives
the bound
• loS not too small.
cosh A
A
for the number
1-
D
from 2.1.5
=
Then the cosine
I a.1.I •
bound
of the small balls, 2
of length:
(using
sinh
<
is an obvious
(1 + sin
of radius
to consider
may be chosen
ratio
only the inner half
in the ball
. -2 2 (1 + S1.n
geometry
the volume
2d(M) formula
sinh
cosh A
A
I a.] I
it is
of hyperbolic
and Toponogow's
I a.] I -
that
+ n , where
theorem)
n >
0
3.
Loops with
small
The following in Gromov's
rotational
fact from the flat case
theorem
1.5) should
The translational of a compact subset
parts
subgroup
with
in mind
(~Zn)
flat manifold
of
(the Bieberbach
be kept
~
theorem
throughout
of the deckgroup
can be characterized
not too large rotational
s =
1.2 or
0
this section:
TIl (M,p) as the
parts,
more pre-
cisely
This
Translations
in
fact follows
from Bieberbach's
classical
proofs.
TIl
Ilr(y) II
In GromoV's
theorem
approach
to the almost
not too long a loop
has its rotational
follows
e
IIr (y) II :::
case and very With
away while
nilpotent fairly
3.1 adapts
=
din)
P
TI C.
1 n(n+l)
(3.02) ~
index
p
a length
in
rotational
since of
Ilr(y) II :::0.48 arbitrarily
(1.3) with
small E
then it in the flat
very
TIl (M,p)
small. subgroup
a torsionfree
by generating
r
from
parts.
steps:
trick
(2.5.6) to
parts
<
elements.
role motion
in the flat case the lattice
in the following
and rotational
which
P
small
basis
a central
If the holonomy
part
is S-flat
no role in the
flat case one can construct
of finite
proceeds
-
3.2 selects in
r
loops with
it plays
flat case:
can be taken
(3.4.2) one obtains in the almost
the short
< p
length
e
,where
if the manifold
subgroup
short
This section
B
small
this result
right
y
but it plays
however
it has a generalization
1 2
<
r
,the set of loops at p 0.48 ; any short basis has at most
p
of
-
such that the translational
have very
small
rotational
that at least
those
elements
part
parts
are fairly
of those
dense
loops
in the ball
T M
P
3.3 shows iterated
commutators
3.4 proves commutators
have
by a length in
their
controlled
y
rotational
induction
are triviaL
33
e rp parts
which much
that all
have
trivial
smaller
d(n)-fold
than
d(n)-fold
0.48.
iterated
P. BUSER AND
3.5 collects short
the consequences
the previous
an estimate
for the index
arguments
3.1.1
(3.4) for the multiplication
of
For any subset
Note
Ac.1T
-
(see 2.2.6)
P c. -
A
lu
3.1.2
y)
E(A)p
Definition.
define
J
61 2
(LoOpS with
rp
rp
Definition.
=
therefore
This
part)
~
(2.2.6, 7.3),
0.48}
p
for a suitable
(3.4.1) .
(Short basis)
bound
nilpotent
3.1.5
lal)
the parameter
34
c
d
that the largest
distance
, 0.48. p
properties
for the number
translational
the following
max(d(A,id)
determines
(3.1.2) will be defined
it is necessary
and the largest
we use here
choice
for example
(3.1.1).
will be proved
For the proofs
cW>.,id)
(2.2.5);
and such that the a priori
in 3.1.2
by
p , then
small rotational
{u , ... ,u } for rp d 1 (ii)) in such a way that it reflects
allowed
inductively
associativity
A short basis
be proved.
the set
Ilr(u) II
rp
3.1.3
these
does not imply
rp
Eventually
flat case
group;
generation)
that one has to be care full with
*'
in the
in the deck
P
If
(6
for example
subgroup
flat case.
(Length controlled
1 {1} u A f.I A-
(E)
to obtain
of the lattice
basis
Definition.
(il
information
are the same in the almost
3.1 The short
¥
(3.3) and
loops.
3.6 applies
u
of
H. KARCHER
parts
have
function
IIA II
in 7.3,
below
«i),
rp
(3.1.4)
of
of its elements rotational
parts
the same weight;
on the group
i'i(A.,B) =
c =
can
°A~8,
of motions
1 IIA- B II .
and since we used
SMALL
c > 8
in 2.3
ROTATIONAL
and 2.4 we have to
(K)
for
the
present
Now the
assume the
<
Ap
PARTS
curvature
0.06
arguments.
elements
of the
short
basis
are
inductively
is If
(ii)
{a1 ...
(3.1.1) Note that also
d
a short
chosen
is
finite
basis
II > 0.48
3.1.4
Proposition. a.
hence
ra.,y] ~~
E
YE
we have
rp
Proposition.
a priori
the
p
rp,-<{a1,
€
ai+l
r
in
..
.,aJ>p
minimal.
Furthermore
is
elements
have
rp
short
basis)
holds
Ilm([ai,y])
II :
moreover, have
The number
d
2.03 Ilm(y)11 •
since [ai,y]
Ilm(a.)11 ~
E
of elements
({a , 1
...
is
minimal
Ilm(aJI in
p •
,a _ }) i 1
in a short
<
Ilm(aJI
basis
(3.1.3)
has
bound
By construction
<
den)
Ilm(a. )11 < ~ -
:=
otherwise
we get
the
int(3.02)
fn(n+l)
Ilm(a. 1) II < 0.48 ~+ -
1 Ilm(a. ~ a-:- ) II J ~
(1()
since
finite.
of the
we also
d
Proof.
is
minimal
then
1)11 is
~+
since
(3.1.2);
<{a1, .. ·,ai_1})p
3.1.5
Ilm(a.
rp
since
rp
(Nilpotency
and all
~
From 2.4.1
the
for
have been selected,
so that
selected:
•
Proof.
rp \
c rp
,aJ
is
Ilm(3)
For each
bounds
> -
cont:tadiction
implies
35
Ilm(a.) J
. Also,
if
II '
a. E 1 J JP -1 aj 1: ai e,
,a _ > p j 1
i < j
, then
,
P. BUSER AND
H. KARCHER
(2.2.5)
hence
(3.1.1) .
a. E J JP
21(m(a.) J
With
m(a.)
0
-1
, m(a. J
1.
-1 a. »
:t=
<
1.
(2.3.1,7.3)
and
Ilm(~)11 <
Ilm(a.) J
0
m(a.)-111
7.6.2
pairwise
Remark.
of 2.5.2
that
possible
since
words
One would
commutators
of which
strong
restrictions rather
n < 0.48
like to use 3.1.4
the inductive
the lengths
on
large
d(n)
euclidean
proof
and 3.1.5
r
in
in 2.5.2
are not yet under
Ap
; the proof
denseness
m > 10
will
P
and assume
curvature
(K)
Then there
vanish.
which
along
This
the lines
is not yet
to use associativity
control. finally
of loops with
be adjustable
7.
So far there
succeed
small
parameters.
w > w(n)
n
only
were
in no
for carefully
rotational
parts)
Put
2014dim
SO(n)
bounds
2wd(M) °A
exists
to prove
requires
dim SO(n) 3 + 2(
L
motions
p
(Relative and
are at most
' Ilm(a. )11 1.
last inequalities.
all d(n)-fold
chosen
Let
that there
these
3.2 Proposition.
1 12811m(aJI
> maxi Ilm(a.)11 J
shows
satisty
(t)
we get from
1.
Finally
3.1.6
t
a number
Po
=
<
n-m-L-1
po(n,m,w)
such that
(hence
(i)
36
Ap
< 0.06 o -
for
3.1.3).
SMALL ROTATIONAL
(H)
For every
v E. T M
(3.1. 2)
Ivl < P
with
P
PARTS
o
(1
m
there exists
such that
<
T] ,
<
It(a) - vi
Remark. For the final theorem one fixed choice of all adjustable will be sufficient.
parameters
However it explains the structure of the proof much better
if we show how the parameters
enter the arguments and fix them later. The
constant
Po
win)
insures that
we did not put that assumption m
will be large enough for the proof of 3.6.2;
into the relative denseness parameter
does not enter the present arguments as a large parameter.
m,
Finally
sine. will
T]
be a very small number such that for the purpose of the proof of 3.3.1 the rotational parts of the loops selected in 3.2 are negligible.
Proof. With 2.1.3 we find for any that to
P satisfying (i) a loop a'e.1f such P It(a') - vi ~ 2d(M) ; but IIr(a')II may not be.~ T] • We shall modify -1 1 a ; a'l'(a") such that IIr(a)II ~ T] and It(a) - vi ~ m-1 P
Define (¥)
Pi
.; .
i 2 2w d(M)m +
(i;l, ... ,L-3) . Then we have
One of the numbers
Pi (i;2, ..• ,L-3)
property:
a' E 1f Pi
For each
such that
d(r(a') ,r(a")
Now 3.2 follows from
(*)
It(a)-vl
<
there exists
a" € 1f Pi-1
21f
< "7.
if we choose
and modify the above a' -1 a ; a'l( (a") ; since then (f)
has the following
Po
(For which
T]
to be that
Pi which satisfies It(a')-vl < 2d(M) ) to
It(a')-vl + It(a")1 :: 2d(M) + P - ::: i 1
IIr(a) II
+ d(r(a') <
Finally, assume that
< (~)
T]
0
r(a")
-1
, r(a'~(a")
(with 2.3 and 3.2
is false. Then there exist loops
37
»
-1
(K)
).
a'
•
• P. BUSER AND
(i=1, ... ,L-3) is at least
with
d(r(a.),r(a.»> 2rr7 11 (ifj) , since for each i there 1. J 2 TI a whose rotational part has a distance> 7 11 from i in TI • Because of 7.6.1 (i) there are less than
one loop
all rotational
H. KARCHER
parts
Pi-1 2 (2) dim SO (n) elements in 0 (n) wi th pairwise 11 defined large enough to produce a contradiction,
3.3 Small Recall
parts
and trivial
that we are proceeding
a much
smaller
Under
rotational
an additional
removed
3.3.1 Let
rotational
to prove
part
(~)
which
> ~
11 and
proves
L
was
(~).
commutators.
that the loops
0.48
than
assumption
in 3.4 - we achieve
d-fold
distance
, the bound
on iterated
r
in a suitable in definition
commutators
have
P
3.1.2
- which
will be
this in the following
Key proposition. 0 <
e
+
<
be an adjustable
1 11 =3'82.1
-den)
curvature
bounds
Each
rp
y E;
(~)
with 3.2
which
The d-fold exists
has a much
(K) and choose
Let
(d:' den»~
rotational
part
be the largest
T M = E ~ E-L such that p angle pick vEE,
J- .
I\r(a)1\ ~ 11 and we can derive
It(a)-vl
a positive
d-fold
commutator
Assume
first
E
iterated for all
m(y) (x) = C x + x
-8':=IIC II~ 0.48
P = Po
the
from
from 3.2
11
in 3.2 as
3.1.5
. Assume
the
• Then we have:
satisfies
Ilr('()1\
Proof.
Choose
::!n(n+1)
den) =int3.02~
and is trivial
smaller
parameter.
<
commutator
rp
a €
than
[ ... [a,y] ,... ,y]
0.48
, namely
d d 82.1 - (n)
(2.3) be the holonomy rotational
angle
of
motion C
of
y
. Let
and decompose
is a 2-plane and CI is a rotation through the 3 E 4' p and choose with 3.2 a loop a such that
Ivi =
~ m~l p . If lower bound
..9
is larger
[... [a, yJ ,y ... ,yJ ' which
that one can even find a loop
38
than claimed
for the translational contradicts
a E
rp
with
part 3.3.1
rea)
in 3.3.1
of the (:1')
id
and
SMALL ROTATIONAL
o f tea) E E
PARTS
- instead of what can be achieved with 3.2; this oversimplified
situation already explains why the proof works. Disregard also the homotopy errors for the moment, i.e. compute the iterated commutator motions:
=
[... [r(a),r(y)], ...,r(y)]
Clearly, iterated commutators
id
fE,
t([ ... [m(a),m(y)],. ..,m(y)])
in the group of
!t(...d-fold ...)!
do not vanish unless
,p =
(2sin ",?)d. !t(a)!.
0
•
To explain more clearly how the a priori bound 3.1.5 and the control of homotopy errors with 2.4 enter the proof, we once more disregard homotopy errors. The result is then applicable to the Bieberbach
A =
0
case: Because of
there are no homotopy errors and the curvature assumtion
trivially satisfied for any
n ;
sition holds with arbitrarily pure translation. mental group
therefore the conclusion of the key propo-
8, i.e.
small
r(y)
=
Since in this flat case we multiply
id
and
m(y)
is a
loops as in the funda-
(2.2.4 (ii)) one does not have to wait until 3.4 to remove the
extra condition immediately
3.2 (K) is
3.3.1
(lO
but can use induction based on 3.1.4, 3.1.5
to obtain the Bieberbach
theorem from 3.2.
The following inductive inequality occurs several times in the present proof: If
(U)
k - y. µ ::::~+1
k - AXk :::: Yµ
(0::::
2A
<:
µ
o
<
y)
then
We now estimate the translational a1
=
a , ~
=
[~-1
'
m(~)(x)
Y]
parts of the iterated commutators
A
assuming
~.
x
+
a
k
and compute in the group of motions
39
."au
3
,I,
= 0
.
Abbreviate
' m(y) (x)
C·
x + c
P. BUSER AND
Since we know the action of E
X = X
C
on
H. KARCHER
E
we decompose vectors
X E T M p
+ x.J... and find
as
We apply 2.4.3 to obtain the bounds
i~+ll
::IIAJlolcl
+llcllol~l
::
Jk
lall + (2S)k-l
211Aliiolci
< p,
(U)
which are inserted in the previous estimate:
I{+ll
Once again
la~
Unless
I
:: 2 sin ~ I{I
(U)
implies
laE
IIAlll < nand -
1
I ->.!.2
p
. J 2 )d-l I a E I
>
(2 Sln
>
p 2~
~2 e
(with
1
0
(2
l
..,J2".) d-l
Sln
(2.1)d-d(n)
0
a e
('17 -
(2.1)d-d(n»
this gives the contradiction
la~1 > 0 . (It is
clear, that this argument is useless unless one can establish d < d(n) .)
3.3.1 (*) for
Finally, the inclusion of the homotopy errors changes only some constants in the above computation, provided one works with a suitable distance function (7.3): It will be necessary that translational important than rotational parts of size 3.3.2
Ilm(y)11
.=
parts of length:
n , therefore we take
max( Ilr(y)11' !lo It(y) I) . p
40
p
are less
SMALL ROTATIONAL
;~
--
i~e ,-of
have for the use in 2.4 the parameter
BAp
assumption this distance
2n ,
homotopy
c
which is implied
for the present
computation
one used in 3.1.3 for the selection Fbr the neglected
PARTS
n > B we require the curvap A by 3.2 (K). Note that the use =
does not interfere
of the short basis.
errors we have from 2.4.1 and 2.4.2
2( It(~) I + It(y) I) - 0.21A( Ilm(~)11 • It(y) I +
As in the
A
= 0
computation
we need bounds obtained
<
It(~+l)
I
2
with the
(~=
t(~»
Ilm(y)11 - It(~) I)
from 2.4 for
k
(2.031Im(Y) II) -llm(al)11
1.006 ( Ilm(~)11 - It(y) I +
Ilm(y)11- It(~) I)
:::(2,03 Ilm(y) II)k-l_ Ilm(al)11• 2.012It(y) 1+(1.006 Ilm(y) II)k-It(a ) l (U)
< (2.03 Ilm(y) II)k-l - p.
These bounds simplify
the homotopy
errors to
As before
we estimate the E-component of the translational parts (with k k 1I~+11l ~ (2,9-) IIAll1 replaced by Ilm(~+l)11 ~ (2.031Im(y) II) -llm(al) II) and find
~nce more using
(U)
we obtain
41
I
P. BUSER AND
H. KARCHER
. we assume contrary to 3.3.1. ~= Ii IIr(y) II > 8 2.1 d-d(n) At last , ~f
=:r
choice of our distance implies 11m(y)11 and also (with 2.03.3 < 4.2 sin Tl = ~ 2.1-d(n»
f,
This contradiction
to 3.3.1
3.4 Nilpotency
r
of
(]f)
proves
then the
IIm (a )11::: Tl hence 1
IIr (y)11< 8 2.1d-d (n)
P
The undesirable commutators
extra assumption
3.3.1 (*) on the vanishing of iterated
in the key proposition
3.4 ..1 Proposition.
will now be removed.
r Po
(Nilpotency of
Assume the curvature bounds 3.2 (K) with from 3.2. Let
{a1, ... ,ad}
r All
d(n)-fold
commutators
in
r
c <{a1,
3.4.2 Corollary. Assumptions
=
Po
(see 3.1.2).
P
moreover
The key proposition
P
be the short basis from 3.1.3. Then
(i)
(ii)
Tl from 3.3.1. Choose
P
...
exist and are trivial;
,a.}) J
P
together with 3.4.1 has the immediate
(Small rotational parts are very small)
as in 3.4.1. All loops
a E
satisfy
TI
P if
Ilr(a)11 <
0.48
then
Proof of 3.4.1 by length controlled
Ilr(a)11 <
8.
induction. We use the distance function
3.1.3 on the group of motions if the size of certain loops is compared to the size of the elements of the short basis; we use 3.3.2 for estimates of translational
parts.
42
SMALL ROTATIONAL
of 3.1.4
Because
we have
J
- J
The first
e
T
rp
since
element
a. J
<
0E
r.±1 [jS,a.±1] ,la.,S
for all
PARTS
from the short basis
)
r
{a , .. ·,a. 1} () 1 JP
a
of the short
1
r (j-1)
3.1.3
basis
(2.4.2).
Ilm([a ,y])11
_-:
P
.
commutes
Therefore
and for all
with
all
all powers
Y E r , in particular one can apply the key p k P k proposition 3.3.1 to all powers a E a n r . We conclude IIr (a1 )11 < 8 1 1 k+1 P P k+1 and therefore Ilr(a )11 ~ 38 (2.3.1); hence, if a t a ) then 1 1 1 ~
~a1~
~1
>
f
(11
commute
with
all
< >
<
r(l)
Assume
=
<)a1
[r(1) r
P'
(H1)
in
J
{l} , a E. r(l)
, P
(ii)
and we will
a,SE.({a , 1
Take
<
Ilr(a)11 ,llr(S)11
y €
a , ... ,a + 1 j 1 because of part
and their (i)
J+ to show
(~)
of 3.3.1
inductively (a)
and
( A/"
p
=
<
A ) -1
p
and
"r
for p
=
a
r
3.1.1
*
therefore
S , which
(j+1)
.
of
18
then
we assume
indeed (a)
also It(a
satisfy
and (b)
* S) I <
(a)
(H.) imply J follows. To
p
and
implies
(Inverses
cause
no problems
.
s~nce
r p = r -1 p .) that we have
1y for products
the inductive
rp
inverses
above;
is satisfied,
following
(b)
S E ( {a1,. •. ,a. 1} ) J+. P
using
and assume
8.
(b) : (a)
(H. 1)
*
nrp
that the generators
assumption
a
(Hj+1)
conclude ,a + })p j 1
...
for all
holds
=> Ilda)11 < 8 •
i < j
now for all
3.1.1.
have
P
P (\ r P . This proves
factors
of length
43
associativity
< p , therefore
(2.2.5)
certain-
-------
P. BUSER AND
[y,a",S]
[y,a]
H. KARCHER
*"
*
([a, [y,S]]
[y,S]) .
Each of the three factors on the right is already in To see that the product translational
is in
r(j)
because of
(a).
we first estimate the lengths of the
r(j)
parts with 2.4.2 and 3.3.2, assuming
(H.)
J
It( [y,S] ) I
1.006 (IImW)1I - It(y)1 + 11m(y)1I - It(S)I)
<
-
1.006 (0 p + 0.48 p) ~ 0.63 P ,
<
1.006 (0 p + 0 p) < 0.29 p ,
-
It([a, [y,S]]) I
therefore
<
(2.3.1, 3.1. 3
(K»
<
p,
1Ir([a,[y,S]]'1/.[y,s])11 <
30
It([a,[y,SJ]~
This gives first which is IIr (a (H
(a). As in
* S)I\ < 0
+ ) j 1
, but
'*
[a, [y,S]] (ii)
d)
[y,S] E r(j)
and then
this implies
. This proves (H
[y,S])I
(b)
(*)
and completes
(i)
of the
,
the inductive proof of
with
word length of elements
a. ) is needed or obtained.
~
(Almost translational
Assume the curvature a,S € rp
S] 6 r(j)
is 3.4.1.
(as products
loops
*'
in 3.3.1 and therefore
Note that in this proof no bound on the algebraic
3.5 Proposition.
[y,a
behaviour
of from 3.2. For
bounds 3.2 (K), 3.3.1; choose It(a) 1,lt(S) I ~
I\r(a)1\ <
-
T
p
we have
0 1.6-lt(a)l, p
(H)
It(a ~ S) - t(a) - t(S) I
<-
0 2 - -It(a)I-lt(S)1
(Hi)
It( [S,a]) I
-<
0 4 - pit
44
p
(a)
I-It (6) I
SMALL ROTATIONAL
~.
~or each
Dr (ak)11~
k ~ p'lt (C/,) 1-
8
1
we have
PARTS
It (C/,k) I < p
(3.4). Then 2.3.3 implies 2 .~::: IIr(C/,k)11 ~ k,lr(C/,)11 k 1\2It(C/,) 12 , or
i
1-
1 k > p'l t (a) max and 3.2 (K)), then (i) follows:
Use
this for
Ilr(a)11 <
Row
(ii)
and
(8p-t(a)
(iii)
+
1 2"
hence
Ilr(a)11~ ~'8 +
i
k1\ It(a) 12 •
and note
2
1\ P
follow with
)
(3.3.1
• It(a) I
(i)
and
<
1.6 -8 'It(a) P
1\2p2«
8
I
.
from 2.3.1 and
2.4.1, namely
< (1.68+1\2p2) /t([S,a]) / ~ Si~A~1\P
rotational
parts and translational
p
(3.2 8 + ~ 1\2p2)
3.5.1 Remark. Note that in proposition
distances
'.!.It(a)I'lt(S)I,
3.4 and 3.5 the statements
about
parts are once again separated.
The
3.1.3 and 3.3.2 on the group of motions and the short basis 3.1.3
have served their technical purpose and will not appear again. We conclude
this chapter with a rather immediate application
which shows already the far reaching consequences sitions. The main application
is the development
formation about the multiplication
of these technical propoof still more precise
Assumptions (a~SmOdrp)
a~S
-1
a,S € IT /2 P
Erp
Justification
equivalent
mod
r
p
of the definition: Take a,S,y € ITp/2 such that a ~ 13 , • First, r = r-1 implies 13~ a • Next, 1\p is small enough p -1 P -1 P -1 (a ~ 13 ) ~ (13 y ) = a ;f' y (associativity 2.2.5); therefore
B'~ y mod r to have
ones.
(Equivalent loops)
as in 3.4, 3.5. Call loops if
in-
of loops in chapter 4.
3.6 The short loops modulo the almost translational 3.6.1 Definition.
of 3.4 and 3.5
'*
45
loa
P. BUSER AND
we obtain from 2.3 and 3.4.2
H. KARCHER
e
IIr(a l( y-l)11 ::2
+ A2p2 < 0.48 , hence
a -'k'y-l E r This proves that we have indeed defined an equivalence relation.p Note how essential 3.4.2 enters: If rotational parts < 0.48 were not automati-
e ,
cally <
there would be no equivalence
3.6.2 Proposition. Assumptions
(Description of equivalence
classes)
as in 3.4, 3.5, 3.6.1.
If
(i)
relation.
satisfy
2 2 < 0.48 - A p )
< 0.47
d(r(a),r(S»
then If
(ii)
a - S mod r
There are at most
(iii)
(iv)
wen)
hence
S-l ~ r
a
*
from
(i)
< 0.47
Whi~h is p
d(r(a),r(S»
e
2014dim SO(n)
2 2 + A p • equivalence
classes
then 2.3 implies (i) . If
Ilr(a*
l
a=y~S,y~r
p
, therefore 3.4.2 gives
and 7.6.1 since there are at most
wen)
p
B- )ll::.0.48 ,
2 2
::: Ilr(y)11+ A p
mod r
; this is a
%
short representative
d(r(a),r(S»
=
<
class is a loop of length < 2w(n)d(M) 4 since 2w(n)d(M) < 2m- °
In each equivalence
Proof. If
d (r (a) ,r (6»
then
p
(ii)
then 2.3 implies (iii)
elements in
follows
O(n)
with
pairwise distance ::: ~ ~ 0.45 To see
(iv) note first that 2.1.5 applied to any loop
decomposi tion
=
a
al ... ~
into a product of loops
•
Therefo~e we take
and choose in each of the
«
wen»~
p
gives a such
'* a.J+ 1... *a.+" J "-
has
TI2d(M) as a set of generators
for
that any partial product of consecutive length ~ It(a) 1
a E TI
factors
a. J
equivalence
classes a representative
TIp/2 which
as a word in the
ai E TI2d(M) has minimal word length. Since all words of word length ~ wen) have a loop length ~ 2w(n)d(M) «p/2 we can use (iii) and the pigeon hole argument to find a number of word length
=
w+l
w::: wen)
such that all words
are equivalent to words of word length
S
w .
Assume by induction that each loop of word length then each
=
£, :::
w+l
a ~
46
£'+1 is equivalent
to
SMALL ROTATIONAL
PARTS
, which is a word of length ~ w+1 and therefore again equivalent
to
of length < w .
Corollary. Assumptions
(Relative denseness of 1
«< Y EOr
p
each
m-1 p , compare 3.2) we have: The translational
are a-dense
IIr wi I ~ 8
Recall
(2.1.4) in the ball of radius
v € T M
Ivl < p/2 -
with
p
such that
-
*w
-1)
I ~
Corollary. (i)
a
we find with 2.1.3 a loop
2 2
P
- t(a)1 2 It(w)1 + A p It(w)1 ' hence a .
(Multiplication of equivalence
classes)
has the following property of a normal subgroup: then
(ii)
T M. P
Iv - t(a) I 2 2d(M) . Then take a shortest representative It(w) I < 2w(n)d(M) . Now a*, w-1 E r and (with -
It(a1C w-)
t (a
parts
in
i.e.
1
Iv -
p/2 - a
(3.4.2) .
(3.6.2 (iv», 2.3.1)
rp
as in 3.4, 3.5, 3.6.1.
If
w.l
mod
rp
EO
[aiJ
(i=1,2)
(3.6.2 (iv»
a
'* y *' a
-1
EO r p
are short representatives
of equivalence
classes
then
is a well defined product. (iii)
The equivalence
classes
mod
r
with the product
(ii)
form a group
p
G
of order
O(n) •
If
IGI 2 w(n) A E
G
and
IIA 112 0.47 - 38
'*
It(a
*'
To see
then
(ii)
defined; if
note first that
w~ - w. mod r l
l
P
0
r(y)
0
of
r(a -1)) < A 2 p 2 ,
(i).
It(w1) I + It(w2) I < p/2 so that are other short representatives,
47
G
A = id •
(i) we have d (r(a~ y a -1), r(a) 1 Ilr(a,¥ y f a- )11 211r(y)11 + A2p2 < 0.48 and 1 y *a- ) I 2 2It(a) I + It(y) I :::p , which proves
Proof. From 2.3.1 hence
-
which is isomorphic to a subgroup
[W1 ~
wi
=
Yi
w2]
*'
Wi
is '
--
--~-
P. BUSER AND
then
2.2.5
H. KARCHER
gives
(3.4.2,
To prove [w]
(iii)
note
* [w-11 = 111 .
2.2.5.
Therefore
To map
G
first
O(n)
[wJ
w*
Associativity
we have
into
that
wand
map
almost
a homomorphism
to
h(
=
1
(2.2.5)
G
and
3.6.2
r(w) E 3.6.2
O(n)
follows
class
8.2
we change
[w ] f [w) 1 from 8.2 and proves
3.6.5
Y in
=
r
P
to describe
enough
n
2 2
8 + A p
is
•
G ->- G c O(n)
; since
r
P
generates
Lie group group
the subset
aid of a suitable
N.
by polynomials a uniform Moreover
with r· p
diffeomorphism
from
diffeomorphic
to the nilmanifold
p
Y , •.. ,Y 1 n a unique
({Y1, ... subgroup
is isomorphic G
,Yi}).
(3.6.4).
r
M r\N
48
to
on
N N,
for
r
From can be
p
representation The multiplication
(1 ,.. .,1 ) IE zn ; 1 n of an n-dimensional
to a normal
subgroup
In the universal
we map onto the subset
metric
theorem.
have
in the exponents
group
which
r-equivariant
r
discrete r
factor
left invariant
of
of the multiplication
of Gromov's
generators
[Y'Yi+1}€
and such that
can be expressed
we have
: G ->- O(n)
h:
properties
the end of the proof
that a set of exactly
the fundamental
M
.
J» ~
We have now derived
•.. *y~n
nilpotent
2 wen)
representative
J),
loops
therefore
h
(iv) and
IGi
one short
. This map
to a homeomorphism
such that the not too long loops
y~l*
3.6.2
d(h([W ]), h([W 0.47 (3.6.2 (i» we also have 1 2 d(h([W h([W J» ~ 0.47 - 38 ,which shows injectivity of h 1 2 the last statement of (iii) .
3.5 we prove chosen
slightly
implies
outlo~k.
of short
h
from
(ii) implies
<
With
(i».
implies
(iii) implies
in each equivalence
[U!..1)
since
-1
of the product
a group
pick
w
3.6.4
r eN.
we interpolate thus proving
of
covering
with
the
this map to a
that
r\M
is
rp
The embedding of finite index in
into a nilpotent torsion free subgroup of
TIl(M).
This chapter achieves the first part of 3.6.5 in the following steps: 4.1 defines A-normal bases in Rn , shows that they exist for lattice subn R and proves estimates which are familiar for orthonormal bases. 4.2 introduces almost translational
Rn
subsets of
and gives bounds for the
deviation of products from the purely translational 4.3 studies in the almost translational representatives
case (but guided bY 4.11 orbits,
and their projections
4.4 proves that the projections
behaviour.
into a hyperplane.
(4.3) of almost translational
sets are again
almost translational. 4.5 uses the above to find a A-normal basis
cl'···
and to represent short elements
some ball in free nilpotent
group
r
rp
,cn
x E T
for an almost transuniquely as
(3.5) using the representation
generated bY that ball; finally
subgroup of finite index into the fundamental group
4.5 into a torsion-
r
TIl(M)
is embedded as a with the help of
2.2.7 and 3.6.2.
Remark.
Most of the more unpleasant portions of this chapter, notably
are devoted to length estimates
in almost translational
in 4.5 and 4.6, but they do not contribute
4.1 Normal bases for lattices in
sets; they are crucial
much to the geometric idea of the
Rn
4.1.1 Definition. By induction over
n
we introduce
A-normal bases for
as follows: (i) (ii)
Any basis for A basis
is A-normal for each A ~ 1 • n {Yl' .•• 'Yn} for R is A-normal if it satisfies:
Rl
49
~=.i
_
(4.2.3),
Rn
=======----
P. BUSER AND
_
--- ----
H. KARCHER
(i=2, ... ,n)
are the projections
4.1.2
Remarks.
{Y , ... ,Y } 1 n
of
Yi
into
Y are pairwise orthogonal if and only if A = 1 . If i is A-normal for Rn, then {Y , ... ,Y } is A-normal for 1 k The
Rk = span{y , ... ,y } 1 k Again,
we study
this notion
3.5) and later modify
4.1.3
Proposition. discrete
Proo~.
n
=
1
r
subgroup
the statement
n - 1 • Then choose
in the purely
the procedure
(Normal basis
Any uniform
For
first
€
errors
caused
case by
8 f
(8 = 0 in 0
of a lattice) n of R has a /2-normal
basis.
is trivial;
by induction
assume
4.1.3
•
for
r
as a shortest element and define a uniform n 1 r' of IR - by projecting the orbits {i1'. 0 y}. (which discrete subgroup 1.€ ~ are sets of points at distance 1Y 1 1 apart on lines parallel to l<. 0 Y 1) ~n-l {Y } = R orthogonally onto . r' is uniform, since for each 1 n-l n x ERe IR holds d (x, r ') :::d(x,f) .
Now define
for each
This definition
of
Y
to handle
translational
1
orbit
y
{Y{O y}
a representative
and the minimality
of
2 0 < < 1y 12 , 2 < 1_1 Y 1 1 1 implies for the projection y' of y . g1.ve
Iy'l
Now has a
~~)
shows
/2-normal
is /2-normal
for
discreteness basis
of
<:
Iyl
r' ;
{Y2' ... 'y~}
r.
50
Yl
hence
<: then,
12
y
(here 2
cos
by the inequalities
1y 11 ~
Iy' 1 .
by induction
and, again
1y-y 1 1)
~ (Yl'y) ::: ~ • This
with
hypothesis,
r'
(*-) , {Yl'Y2' ... 'Yn}
NILPOTENT
For handling
the error
Proposition. (i)
(iii)
terms
det (Y .... ,Y ) > A i n -
with
the same
• IY
i
(i)
and
(n-i).
7r
1 (M)
the following
of A-normal
I • ... • IY
n
is usefull:
bases)
I
Ix.y.1 ~ ~
(j=l, ... ,n)
I
in both decompositions.
~
the components
(ii)
OF
n x = x Yi + X.y. 1 i=2 ~ ~
conclude
for
0
! i=l
x. (i=2, ... ,n)
allows
i
8
properties
-In(n-i)
then
(ii)
by
~ 2-1(j-i~A-1j(j-l).
x'
If
FREE SUBGROUP
caused
(Uniformity
I! x·y.1 i=l ~ ~
(ii)
TORSION
of a vector
more
are trivial
to be much
longer
than the
than
for
n
and are assumed
by induction
Then
Idetn (Yi'··· ,Yn) I
>
IYi I • A
-!(n-U(n-2)1 ~
z .....
Y I
IY~ I
(induction)
>
gives
(i).
Ix +
To prove
yl 2
x
>
Next
define
previous
(ii)
(A-normal)
first
observe:
min (sin
~ (x,y) ~ (O,n)
then
#
!' !)( cos
Ix I + Iy I)
, span(y , ... ,y _ )) j i j 1 inductively
51
-
¢
2
¢j := ~ (Y
inequality
If
E [O,~]
::: 2 -
.i
sin ¢.
and obtain
(Ix I + Iy I).
from the
-----
P. BUSER AND
j-1
I
Ix.y. + x.y.1 J J i=l 1 1
H. KARCHER
>
t j-1 2 -.t sin ¢. ( Ix .y .I+ I x. y. I) J J J i=l 1 1
>
2- Z
I
~ (.
j
1)
J-
11-
•
j
I
sin ¢..
i=2
Ix. y.
i=l
1
I
1 1
I •...•
Idet (Y1 '... ,y .) I=,..: 2 sin ¢. • IY1 Iy .I and since J 1= 1 J {Y1'···'Y.} is A-normal in span(y , ... ,y.) (4.1.2) we can use 4.1.4 (i) 1 J . J to elimi.nate 2 si.n ¢. from the last inequality, proving (ii). "i= 1 (iii) is clear since Y - Yi € /R. Y (i=2, ... ,n). i 1
Finally,
since
...i!
4.2 Almost
translational
Additional
problems
orbits
arise,
are not straight
defined
the first
sidered
inductive
corresponding
to 4.1.4
the arguments
subset a ~
io
(i) (ii)
f
p
o e
T
p
For all
are
translational
deteriorates.
behaviour
to work with a normal
can no longer
basis,
arguments.
decreases
be con-
the representation
The following
notion
set
T
C ~n a 8-translational, p (with the parameters restricted
to
8 ~
then
, Ix I
Icl ~ p
1 ~2 p
.
, there
1S some
such that
Ix - cl < a (iii)
For all
a,b
is defined; -1 a (iv)
E Tp
E.
T
P
a E T
for each
Associativity of all products
(a
< -
, la + bl
such that
*"
helps
a-dense
:
if
x € ~n
Products
0
length
further
if it satisfies
for convenience)
to project.
-I
8
from 4.1:
We call a finite
of radius
difficult
one has found
(iii) needs
4.1 to the case
only and the admissable
step the elements
Even after
Definition.
to generalize
more
elements
also the almost
to generalize
4.2.1
~n
if one tries
short
steps;
as loops.
of
and therefore
for sufficiently
in the inductive After
subsets
a
*a
p
p.
(1-8)
a product
, lal < p. (1-8)
-1
there
*
b E: T
exists
P a unique
-1 = a
*
a = 0 •
b) .. c = a .. (b -If- c)
involved
a
follows
52
from
holds,
(iii) .
if the existence
;0 '
NILPOTENT
(v)
TORSION
FREE SUBGROUP
OF
711 (M)
The product satisfies lait-b-a-bl
< ~p lalolbl ,
I [a,b] I
<
28 Ia I Ib I • 0
p
4.2.2 Remark. Because of 3.5, 2.2.5, 3.6.3 we have with
that the
set
T
p
=
{a = t(a);a €f
together with the product
t(a)
p
o
* t(S)
(3.5),lt(a)1 < p} := t(a ~ S)
satisfies 4.2.1
(i) - (v).
The induction starts from this example. In order to control
the deviation
some immediate consequences
4.2.3 Proposition.
(i)
=
of definition
4.2.1 in the following.
Notations and assumptions
Shortest elements [c,c1]
of products from vector sums we collect
0 , if
a
Let
a,b € T
p
<
o ~ c1 € Tp satisfy [c,c1] is defined.
(ii)
(iii)
as in 4.2.1. Then
with
-1
satisfies
lal,lbl,la-bl < (1-38) (1-82 )-"Zop, then -
I I 11" b -11 < -8(1+8) b -1 - (a-b) I < -8 Iboa - p p 1 la - bl < (1 +~ Ibl) °la*b- 1 , Ia
If
-
2C1 ,
Ib I Ia-b I , 0
p
1 la-lfb- 1 < (1 - ~ Ibl)-lo la-bl < (1+8) la-bl . p (iv)
Let
a1,.·.,akE
defined,
Tp , I~=l lail ~ (1-8)p • Then
la1*··.*akl ~ I~=l lailo(1+8)
*"~
a1~' ..
is
and
is defined for
53 _
lUll
.at
P. BUSER AND
<
1
(v)
j
<
k
and
Ia j - J.-a I
~
8(1+8) p
. I Ia -j + Ja
~
8(1+8)3 ~
Let
c , ... ,c ~ T 1 n p
I
\,n L·_11.c. ~~ ~
If
H. KARCHER
I
lal
be a A-normal
(1-8)ljal
basis
c.li ~
then
< __ lajl < (1+8)-ljal
(1-8)ljal
'
lal2 '
- j(j+1) 2
(1-28) 2 p
<-
2
- j(j-ll 2
with
A
c 11 1
and
<__
la-jl
<
2 .
-
*... *c nIn
<
'
(1+8)-ljal
are defined
and
n Ict1~ ...~cln n Proof.
(i)
L
-
is clear
<
l.c. I ~ ~
i=1
from 4.2.1
in 4.2.1 (v) . To see (iii) -1 are defined: (a b ) *b
SILl.c. I ~ ~
-
(ii) check
and first
(v) . For
(ii)
a
a
* b-1
that the products
*" a
-1
use
= 0
and
*
la + b
-1 I
~ la - bl +
1-1 b +
1 la :/fb-
Therefore
Ia
1
+ bl ~ la + b-
b
I ~
+ bl +
(1-8)p
and
% jal_lb-1
1
::
lal +
%
1 (Ial+lbl) _ Ib- 1
we have
* b -1 -
this inequality
(a-b) I
implies
8
(1 +-Ibl)-Ia*b p
and both
< (1-8)P.
estimates
by induction, (a1,.·..*am)
k = 1
give the remaining being
trivial:
"* (am+1 *... *ak+1)
inequalities
For
exists
54
1:: m ~ k
since
of
-1 1
(iii) . (iv)
the product
is proved
NILPOTENT
TORSION
m ~ (1+8)' (iv)
I
(
FREE SUBGROUP
lam+11f-· • ·~~+1 k+1
I
independent from 4.2.1
(iv).
Jr
1 (M)
I
<
Ia.I + la.l) ~ i=m+1 ~
i=l
OF
(1+8) (1-28) p
of the brackets,
The estimate
follows
<
(1-8) p
since we have
associ-
from
k+1
- I
~ (1+8) P
application
to
j
a
Ia -j + ja I
first
~
la1* .. ·*~1
+
- (1+8) p
• 1~+11
<
k
8
a
for
I
i=l -1
observe
(1+8) j-(j-1) Ia -112
8 2P
+
8 3. 2 j+1 - (1+8) • IJal • y P J
and their
estimate
+
Ia + a -1 I
j'
~ (H)
The powers
~
is direct;
<
<
a.1
i=l
product
are defined
by
(iv) , from which
also the
follows:
IliCil'lljCjl'(1+8)2,
11. . -T-. 1+1
Finally
4.3 Orbits,
4.3.1 Let
representatives,
Almost T
(ii)
use 4.1.4
straight
be as in 4.2.1
(4.2.3).
(1-38) p -
To follow
.
< 2
projection
orbits.
P c E Tp , Ic I ~
A
with
2 Icil
and let
20-,
the orbit
0
f
c
1
E T
the products through
c
be a shortest
P +1
1
c
towards
*c
element.
are defined
the hyperplane
and
{c y 1
the difference
vector
between
consecutive
55
A.1i
A
[
,a,
orbit
points:
For each ] = 0 1 consider
.L [c,c
n., P. BUSER AND
w
and compare
+
*
c - c ,
if
>
'*
c - c ,
if
:::0 ,
-1 1
=
c
=
c1
w
them with
+c
using
1
(i)
H. KARCHER
4.2.3
0
,
(Hi)
<
<
Therefore
(H)
Isin ~ (w+,C ) I 1
8 -p -Icl
<-
<
8 := sin
-
Isin ~ (w-,-C ) I ::: 8 p (1+8) Ic-c11 1
The second
angle
estimate
and the difference obtain
that the lengths
is enough (ii)
of the vectors
for the above along
the orbit
through
c
(i)
that the scalar
product
with
=
8
0
case,
c
< -
is : ~.
c
1
through
c
towards
(1-8 2 )_"~,
Therefore
the hyperplane.
changes
along
between
In particular
(0-38) p -. 2~ (iii).
towards
8 = sin'"
so that the angle
of the orbit
1 Ic I· --.. cos." application of 4.2.3
hold
in the
of
of any two orbit points
{c1} -l. do not exceed
hyperplane
from
is independent
vector
<
.9- ,
4.3.1
c we
the which (i),
We conclude
the orbit
1
now
almost
as
namely
(iH)
4.3.2
Representatives
Consider
c 6 Tp
of orbits.
such that
that we find on the c1-orbit max (-k,k+l)
Icl ~
(1-48)p
through
c
- 20"; then 4.3.1
after
Ikl
~ 1 +
characterized
l1 • (1-8)-1Ic11-2 , a unique by the same inequalitties as in 4.1:
(")
MQreover,
with
4.2.3
and 4.3.1
(1-8) Ikc11
<
we have
<
56
the estimates
(1+8) Ikc11 ,
steps,
(i) _ (iii)
show
where
representative
c
which
is
NILPOTENT
TORSION
FREE SUBGROUP
> ~ (1+8) Ic 1 12
a < (c l;c
Isin ~ (c-e,c1) I < 4.3.3 Projection
claim
projection
T
p
<:
lei
<:
>..0Ic'l ,
by
>
We estimate
(e-d,c1
the same computation
1
Tp
a
proves
< (}
-
28)-i
1-
2"
+ 28).1.,
+
T'
is injective.
by modifying
works for
lei ° sin ~ (e,cl)
where>..
e ~ 21 E Tp
representatives
which shows that the projection
instead
4.1.3 to get the second of
d
if
inequality;
e ~ c
the first inequality.
' then 1 We can, after renaming,
that
(1+8llc11
2
(1+8) Ic11 Of course only the case where By
of representatives
onto
cos ~ (e-Cl,c ) < 1
=
Ic I ,
(compare 4.1)
and for different
Ic'l
:
~
into
Ic'l
assume
1 (M)
sin.:!.
{c } • 1 as in 4.3.1, 4.3.2. We map the set
Assumptions orthogonal
<
p
CoS..T
7r
Ie I ° cos.J
,
o..!£!
~
OF
choice of
c1
and
e ~ a
le-al
or 2
is not too large might cause problems.
we have using 4.2.3
and Ic11
:
Ic~l ~
(e il-crl) I
:
57 AU
Hi
s.
I(e - d) - c11 + 81c 1 + 81e - 211 1
117:
n ~-
P. BUSER AND
H. KARCHER
hence
=
2(e-d,c1) <
Multiplying
:e - 2112 + Icl12 - Ie -
(1-8) Ie - 2112 + 381cl12
4.3.4 Denseness of
4.2.1 there is for every
some
such that
c'e
T'
we obtain
cos
2
~
Of
~ (c-u,c1) ~
Ix - cl < 0
x6
{c } 1
..L
=fR
n-l
Ixl <
with
The representative
e
<
Proof. The worst case happens,
(1+38)0
l
p
- 2
if
c
and
0' .
=:
C
are on different
But the angle estimate in 4.3.1 still gives with 4.2.3
Ix - c' I
o
<
4.4 The product in 4.4.1 Definition.
cos~
<
+ Ic11 • (1+8) - sin"
sides of
(i)
0(_1_ + 28(1+8)) 1-82
.
T'. a' ,b' E. T'
Let
be such that
la'l,lb'l ~ p'
la' + b'l ~ (1-8)p' • From 4.3.3 we have for the unique a,B E Tp
the bounds
(!)
:=
112p
(a *B)-
Therefore the following definition
is justified:
and
preimages
lal ~ A.-Ia'i , IBI ~ A.lb'I , so that the product
is defined and the representative
a*" B
can be obtained with 4.3.2.
a'7f-b'
4.4.2 Proposition. la'*b'
(Bounds for the product)
- (a' +b')1
~
~, la'llb'l;
(a,)-l E T' ,if (iii)
(a'
*" b')
28 .
and its projection
exist, we claim
Ix - c' I
(i)
21 +
T'.
By assumption c E Tp
cl12 <
(1+28 + 782) Ie _ 2112
~
these two bounds for (c-21,c ) 1
a -
~c'
=
a'
* (b' *" c')
follows from 4.4.1.
58
la'jfb'l
< p'.
la'i ~ (1-8)p' . , if the existence of all products
NILPOTENT
(iv)
<
l[a',b']1
TORSION
FREE SUBGROUP
2~ kl'lb'l p 4.3.4 this
Together
with
cr'-dense
and of radius
OF
7r1 (M)
. shows that p'
T'
is
=
{c1"}.L.
p
Rn-1
in
0-translational, 4.2.3
is
therefore
applicable. k (v)
L
if
la~1 < (1-0)'
Proof.
(i)
First
second
term
Secondly,
from 4.3.2
with
4.2.1
and 4.3.3
I-I I-I -o a 'Ib p
<
(ii)
we have the
angle
<
---,
the
1«a*E)-
Ix I 'sin
<
- a - E , c ) I 1
+
2(1+0) Icl1
1=
1=
(x;y)
(x,y)
I «a "* E) -
-
<
+ Ic11
- cos
a ' E: ~ (
(a *" E»
I
o
cos.J-
ka
-}f
E - a - E , c ) I 1
:: a'lyl
5.2 • It
decrease
in
2
< 12 , the
the
radius
hence
<
__ 0_
-
cos2
<
2 ~
<
4.2
ra*EI
-8-
(2(1+0)lc11
p
lal'IEI
p
increase p + p'
-
, then
2
Since
<
lal'IEI
<x,y)
"
x,y
,
%
'
and
:: E:
estimate
product
c >1 1
2
sin
scalar
-a*E,
* E) -
I«a
Now, if
•
By definition
we bound the
<
p'
~ -
i=l
=
op
-.
in
I_I I-I a 'Ib
the
<
estimate
112P ; this
59
%
+ 2
proves
+
%
lal'IEI)
(1-0)-1(lal'lc11
4.2
is
o
- It p
2
' la'I'lb'l
compensated (i).
+ IEI'lc 1) 1
by the
• P. BUSER AND
H. KARCHER
o for the second
equality
every
For the same reason
element.
(iv)
can be derived
we used
4.2.3
from 4.4.1
(i),
(iii)
(i)
namely
that
c
commutes
1
with
is true:
or quicker
from 4.2.1
(v)
[a' ,b']
(v)
follows
and
T
inductively
is available
4.5 The A-normal Rather
large
we summarize
from the definition
and, again,
basis
constants
for
T
will
appear
from where
elements
we know
cr:,:3w(n) • d(M)
one needs
consideration which
satisfy
of 4.4 looses products
up to
- which
c
since
commutes
1
associativity
with
other
in the assumptions
Ic11 ~ 2cr
=
elements.
of the following
(4.2.3
(i)); for the denseness dim so(n) w(n) = 2'14 ; the
theorem;
parameter
(3.6.3), where (n-ll-fold n 1 a factor 12- + in the radius; to use 4.2.3 (v) n2 2 times as long as the elements x under
of course
should
at least
include
the generators
c.
l
we select.
(iii)
in
T'
P
Theorem. Let T satisfy 4.2.1 with the additional assumptions n+6 ~ n2 P -n-l -n - 1. n2 cr• 4 • 2
m
in
they come:
Shortest
application
since
4.4.1
below. (When applied to 4.2.2 this (n+3) 2 . w> 2 w(n) In (3.2).) Then:
10,
(i)
(ii)
T
has a A-normal basis {c , ... ,c} 1 p n Ic I ~ 20"((1 +38) • A)k-l:,:2k(J'. k - for example is defined
can be satisfied
with
<
1
2" -
is used
-I
28) -
z ,
- then
if
and associativity
60
A
R
by choosing
holds.
NILPOTENT
(iii)
Each
T , Ixl < R , has a unique p --of the basis; for the images
't· 1.ons proJec (k)
(iv)
FREE SUBGROUP
E
x
in terms
haVle
TORSION
x
=
T p -+ T'p' - T (2) -+ T(2)' (k) lk (ck )
*...
*(c
We have n Ix(k) -
x(k)
l.c~k) I i=k 1. 1.
<
8·
I I i=k
x
x = clll~..•*cln 1\' n the iterated
under
(4 ..,. 3 3
4 4) we
.
W1.th the same exponents. -hence
by 4.2.3
n
I
of
-+ • • . - ~(3) 1
(k) 1 ) n
Pk
trl (M)
representation
n
<
OF
1. c (k)
1. i
I
8 _I 1-8
<
x
(v)
(k)1
•
1, ... ,i < j , hence
(vi)
For If
x,y E T with Ixl < (1-28)R P _ 11 Ij x - c1 .,1f ••• *c then y*x*y j
, Iyl -1
<
"'n2 2.(. R
ml c1
we have:
* ... *cjmj
Proof. (i) The assumption on ~ allows to go through (n-l) induction steps ---P of 4.1.3 - of course using 4.3 and 4.4 - to find {c ' ... ,c} . 4.3.3 shows 1 4 n 28)-z and with 4.2.3 (i) gives the bounds for the Ickl .
.\~(t (E)
restates
4.2.3
(v).
To prove
(iii) we have to go through the induction once more and carry the .. . (k+1) (k)(k) est1.mates (1.v) along. Note f1.rst Ix I ~ Ix I ~ (1+8) Ix I ~
length
< ...
<
(1+8)k.
Next,
in
T(n)
Ixl
all elements are (in a unique way) powers of the selected (n) (n) 1 (n) k shortest element: x = (c ) n ; namely: let bE T and a the power n of a:= c(n) closest to it, then 4.2.3 (iii) gives n 2 3 k 1 la .. b- 1 < (1+8)lak _ bl < (1+8) lak+1 _ akl < (1+8) lal, hence 2 2 n-l o . Finally IlnC~n) I ~ (1_8)-11 «n))lnl :: (17~~8) Ixl by 4.3. 2 (~*).
Now assume (k)
x
by induction (k)
=
(ck
that we have in
lk
)
(k) "it" •••
*(c
1
) n
n
T(k)
a unique
.
together
W1.th the est1.mate
n i
I
=
< k
61
•
representation
.
• P. BUSER AND
This
implies
H. KARCHER
(4.3.3)
n
n
l:
<
i=k Consequently
y
l:
A"
i=k we have
4.2.3
(v)
available
for the product
(k-1)
hence n ly(k-1) I :: (1+8) li~k
n
l:
(k-1) I < (1+8) lici i=k
Therefore we can use 4.3.2 to find the representative y(k-1) on the (k-1). { (k-1)}-L . ck_1 orb~t towards c _ ; moreover we can compute that representat~ve k 1 (k-1) lk (k-1) 1 . from the product (c ) jt- ••• :;Jf-(c ) n : s~nce the k n multiplication in T(k) is defined with representatives in T(k-1) (4.4.2 (v» we see that y(k-l)must be the unique representative . (k-1)- _ (k-1)l-. (k-1) _ ((k-1»1 ~.e. Y - x • O..,serve y - c _ k 1 4.3.2 (~*)
*
(4.3.3) above x(k) (k-1)d.th y an w~
similarly
x
(k-1)
(c (k-1) l'
k-1
<
11' c(k-1)1 k-1
x
if.
)
(k-1) and
lk_1 = 1 + l'
<
(1_8)-1Ix(k-1)I
-
<
(1+8)klxl
-
-
~
2
J:..-2-~n" 12 Pk-1
Therefore n (11' I+ Ill) Ic (k-1) I k-1 contains (k-1)y about
=
the bound (k-1)-
x
= '"
(y*x*y-1)
(vi)
T (k-1) . Since
for
(iii)
and
are now clear
(recall
we observe
Iy,*x ll-Y
has the representation
= mn = 0 (j+1)
< 2"
i=k
we have proved
the commutators
(k=l, ... ,n) ). In -1 hence y¥-x*y
m.J+1
(iv)
., 2
- ~n
1
l:
+
1
2
and
(iv).
Pk-1
l'
The statements
r~. (k) ,c.(k)1 -1 ~ I =
c~l
* ...
are unique
r
~.,c. J ~ J I[y,x] *xl :: R
*c:n
=
• But
]
and (v)
(k)
(4.2.1
(iii)
(v»,
implies
since (j+1)
y
*
(j+1)
x
*
62
(-1) (j+1)
y
o .
NILPOTENT
The embedding
now apply
proofs
T
r R
into the fundamental
4.5 to
p
to prove
r
is injective,
that
R
rr
r
normal
i.e. that the natural
=
W(rR)/N(r ) R R of equivalence classes
G
(any
- frequently
words
map
r)
is an embedding mod
of 3.6 to p The injectivity
'IT •
of
R
can be embedded
the transformation
of 4.6.1 we prove
associativity
that
r
in a product
group
suitable
for products
r . This is consistent:
preserving
of a suitable
which
the relations
in
r
Proposition.
(Injectivity
of
r )
set _
o " 2 (n+1 )
is a torsion
free nilpotent
relations
< n); in particular ~n suffices
(n+2) 3
w(n)
r
• The condition for section in 3.2
R on
y~
* Y2
commutator
*Y3
beyond
R
R
<
rr
(much)
is used
to
2
2- (n+2) P . Then
group with
is stronger
rR
the generators
_ .k1 [y. , y .] -y. 1 ~ J to a nilpotent
(v):
is isomorphic
is trivial: words
All loops of loops
of the generators
' then we have
63
•
4.5
<
5 and can be achieved
as normal the only nontrivial
loop length
the
*" •••
.ki-1 *y.1 ~-
group
than in 4.5 for convenience. by taking
ill
=
10 ,
(0 < 3w(n)"d(M».
n < 3
The case
3
3.2 and assume
and the commutator
have
preserve
r
R
fR
embeddings
the information
about
with
> 2
with
is injective.
applications
It also
(M,p)
A
group
rR implies injectivity on the following obvious fact:
If
4.6.2
'IT
group
It(y) I . The representation
to its presented
a group
1 (M)
(3.5, 4.2.2) using that the map t from p of translational parts is injective and preserves the
=
are based
7r
r
Iyl
rR
OF
1
Then we use the group
rr
FREE SUBGROUP
We abbreviate
from (2.2.7).
of
theorem
to the set
TORSION
sufficient
Y
of length
y =
ll* l2 1 2
is given
by
associativity
to prove
If
P. BUSER AND
This
shows
that the loops
(~
H. KARCHER
y
do and proves
injectivity
with
of
4.6.1
.
o
(ii)
The case
n
=
Let the nontrivial
=
[Y4,yJ
G (= z;3,.,) 3
illustrates
commutators
the
be given
l-th
1
(for
0.
=
l
power
=
[Y3' y 2J
by
in the general y~
, [y4'y 2J
=
case. yi
'
group
of an automorphism
+ G
G
g4
3
3
. Then the product
by
x z;
G
3
(YolO)
into a nilpotent i = 1,2,3
,
l
(y • torsion
commutator
Again
y = y~l ~ ... *y!4
each
loop
(1 ,... ,1 ) C G 1
4
lengths
S
injective
4
relations
; associativity
2n2/2
{y.} ,
1 + m)
G
free group
4
- shows
with the generators 1. . . g4 lS conJugatlon:
are
ErR
is mapped
- holding
- injectively
for products
that the map is product
of eight
preserving.
- onto loops
Hence
of
r
R
is
n = 4 .
for
The last step also showed
generators
g!
04 = (0,1) • In this group
and the defining
Remark.
of the proof
e. ~
(y,l) • (y' ,m)
turns
a part
, call the above matrix
and define
to obtain defined
* y;
y~
4
°
, ...
1
,°
4
that a torsion
and the commutator
64
free nilpotent
relations
(*) exists
group
with
for any
NILPOTENT
V,µ,A,X
€ ~
TORSION
FREE SUBGROUP
. This conclusion
fails
OF
for
71:1 (M)
n ~ 5
(e.g. because
of the
identity).
For the general 1ength
case we define
(Yl' ..• ~Yj)r
~ r ~ R ; they all have normal in these
products
induction
of the fundamental
(iii)
may involve
step yields
R . This restriction
words
*...
loops
therefore
is removed
group
to be the set of loops of 11 1· Y1 *- Y jJ ; note that of length
injectivity
in 4.6.4
where
up to
" 2r 2 zn
only
for
f , R
is known
fR'
with as
M.
of
Assume by induction that f. := (Y , ... ,Y.) is injective and 1 . J J r (ZJ,o) is nilpotent without torsion - which we know for j ~ 4 from (ii). Then the same holds
for
>
and
(Y , ... ,y. 1) where cr is 1 J+ cr _(j+1)2 C := 2 comes from 4.1.4 0
y. 1{ . } : f. ->- f. via the presenJ+ J J of f .. Observe that f. can also be presented with elements much J J -1 A shorter than r (Y E f. does not imply y. 1 Y y. 1 (;.f. ): In fact f. J J+ J+ J J assumed torsion free and has therefore the presentation We have to define
"conjugation"
A
A
*' *'
J 4.5
=
W ({Yl' ..• ,Y .})/N' J
=
free group
of words
modulo
the commutator
:==
(i=l, ... ,j)
a homomorphism W({y , ... ,y.}) ->- W(f.)/N(f.) 1 -1 -1 J J J [Yj+1'" Yi Yj+1'Yj+1 Yk Yj+1] = Yj+1 [Yi,Yk] (4.5 (v), (vi» projects to an automorphism
defines
'* *
*'
Y·+1{ J comes
r.
• } : J similarly
=
W({Y1,. .. ,y.})/N' -1 J y. 1). J+
*'
->- W(f.)/N(f.) J J
f. J
from
the factors (with 4.1.4
(ii), 4.2.3
condition. (which computes
-
relations
(v). Hence
(v»
of the normal
and then on
the above
1
65
*
(The inverse
word decomposition
shows
automorphism
which because of -1 Yj+1 E (Y1'· .. 'Yj)r
operator
of
a
that the compatibility
loop-wise
on rather
long loops)
P. BUSER AND
holds
for all
4.5
r. ,
a. €
1 E:l':
H. KARCHER
< R
satisfying
J this hypothesis
(Note that under
-
'*
y~ 1 J+
I
r.
Ci.
Now decompose
each
and interpret
y
by
J
(v), (vi».
y
as transformation
Y
T
r. x.z;+r. J
Xl'.
J
by
yT(d,m)
The action y + yT
on the identity
is injective
S = S(j) *Y~+1
€
yT(Q,Q)
=
(~(j)'l)
shows
(Here we use the injectivity
rj+1
it
follows
from
of
[S(j)['[Y(j)[
that the representation
r.).
Moreover
~ roc1
for
and from
that
(*)
Hence
Therefore
y -+yT
extends
to an isomorphic
embedding
A
transformation
group
of
A
r. J
is assumed
that proved
torsion
r'+
In particular
it follows
J l
from
1 acts without torsion. - Since J+ _n3 4.6.2 for R' = 2 R; see 4.6.4.
subgroup properties
in Gromov's
of the fundamental of
product
rR
*
which
group.
we have
1.5 which
theorem
The important
have been detected
is replaced
point
concerns
of
so far
the nilpotent
is that the algebraic
so far remain
by the product
66
r'+
into the J l is injective, and since
r.
We now turn to the part
Gromov
r. x.z; .
J free,
of
TIl (M,p)
the same if the
NILPOTENT
denotes
the group
set of shortest
TORSION
FREE SUBGROUP
of equivalence
classes
OF
mod rp
7r
1 (M)
from 3.6 and
~
is a
representatives.
4.6.3
Prop9sition.
(Injectivity
Onder
the asslli~ptions 4.6.2
of
TI r 3
injective
then
If moreover
TIr/7
2-n R
and
is injective
the representation
R' < r < 7R -
and
=
y
ITr/7
to an isomorphism
{Yl'···'Yn}
Remark.
in
The implication of 4.6.4
for
4.6.3 r
=
rand r is a normal
with
as follows:
Since
and therefore
with
:
~
each
r
x
.. .. f l.n)ectJ.VJ.ty 0
r r·
isomorphic
of 4.6.2
a E TIr/7
+
r
~
the interpretation implies
3.6.1,
3.6.2
to rr/7
for
r = R'
and
[w[:
:=
a + aT
w IE ~
(w,y
*
r
r
= a
r/l00
*w
*c
(3.6.2) we have
in mind
a
*w
€ TIr/6
(!) decomposition
Y E
,
rr/3
c)
is again
To prove
x
as a transformation
aT(w,c)
the unique
w*y
The representation
TIl(M,p)
A
subgroup
can be used because
w G ~
aT (w,c)
is
r
7R •
To use 4.6.1 we represent
aT
to
r
in of Y Err as normal word n (Zn,o) then the group r generated
between
TIl(M,p)
is isomorphic
If
*...*
ylll A
extends
holds:
()T a 1
injective
*" a
2
=
aT 1
- consider 0
aT 2
aT(o,o)
and use
. wrJ.te
hence
(Associativity
67
in
TIr !).
nEr P. BUSER AND
r
H. KARCHER
Then
Thus
a
+
aT
extends to an isom~rphic embedding of
formation group of the set and 2.2.7 implies
TIr/7 into the trans-
~ x rr . Now 4.6.1 proves injectivity
TIr/7C TIr/7 ~ TIl(M,p) ! Next, r
of
TIr/7
is because of 3.6.4 a
norm~l subgroup of of in
TIl(M,p) and because of Y E rr/7 C TIr/7 a factor group i ~r/7 - there might be more relations between the Y in TIl(M,p) than i rr/7 ; however, the elements of TIl(M,p) were constructed as trans-
formations on
rr
~ x
and since
rr
is isomorphic
to some group
we see that no nontrivial normal word in the generators
0 ...
of
r
(Zn,o)
vanishes:
0
(0,0)
if and only if all
l.
l
a .
4.6.4 So far 4.6.2 and the usefulness of 4.6.3 have been proved only for 3 n R' = 2- R . This restriction will now be removed: Since fR' is isomorphic to
r = ~ TIl(M) , each
Yil •.•y~n
and the assumptions for
Y
e
Y E r
with respect to the product of on
R
has the unique representation TIl(M) ! On the other hand 4.5 (iii)
in 4.6.2 imply unique normal word representation
r7R . Therefore the map
Y~ 1
*"...
*
y~n + y~n .•.y~n
is an embedding
of
r7R into TIl(M) which is also product preserving - trivially, because short homotopies are homotopies - hence r is injective by 4.6.1 . Now _ 7R canonically as in 4.6.3 - r is a factor of f7 and is a factor of 7R R rR, ,therefore r = rR, shows r= r7R . This completes 4.6.2, 4.6.3 •
r
A
A
4.6.5 Theorem. Choose
R
A
(Structure of the fundamental
group, summary)
as in 4.6.2. Then
(i)
TIR is injective and
(ii)
n rR ~ (L , 0)
is isomorphic to a nilpotent,
subgroup
of finite index in
r
TIR ~ TIl(M,p) •
68
TIl(M,p) .
torsionfree
normal
NILPOTENT
r
Y
TORSION
is generated t: r
by
can uniquely
these
generators
n
FREE SUBGROUP
loops
as a normal
Loops
in
to the nilpotent
they are equivalent 3.6.4)
are equivalent
TIR C TIl(M,p) mod
is isomorphic
mod
that each element
word
Y
to a subgroup
11 1 Y 1 0 ... oy n n
< j < n)
Le.
•
if
(3.6) if and only
rp
G ;
r , Le.
=
structure,
(1 < i
(iv)
(MJ
:rr 1
such
Yl'·"'Yn
be written
are adapted
OF
r\TI (M,p) = 1 O(n) of with
here
IGI
G
(from
< 2014dim
SO(n)
(3.6.2) •
(iii) restate consider mod
the map
choosen
before
TIR + G
~:
rp • This map
the results
is injective
4.6.3
which on
of 4.6.2
sends ~,
a
* a2)
=
To prove
(iv)
to its equivalence
the set of shortest
~(al
. We also have
each
to 4.6.4.
class
representatives
~(al) 0 ~(a2)
if
a ,a ,a ~ a € TIR (3.6.4), therefore the extension of ~ to a homomorphism 1 2 1 2 of the free word group W(TI ) onto G projects to an epimorphism R
G
which
contains
therefore
-
r
=
r kern
in its kernel.
However
~
69
~
is still
injective
on
~,
--_.
The nilpotent
Lie group
diffeomorphism
~is
F
M
:
N
-.- ..
-----------
and the r-equivariant
+ N •
chapter finishes the proof of Gromov's theorem in the following steps:
5.1 embeds
r
(4.6) as a uniform discrete subgroup into an n-dimensional
group exponents of
y
using that the multiplication
N =
11
M
r
is polynomial
in the
1
Y1 •... ·ynn •
5.2 constructs a r-equivariant covering
in
nil-
to the Lie group
differentiable N
map
from the universal
F
by interpolating
local maps with the averaging
5.3 estimates the curvature of a left invariant metric on with the aim of making
N
which is defined
as almost isometric as possible; the estimate is
F
based on and similar to the commutator estimates 3.5 (ii). 5.4 proves with 5.3 that the map
has maximal rank and hence is a r-equi variant
F
(5.2.6) . 5.5 proves the corollaries
of 1.5.
5.1 The Malcev Polynomials The nilpotent torsionfree with generators commutators
subgroup
Yl' •.. 'Yn;
rc
TIl(M,p)
the nilpotent
was obtained in 4.6 together
structure is determined by the
(compare 4.5 (v»
(1
The unique representation allows to identify
r
of
Y Gras
<
i
<
normal word
< n) .
j
Y
=
y~l
* ... *y~n
with a nilpotent group structure on the integer lattice
r ~ (Zn,*) ; in this context we also denote
y
by its exponent vector
(ll,···,ln) . We _therefore have integer valued functions
5.1.1
-
P.
~
such that
71
-
• P. BUSER AND H. KARCHER
5.1. 2 Theorem . (Malcev The functions
Pi
[ 22])
Q
and
in 5.1.1
i
are polynomials.
Proof.
It suffices to show polynomiality of the P. since then .L -1 _ -In -11 shows the polynomiality of the (ll, .. .,ln) -Y ... Y n 1
(i) The proof
is by induction
If
(ii) The case
=
[Y4,Y3]
Y~
and together
+
n = 4
*
Y~
with
(as in 4.6.2)
n < 3
the case
is trivial:
-"
1
illustrates
(iii) Assume
conjugation
(i)
yields
+
by induction
reduce
the induction
Lemma.
For each
the induction
then
2 VK;14m3(m3-1)
From this we first
sucht
and
then
with
(mixed notation
1
2 µK014(14-1)m3
that
conclude
5.1.2
step to this
step. is given
If
[Y ,Y ] 4 2 explicitely
for better
, 12 + m
2
holds
that powers
(m , •.. ,m _ ) 1 n 1
Y!
Y~
,
in 4.6.2
with
,
(n-l)
by polynomials
generators. and then
fact.
there
exist
polynomials
Fi(l)
(i=1, •.. ,n-l)
that
(m ,·
1
•• ,m _ ) n 1
1
Fi (1) are for fixed
Moreover,
the
induction
hypothesis
1
(iii).
72
polynomials
in
(ii)
readability)
+ K014m 3
for groups
are given
=
m , ... ,ID _ 1 n 1
by the
--~--------------------~~~~=~~== THE NILPOTENT
Proof.
LIE GROUP N
Consider
the factors (Y1""'Yn-1)/
(m , •••
1
,m
For each
1 EO r.
) 1+1 n-1
(F1 (1) +
hence
F 1 (1)
Ii (1)
where
is the polynomial
>
(1
is given in
-
Ii (1)
P 1 (O,F 2 (1) ,... ,Fn-1 (1) ,m , ••. ,m - ) 1 n 1
obtained
the proof
1) 1'" ., n-1
We can apply
+
Ii (1) ,F 2 (1+ 1) ,... ,Fn -1 (1+ 1 ))
from the first
(using the induction
Now we complete
and
holds
* Y nIn
the induction n-2'Y n,>;
by polynomials
hypothesis
of 5.1.2.
*' (m1,.
in
polynomial (iii)
P1
of the group
above).
Decompose
.. ,mn_2
hypothesis
therefore,
Malcev
)
*-
-In Yn
(iii)
*
In Yn
*
mn-1 Yn-1
to the groups
"*
-In Yn
< Y l'
*"
Y
nIn+mn
... ,Y n-1
>
if we show that
In,m _ n 1
then the whole
11, ..• ,ln,m , ... ,m 1 n
73
product
(*) is polynomial
P. BUSER AND
-with
the power
lemma
applied
with
induction
hypothesis
H. KARCHER
(Yl'"
to
(iii)
>:
.,Y
- ,Y n 2 n
applied
to
Y n 1
> :
~(F1
(l), ... ,F _ (l» n 2
~
(Y1,···,Yn-1
(A1 (l), ... ,A _ (1» n 1
, Ai(l)
polynomial.
Finally
( ,... ,A _ (1))m (All) n 1
is a polynomial
in
5.1.3
(Embedding
Corollary.
Use the polynomials to
~n
. Then
(~j,*)
N:=
(j=l, ... ,n)
(~j,*)
with
1
and
m
because
r
of
of the lemma
into
applied
to powers
N)
P. of 5.1.2 to extend the multiplication l n (R ,*) is a torsionfree nilpotent Lie group as normal
the unit cube
in
subgroups.
(zj,~)
is a discrete
{(t , ... ,t.) ; 0 < t. < 1} 1 J l-
*
from
zn
with subgroup
as compact
of
fundamental
domain.
Proof.
All claims
(e.g inverses,
statements about polynomials n n n on Z,E x Z etc..
5.1.4
Definition,
The A-normal X. l
Yi
associativity, n
on
(Left invariant
generators
n
IR,IR
metric
on
for
r
Y1""'Yn
vanishing
of commutators)
etc. which
are known
to be true
N) determine
a basis
:= t(y.) € T M , i=l, ... ,n (2.3), they also determine a basis ~1 P := exp Yi E TeN = L for the Lie algebra L of N. We introduce
product
on
L
such that the linear
are
n
x IR
map given
a scalar
by
Xi + Y is an isometry i T M and L , and extend this product by left translation to a p Riemannian metric on N . - Clearly this definition is designed to make the between
"lattices" show that
r· p
c. M
and
this aim is indeed
rc
N
as nearly
achieved.
74
isometric
as possible;
5.4 will
THE NILPOTENT
5.1.5 ~.
For any simply
and the terminating describe From
power
connected
series
the multiplication
this description
Lie algebras nilpotent discrete
by polynomials
extends
to a homomorphism
Lie groups.
However
polynomials
one can use 7.7.4 formula
2n
of the corresponding
this description
(7.2.7) to
in the Lie algebra.
that any homomorphism
r
(5.1.2) on the other
of
endomorphism
of the Lie group
is polynomial
polynomials possesses
statement
5.1.6
Lie group
of degree
one sees immediately
endomorphism
which
nilpotent
of the Campbell-Hausdorff
is not well
of nilpotent
simply
connected
adapted
to uniform
subgroups.
The Malcev
Malcev
LIE GROUP N
show immediately
and consequently
N
are available
a uniform
hand
constructed on any simply
discrete
subgroup.
extends
in 5.1.3.
to a polynomial
We show in 5.1.8
connected
Hence
that any
nilpotent
we can rephrase
that
Lie group the preceding
as
Proposition.
(r r+ r
determines
N)
A
Any isomorphism
between
uniform
discrete
subgroups
of the simply
A
connected
nilpote~t
Let
cor~llary. M,
M
covered
N,
N
respectively
N + N .-From this we immediately
isomorphism
5.1.7
Lie groups
(TIl (M)
be compact
determines
r\
to a unique
deduce
N)
n-dimensional
by nilmanifolds
extends
Riemannian
manifolds
which
are finitely
A
N
fl \ N . If
resp.
M
and
M
have
the same
A
fundamental
Proof.
Observe
subgroups in
N
group
of
then
TIl i~ again
as well
fl
=
N
fl'
that the intersection
as
N
5.1. 8 Proposition. Let
N
(Malcev
be a uniform
a finite
index
and we can apply
[22
discrete
J;
rA
fl
of the two finite
subgroup.
5.1.8,
uniform
subgroup
:=
Consequently
fl'
index is uniform
5.1.6.
discrete
subgroups)
of an n-dimensional
simply
connected
A
nilpotent
Lie group
°
N.
Then
fl
has a "triangular"
set of generators A
1, •. .,0n
written
as
(Le . 0~1.
[oi,ojl E
multiplication
of
(with the exponents
N
; y Go· fl
is given
by the Malcev
as coordinates).
yeo N
can uniquely
be
(t , ... ,t) €. zn and the n 1 polynomials of fl =
if and only if
75
-
. Each
P. BUSER AND H. KARCHER
Proof. (i)
The fact that
Consider
exp
is a diffeomorphism
a left invariant
metric
on
N
(7.7.4)
is used extensively. Ilad II < ----11 where OOr (7.7.3). Then
such that
A
r
is the diameter is r-dense
is still
of a compact
fundamental
domain
(2.1.4) in the ball of radius
(2r)-dense
in the ball of radius
lOr lOr
of
-
6\ N
around around
e ,hence
the origin
exp-16 of
A
L
=
TeN
(7.4.4).
In particular
B
(0)
10r
(ii)
The triangular
trivial.
exp
(l
Then we have
spans
set is obtained
For the induction
1 exp- 0
:=
-1 6
Ye
for all
inductively.
step take
f
01
has minL~al
1
the Lie algebra
e
length;
exp-16
identity
on
spans
L,
Nl 61\ 6
6
(l
that
from the center
put
Proof.
{n
D
is in the center
Lie group
€
N
that
N
{N .• n,n E l.
NiO
of
such that
map
power of
L,
Exp ad Y
1 Log
series
is the to
hence
N
is a generator
subgroup
maps
< d(n,6
for
of the simply
6
1
. We claim:
connected
D}
the compact
- {e})}
which
therefore
Dirichlet
(for
fundamental
N mod 6)
contains
a set of
of (N \N)mod (6 \ 6). The orbits 6{0(0 € 6) lie on the 1 1 (oE 6) in such a way that the segments between consecutive
r.epresentatives "lines"
6
N \ N 1
d(n,e)
A
01
discr~te
N +N \ 1
onto the set of orbits
(terminating)
is in the center
1
and observe
The projection
of
is
Nl
that the linear
the
Y
is a uniform
nilpotent
dim N = 1
exp Y .
it follows
N1
N
exp-16
L , and - applying
Exp ad Y1 - we obtain
of
The case
exp ((Exp ad Y 1) • Y)
Since
L
76
domain
LIE GROUP N
THE NILPOTENT
points
III
of
have
Ily 1 II ;
lengt~
also,
there
are only
finitely
many
0 E. Il
IIY II} ,
in the compact
set {n EN; d(n,D) ::: so that only finitely many 1 exp( [O,lJ • Y ) ·0 (0 E Il) meet D. This proves discreteness 1 A A A Il in Nl \ N • Now the projection N + Nl \ N is continuous, hence
"segments" of
Ill'
the image
Now take
compact.
a triangular
Then
0
together
N/N
to
N
1 triangular
for
set of generators
with
a set
02' ... '0~
02' ... 'On
Il and the unique
of
Il/IllAin
of preimages
normal
word
N/N
1 is
N nIl
in
decomposition
carries
over
from
A
(with all
1 as in 5.1.2
0 E Il having
the product
in
integer
Il is given
exponents
by the Malcev
since
polynomials A
to these kernel
generators. ffiY
1
in
The differential
L,
The
between
-1 0i
(terminating)
(t , ... ,t ) n 1 namely:
In particular, coordinates hence
is polynomial
]
the transition
coordinates
t. + R.(t. 1, ... ,t ) (j=n-2, ... ,1) ] ] J+ n
=
of
by the Malcev
identifications
5.2.1 Actions
of
p EM.
and exponential
that
L.
N
is polynomial
polynomials
(5.1.5)
on the integer
also in Malcev lattice
(ii),
everywhere.
5.2 The local
Let
shows
of
by 7.2.7
the multiplication
and given
formula
basis
n
x. ]
R.
has the
•
x
each
A
1
is a triangular
Campbell-Hausdorff
is polynomial,
eXp-l(ot1 1
where
Il = <0 »; 1 with respect
N + N/N
of the projection
1, ... ,n)
(i
Malcev-coordinates
(x , ..• ,x ) 1 n
n
hence
exp
(iii)
N1
M
r
and preferred
be the universal The group
and their
r
average
bases.
Riemannian
being
r-equivariant
covering
a finite
77
index
of
M
subgroup
and
PE M
a point
above
of the deck trans formation
P: BUSER AND H. KARCHER
ITl(M,p)
group
uniformly
(4.6.5) acts uniformly
on the simply
connected
(5.1.3). We use the two actions
M
on
nilpotent
r
of
by isometries. Lie group
to transport
N
r
Also,
acts
by left translations
the bases
=
{x1,···,x } C T-M (identified with T M ) and {Y1'.·.'Y } C T N L of n p p n e 5.1.4 to the tangent spaces T -M resp. T N and consider all these tangent yp Y spaces identified via these preferred bases.
5.2.2 The injectivity 1/2 R
The number of
of
of 4.6.2,
M.
4.6.3 is a lower bound
for the injectivity
radius
exp- : T-M ...M p p
Proof.
Since
injectivity
P
radius
in
M
the maximal
a
as
loop in a nontrivial element
of the
a
~
Because
homotopy
p
of
exp_ is (much) larger than R, p the existence of a geodesic loop
imply
This loop projects
(fixed point
to the endpoint lift of
would
< R
of length
the same length
rank radius
< ~ R
radius
of 4.6.5, the loop
free)
loop
a
a
deckgroup a
to
at
[aJ
of the covering M
p
in
is the shortest
[a] € ITl(M,p) • Now
class
of the lift of
is the closed
to a loop
an
a
M
geodesic
considered
M'" M , maps
- a contradiction
since
at
of
as
p
the
a.
5.2.3 Local diffeomorphisms. We define
for all
Y E
r
local maps
F
F
Y where
exp TN.
y
BR/2(YP)
y
W = Ly 0
W
_
Y
(The identifications
are suppressed
Y
T _M
yp
N
R/2
around
-1 exp _ (x)
yp
yp
on which
the exponential
(5.2.2), and
has an inverse
exp 0 (dL ) 1 : TN'"
o
:=
is the ball of radius
exp _ : T _M ... M
map
(x)
Y
is the Lie exponential
++ T N
y
left translated
from 5.2.1 via the preferred
bases
in the notation).
5.2.4 Local equivariance. The compatibility
of the local maps
with the two actions
from 5.2.3:
78
of
r
is obvious
to
THE NILPOTENT
5.2.5
Average
of the local maps.
We now interpolate obtain
a
image
F(x)
r
the maps
equivariant
Fy(X)
D
, ye
since
compact
is
radius
support.
y' E r
F(x)
y
*
F(x)
of Gromov's
5.1.4 played rank
is a differentiable
for all
N
- hence
a
diffeomorphism (3.6»
..
X€
by working
M
the
weighted with
the
•R
(3.6.3, 4.6.2)
we take as weights
and the sum over
discontin~ous
M
on
and
\jJ
r
is
has
(5.2.4, 8.3.2)
(8.3.1)
and differentiable
(8.3.3).
theorem.
no role. x ~
M . +
map of maximal
It will be used in 5.4 to show that If we anticipate
this now, then
f\N
rank between
compact
manifolds,
hence
a
Its lift
F
a
of 8.3 to
~{Fy (x),¢y (x)}.
:=
r\f.i
is therefore
• For
of the various
slightly
from 8.3.1
acts properly
F(yx)
So far the metric
covering.
of mass
technique + N
Now the map
5.2.6 End of the proof
finite
M
,/,(d(x,y'p) -1 'I' D »
cut off function
M+N
has maximal
M
of
L
• (
r C ITl (M,p)
r-equivariant:
dFx
F :
8.3.1
2-(n+1)3
<
(j
D
is the smooth
F
as the center
<
d(X,yP»
\jJ
map
r\ M ,
of
\jJ(
finite
the averaging
r . Modifying
than the injectivity
where
with
y
differentiable
D/2
rather
F
will be obtained
image points diameter
LIE GROUP N
:
r - equivariant r- equivariant between
+
finite
r\
covering
covering
N •
79
"
N
diffeomorphism.
the finite
and the nilmanifold
f.i
I
of the simply
Now
r\ M
r\ F of
connected
space
is the desired M
(with deckgroup
'" G
---
-
--
.--
-
----
-------
P. BUSER AND H. KARCHER
5.3 The left invariant metric and its curvature. In this section we derive from the commutator estimate
(in terms of loop
lengths)
It[a,s] I
:: :6 It(a) I It(S) I
a similar bound for the Lie bracket of terms of the Lie algebra
L
L.
(3.5 (iii»
In order to restate 3.5 (iii) in
we need
5.3.1 Definition. n
G(X)
:=
I
i=l Note that each
n E N
word, and (with 7.7.4)
x.y.
~~
if
exp
X
has the decomposition G :
L. ~
:=
n
span{Yl""'Y'}
= ~
Y tl 1 .. • • • +
Li
*' Y tn1
as norma 1
is a diffeomorphism.
Moreover
(i
We have from theorem 4.5 . kl wr~tten as y = Yl *...
*
(iii) that every loop k . Yn n thw~
It(Y) - 2: k. t (y. ) I
~
and the norm in
L
~
<
-
y
2, ...
,n)
of length < R
6 12: k. t (y. ) I
~
.
(4.5
~
can be
(iv»,
has been defined such that
Finally recall from 7.1.2, 7.2.7, 7.7.4
5.3.2
exp H((Exp ad
X) •
Y,-Y)
[exp
X,
exp YJ
Therefore we can rewrite 3.5 (iii) (with a correction
6 :::. 1/50)
as
80
N
•
factor
(1_6)-1(1+6)2,
LIE GROUP N
THE NILPOTENT
5.3.3
IG(H(Exp
if
ad X) • y,-Y)
exp X , exp Y
I
<
4.38 IG(X) I·IG(Y) I p
rR
E
In order to obtain a bound for
I[x,YJ I
from 5.3.3 and 5.3.2 we need in the
inductive proof and again in 5.4 also a comparison
rR
X E exp-l
and the corresponding
between the vectors
lattice vector
G(X)
which is similar to
4.5 (iv).
5.3.4 Proposition. Same assumption
(i)
(Norm of the Lie bracket)
as in 4.6.2. For
Ilad xl
II <
L.
1
i
p
if
zl
<
X E L. 1. 2 2 z·n
4.98 P
with
L
n
L)
Y1' .•• 'Yn is a normal basis (with 4.1.4 (iii»
Ilad xl
L.
<
•
Note that
(iii)
or
Z
:=
implies
holds
(see also 7.7.1)
1
IG(Z) -
l, ... ,n
Ixi
58
if
(ii)
=
II <
(5.1.4, 4.5 (i»;
for all
X €
L
therefore
(i)
•
1
Remark. The induction estimate constants
is needed to control error terms whereas the main
(5.3.3) is induction for all
To interpolate
free. Therefore the inequalities
5.3.3 from the discrete set
following
81
~
have the same
i
we also need the
P. BUSER AND H. KARCHER
5.3.5
Lemma.
Let
DeL
Br(O) C L . Let
W.
space
A:
be a set which L + W
be a linear
Then
11A II
Proof.
IAI
D 11 B
. However
r
(otherwise
map
(2.1.4) in the ball
from
L
into some metric
X
from the convex
vector
Put
max{ !. r
---
is a-dense
IAXI
a supporting
cut off so big
r
<
is also the maximum
D
XEDflB}.
W
a piece
of
hyperplane of
B
for
of
r
con(D D
that
(l
could
hull
B) would meet Br-2a r not be a-dense in B ).
of
and
r
Therefore
r-2a r
>
Proof
of proposition
is in the center
5.3.4.
of
L,
The start
we have
IIA II •
of the induction
ad Xl
= 0
is trivial:
for all
Since
X ELand
L1 G(Z) = Z
To prove
on
L2 . Now assume
(i) on
Li+1
5.3.4
we first
(i) and
consider
preimages
-1 i+1
(Ii. j=1
X
=
G
Y
=
G-
The assumption
R
k.Y.),k.E j=l J J J 2-(n+2)2
exp X , exp Y €
IHuExp
of lattice
points:
2" R ,
or
X = Yi+2,· ooY n
1
(I
IG(Z) - zi < 10-31zl
therefore
i
1
Y .), 1. E z, Ixl:: J J J
1 i+1
< p.
(ii) for
Z,lyl
::2"R
(4.6.2) and 5.3.4
for all
fR
implies
satisfying
and we can rewrite
ad X)Y,-Y) I
.(ii)
<
82
4.48
p
Ixllyl
5.3.3
.
as
IZI
that
<
R
L1
LIE GROUP N
THE NILPOTENT
Next
IIad X IL
II, IIad Y I II < 10i L. (iii) to obtain ~
7.7.5
I[x,yJI
3
by 5.3.4
:::.1.01 I H«Exp
(iii), therefore
we can apply
ad X)Y,-Y) I ,
hence
I [x,yJ I Finally
we recall
B / (0)
in the ball 10
i+1 -(n+1)3 (2 R)-dense I . 1 ill.Y. are J= J J (5.1.4,4.5 (i», so that exp-1(r / ) is R 2 by (ii). Now we apply 5.3.5 twice to obtain
that the lattice
R 2
-4
R-dense
B /
in
R 2
I [x,yJ I
(~)
for all
X,y € Li+1
To prove
.. (~~) on
<
points
4.68 p
X = Yi+2"
or
.. 'Yn
.' we cons~der
Li+2
Ixllyl
_ ~i+2 Z - Lj=1 ZjY
j
. w~th
~i+2 L =1 j
I <
I ZjY
j
and define
(j
=
1, ... ,H2)
Then
I G-1 (Z) (7.8.2
(ii):)
-
zi
i+1 IIj=1 H(Z(j),Zj+1Yj+1)
<
-
i+1 Ij =1
I
[Z(j) ,Zj+1Yj+1]
- Z(j)
I
- Zj+1Yj+l1
I
• f( Ilzj+1 ad Yj+1 L
II) H1
( (*) above
(5.3.4
(4.1.4
:)
(ii):)
:)
<-
4.78 P
IH1 j=1
I Y I· IZ I Zj+1 j+1 (j)
< -
4.88 p
H1 Ij=1
j IZj+1 Yj+11 • IIk=l zkYk I
-<
4.88 P
2
1/2n21
I z·
83
Ij+2 k=l
I I zkYk
2
n2/2
R
• P. BUSER AND
This proves
5.3.4 .(observe
5.4 Maximal
rank of
5.4.1 outline. maximal
rank.
arbitrary that
curve
dF
section
Xo €
through
has maximal Xo
I
Since
Y €:.
M ,
5.2.6 by proving
we complete
abbreviate
L
t
(8.1.4) it therefore
d (dt ~(d(xt,yp)/D)t=o
• exp
of
sums of images
different taken
E
(~)
~(d(xo,yp)/D)·
• 5.4.2 Lemma.
(:t
exp
suffices
to show
proceeds
in the following
(*) is compared Fy(Xo)
-1 0
Fy(Xo)
F
Yo
r
y
steps:
(5.4.2). The distance
is shown to be small
local maps are close
together
-1
~
L Y
local
has
t
o
+
both
dF
:= F(X ) and let t + x be an o t x with I~I 0 It is shown in 8.3.3 and 8.3.4 o d rank if and only if (dt U(xt'Yo»t=o to, where
YEr
The proof
that
Yo
r
0
(*)
I) .
F
In this Fix
H. KARCHER
1.01IG-1(z)
\zl:::
IIIIIil
(x » t t=o The number between
of terms
the different
(5.4.3). The differentials
to each other
(5.4.4). These details
of the are
in 5.4.5 .
(Number of terms
in
(-1-»
Define
then
Proof.
Around
each
yp E
M
we have
the fundamental
84
domain
in
THE NILPOTENT
with volume D:=
V:=
vol(r\M)
diam(r\M)
LIE GROUP N
and contained
<
gives
(in view
<
of the strong
(sin~ Ar)nl
curvature
assumptions)
• (Si~ Ar) -nl r = llD
5 .. 4.3 Proposition. Same assumptions and
OP
E rp
(Distances
is a point
If
closest
x
o
to
<
(x ),F~(x»
you
r
between
as in 5.2.3.
d(F hence
of radius
. Clearly
>
Now 6.4.1
in a ball
0
the local
€
M
and
7D
images) y,o E r satisfy
,yp) < lOD
d(x 0
xo
1.6
(closer than
-n
D)
then
Dill
also
(8.1.4,
8.1.8),
(7.4.4,
8.1.9)
-1
I exp F (Xo )FY (x0 )
.
Proo!..
!~. y
• yp
/\.
• op
The local maps tangent
spaces;
between
these
exp,
~.
x0
0•
in
M
F
(5.2.3) are defined
y
to prove
tangent
expo X
X
in
5.4.3,
spaces,
5.4.4
namely
85
with
Y
N
exponential
we therefore the following
need
maps fairly
from different canonical
maps
P. BUSER AND
(i) Identifications. isometrically bases.
Using
Recall
identified
H. KARCHER
that all tangent
in 5.2.1
T -M,T N (y € f) were yp y way via the preferred
spaces
in a f-equivariant
dL : TeN + TnN we extend this canonically to n of all tangent spaces of N. Such a canonical procedure is
left translations
an identification
x and 0 are fixed throughout the proof o T M (x € B (x» via radial parallel transx laD 0 p lation (from x to op ) onto T~-M. Note that P ,: T _M + T~-M up raa yp up rad does not agree with the above f-equivariant identification if the rotational -1 part (2.3) of the loop is nontrivial; however 3.5 shows for o y at p E M not available
on
M,
but since
we will map all tangent
all
with
y
d(x 0
spaces
,yp) < laD
IIidentification
(ii) Errors
On
M
caused
we have
It(O-ly) I < 11 D):
(i.e.
error
P
of
ra
by the nonlinearity
from 6.6.1
and 6.4.1
T -M + T~-M II < 18 8 Dip. yp up -
d
of the exponential
and 3.2
maps
of
M
and
N.
(K)
<
<
On
N
consicer
observe
2
.!.. A 3
• 11 D
vectors
Y
E
12 A D 12 AD • sin 12 AD
TyN
, X
E
= expy H(-Z,X) (7.2.7). Ixi ~ D , Iyl ,Izi ~ 12 D -
d(exp
o
If there
X, exp
y
y)
d(exp
y
H(-Z,X),
f(14 D • [lad II) • IH(-Z,X)-yl
<
f(70 8 Dip)
<
1.01
would
(IX-(Z+Y)I
be no further
(ii)M-would
complete
+ 5 ~
p
, Z := exP
Then 7.4.4,
<
(IX-Z-yl +
<
81y
ToN
expo X
assuming
from
sinh
exp
y
7.8.2
(figure above) and 5.3.4
and
imply
y)
I[z,x] I· f(5 8 Dip»
Izl·IXI)
errors, then the proof.
86
.
(ii)N - wi th
Ix- (Z+y) I < 45 A2D3
-
THE NILPOTENT
(iii) Errors
caused
In the definition (figure above) ferred
. In
r
Therefore (E)
vectors should be the vectors X and Y in -1 _ d exp - x E T~_M instead of the r-transport ra yp 0 up the identification error (i) <
-1 M exp - x , i.e. we made yp 0 1 < (18 8 Dip) • lexp-- xl. yp 0 (E)
in (E) we have to be able to compare the vectors ~1 . N o-ly expo y . S~nce is a loop (in r) of length < 11 D we -1 kl k the normal word decomposition o Y = Yl k ... ynn as a product of loops
*
(4.5) as well
and 5.3.4
gives
L
Note that longer
than
initial bounds
r
as in
1 exp
~
-1 -1 (0 y) - ~ k.Y.
tangent
(x you
have much
1
~~
and
t(O-ly) by
<
all errors
), F ~ (x
)) 0
error
P
are canonically
identified
error
since
is caused
P8
(11 D)
vector
in (i) and not -1 eXPoj) yp is the
of the loop, but
3.5
2
in the described
way in
(ii)N
to obtain
<
(5.2.5) and
smaller
I~ kiYi 12
n 58 2 2
the final tangent
1.6
<
D < 2a
(v) implies
(Y.
the difference
d (F
(4.6). Now 4.2.3
k. t(y.) and L k.Y. ~ ~ ~ ~ 11.1 D . Another small
(iv) We insert
where
M
from
P
N
Next, to use -1 eXPoj) yp and
transported
-1 -1 exp~_ x and exp _ x up 0 yp 0 to N via the pre-
these
we have
of
have
is not abelian.
of the local maps the vectors
are r-equivariantly
bases.
(E)
since
5.2.1
LIE GROUP N
1.6
-n
Dill,
Dip ~ 2
estimates
_n2
• 40
-n
than needed
87
(4.5) were used. in the present
(Obviously
proof.)
we
(i)
• P. BUSER AND
5.4.4 Proposition. Assumptions
(The differentials
as in 5.4.3,
Id
exp
H. KARCHER
-1 0
dF
o
Y
Yo
=
Yo
of the
F
~
almost agree)
F(Xo) , compare 5.4.1 (*)
x - d exp
-1
Yo
0
dF
0
0
0
<
X1
Ixl
0.1
Proof.
M
in
in
N
Recall F (x) = exp 0 identification(T -M + T N) 0 expy-~ x . We have to y y yp y p compare the differentials of the exponential maps involved with respect to the identifications
5.4.3
(i). On
N
N , Iyl < 11 D , n -
:=
occurs only one type of error controlled by
7.4.4 and 5.3.4: For
YET
y
exp Y Y
holds
II(d expy)y - dLny-f II ::: f(ll D Ilad II) - 1
and with 5.4.3 and
W
1 II(d exp- )w Yo y On
M
:=
Yo -
F (x ) Y 0 -1 II <
dL y
o
exp _ Y yp
0
F
y
(x
)
f (D· IIad II /10) -1 2-f(D·lladll/10)
(K
=
y
o
<
8 D/p •
N,
it is
0
0 , IKI ~ A2) : For
on
y ~ TypM , Iyl < 10 D ,
holds
I(d exply - PYi? + xl For
-1
exp
55 8 D/p
we first have the same type of error just discussed
controlled by 6.4.2 X
:=
Y
:
<
sinh 10 AD _ 1 10 A D
this is the only error; for
88
y
F 6
<
there are two more:
r LIE GROUP N
THE NILPOTENT
since parallel to use radial identified between
M
in
is path
parallel
translation
by parallel
translation
this
6.2.1,
translation
radial
to
identification
6.7, 6.7.1
(for
we have
this adds the identification
Now add the error
I dexp
-1
on
Ox
odF
Yo
Proof
Since
of 5.4.1
had to be
(here for the use of 6.4.2).
The error
P yp + x
is controlled
P _
yp + x
I
We have
5.4.3
-1 0
dF 0
Yo
(i).
errors
0 0
under
I
~
more
x1
on
<
better
than needed
I~I/D
we have
specific
1.5.1
to find an upper 2 IKld (M)
bound
the assumption 3
(En). wen) E
(m=10)
T _M + T.\'_M W up
transport
M
to prove
(74 8 D/p
5.4.4:
2 2
+ 30 AD)
o
I~ I .
here.)
(*).
of Corollary
w = 2(n+2)
by 6.1.1,
from 5.4.2,
5.4.3
5.4.4
5.5 Applications
Proof
(i)
<
>
5.5.1
in 5.4.3
spaces
and the three
- d exp
Id~ ~(d(xt'YP)/D)
and from 5.4.2,
-
error
are much
tangent
to the r-equivariant
0
Y
(Again the estimates
5.4.5
N
where
we decided
< 1.1 D) :
d(op,x)
to come back
op
and
Ip (T -M + T.\'_M + T M) rad yp up x
Finally,
depending,
0.8
Nl
0
I~I .
assumptions
for the injectivity
radius
at
p ~ Mn
under
< E < E if r C IT (M) is not abelian. Take n 1 in 4.6.2 and 3.2. Then we find with 3.2 P > 2wd(M)m4
48
0 -
I[a,S] I <-laloISI for all loops a,SE r with - p 1 lengths lal,ISI < -3 P0 (3.5 (i~i». By 4.5 (i) there exist two noncommuting n loops a,S E r of length < 2 0 (J = 2n 0 3w(n) d(M) hence [a,S] is a non-
..
suchthat
89
.. P. BUSER AND H. KARCHER
trivial
5.5.2
geodesic
Proof
Consider (1.4.
of Corollary
the family
(iii».
(\ I ••• lap geodesic
loop at
If
in
r
loop. As
gt
r
p
of length
such that 00
d(M)
•
1.5.2 of E-flat metrics
has nilpotency
t +
<
S
degree
r\l' ~o
on
P
I
I
e
-2bt
<
_ gt < e
-2at
then we find loops
[a1,a2]; ••• ,a J is a nontrivial p E + 0 ) we obtain from 3.5 (iii)
:= [ .••
(i.e.
<
where the constant at eI and
lsi
2
for
b > pa
is independent is of order
of ~
t bt
e-
.
90
and where as
t +
00
the •
la.~I
This
are of order
is only possible
<
lid II
6. Curvature
controlled
6.1 Curvature.
The covariant
t + c(t)
curve defined
is denoted
near the image
covariant
differential
of the second
R(u,v)X
constructions
derivative
of a vectorfield
:t X ; if
X(t)
of
c,
then
of
Y
The curvature
covariant
, Y
X
v,u
X
i-
D
u
DY
is called
is the skewsymmetric
(D X)
v
D X. D V U
If
D
is the Levi-Civita
sectional
of a Riemannian
metric,
we have
sectional
(R(U,V)V,U)
curvature
RK(U,V)X
Sectional
curvature
is a symmetric
orthonormal.
implies
K - «v,x>-u
bounds
bilinear
u,v E T M P
K
- -v)
8::: K::: /'; imply
.
(since
b(u,w)
:=(R(U,v)v,w)
form and since
6 R(u,v)w
R(u,v+w) (v+w) - R(v,u+w) (u+w)
R(u,v-w) (v-w) + R(v,u-w) (u-w) ,
see
6.1.1
[5
J). IR(u,V)v
- R/,;+8(u,v)vl
2 ::: /';;8lui Ivl
2 <
~ (/';-8) 3
for orthonormal
u.
~
2 max(/';,-8, "3 (/';-8»
<
i
91
2111111
the
curvature
K
Constant
deri¥ative
maxlKI
a
a vectorfield
:t X = Dc Y , where tensor
along
differential
X - D2
:=
= Y(c(t»
t + X(t)
part
P. BUSER AND H. KARCHER
Because
of 'these inequalities
sufficient
control
6.2 Parallel lations;
Translation.
curvature
proof
of Gromov's
Assumptions. a homotopy
and for all
Let
E
of the curvature
the definition
cases
C.
~
t E
dependence pinching
[0,1]
:
between
->-M
[l,2J
over
M
with
translation
6.2.1
Path dependence
along
of linear
parallel
trans-
implies
This is basic
for the
see 2.3, 2.4.
of the paths
c,
~
with
c
t
a fibre norm
and a covariant
is a norm-isometry.
Let
(0)
the same initial
vector.
Then:
translations.
(Riemannian
(general
Let in the Riemannian
case
s ->- Y.
~ (s)
be vectorfields
which
are called
in both
from p to q and c (1 < t < 2) t for the area of the homotopy
o , Xl
~
vectorfields
give frequently
immediately
on the path.
for the lengths
bundle
and affine
tensor
theorems,
s ->- X. (s)
be parallel
linear
Assume
such that parallel
bounds
be curves
and
be a vector
derivative
We consider
and other
curvature
tensor.
of the curvature
controlled
Let
sectional
"affine
92
parallel".
Then:
case)
case) •
CURVATURE
6.2.2
Path dependence
CONTROLLED
of affine
CONSTRUCTIONS
translations.
(Iy. (0) I+L).
IIR II· F
1.
Remark.
The affine
"development to
c(o)
of a curve":
Let
, then the image
the euclidean
geometry
in the Riemannian
Proof.
translation
Let
let along
=
yeo)
0
related
and Levi-Civita-translate
of
since
c ' t
along
€
E
<
I
1
X,(1})
TqM
2
I~
Similarly parallel
o .
2
I
<
-
IE
I :tX(1,t)1
dt
1
q
:t X(s,t) I ds dt
d
1
D
(since
dt X(o,t)
=
(since
.E.- ~
0)
d
IR(as- c, dt c)X(s,t) I ds dt
0
2
<
is paral-
0)
0
I I 1
I}
1
r
which
joins
q
IXl (1) - X2 (1) 2
the homotopy
i.e.
D
14:(X,(1),
back
M.
ds X(s,t)
t ->- X(1,t)
yes)
P
of
X(o,t)
Then,
of the
T M is called development of c; it has in p T M the same Frenet invariants as the curve c
be the unit vectorfield
the paths
to the old concept
in
geometry
X(s,t)
is closely
IIRII· Ixi •
let along
Y(s,t)
d I 11 Ias c " 1 o
X
=
d dt c I ds dt
be the vectorfield
the paths,
dt ds
along
the homotopy
which
is affine
i.e.
D
ds Y(s,t)
Since
the difference
field
it suffices
of two affine
to assume
special
parallel initial
93
vectorfields conditions
is a linear Y(o,t)
=
parallel
0 E T M
P
10
P. BUSER AND
H. KARCHER
Observe
1
<
IY(s,t) I
D
J 1Y(s,t) ds
+
IY(o,t) I
<
Ids
<
L
o Then,
t + Y(l,t)
since
joins
Y (1) 1
to
Y (1) 2
T M
in
q
2
J
<
I!
Y(l,t) Idt
1 2
1
J J
<
1
la~
(!
a Y - at ct) (s,t) Ids dt
0
D
(since
0)
dt Y(o,t)
a
at c)Y(s,t) I ds dt D dt
(since
<
6.3 Jacobi
fields
of geodesics these
gives
~..E..
J
ds ds
together
These
with
initial
constructions.
rise to Jacobi
They are determined
6.3.1
the initial
conditions
information
are the Rauch
entiating
the radial
Any construction Jt(s)
by the Jacobi
+
=
with
:t ct(s)
families along
equation
o
are specified
integration
the geodesic
procedure
0)
conditions
then one further about
) t
ds
fields
by the geodesic
if they are such that one has good estimates equation,
~c
(IR II• F
and geodesic
ct(s)
geodesics.
L·
D (dS Y(s,t)
estimates
family
for the solutions
translates
construction.
at
exp r· p
94
the estimates
The best known
for the radial
of geodesics
construction
t
=
exponential 0
(v+tw)
in question;
of the Jacobiinto
example
of this
map: By differ-
r
CURVATURE
one obtains
a Jacobi
differential
of
6.3.2
J(o)
Under
CONTROLLED CONSTRUCTIONS
field with
clusion
that the length-distortion
6.3.3
~)
of the constant
cases,
Observation.
of the metric;
use
J(r)
'0 ~ K ~ ~
assumption
the values
resp.
wand
dr J(o)
between
(0
initial
data whose
one can show that
curvature of
exp
see 6.4 for a more
The tangential
< J, c ,)" =
all further
give
the
solutions
estimates
•
IJ(r) I
lies
with the final
con-
is between the constant p explicit statement.
component
0
(d exp) ° rw p rv
of a Jacobi
field
curvature
is independent
to obtain
(a+sob) ° c' (s)
Therefore
values
D
o ,
the curvature
simple
will
be stated
(a,b E R)
for normal
•
Jacobi
fields
J(s) .J.. c' (s) .
6.3.4
Abbreviation.
Jacobi
equation
In the case of constant
reduces
to the scalar
fn
If
K
those
is constant solutions
+
Kof
or a continuous
which
have
f(o)
1, f' (0)
=
if
f(o)
0,
K
the
O.
function
o
curvature
equation
the same initial
If
sectional
denote
we use special data as
cos
symbols or
by
the solution
to denote
sin:
C K
6.3.5
Lemma. K < ~
Assume solution f~(s)
(Estimates
> 0
of
and
in terms
Ic'l
f" + ~r2f for
f' (0)
=
denote
of upper
r . Let
f~
=
by
the solution
curvature IJ(o) I ° C
SK
bounds.) 2
+
IJI' (0) ° S
~r 0
with
the same initial
conditions
s E (o,t) • Then~
(i)
on
95
(o,t)
2
be the
~r
,
as
IJI • Assume
P. BUSER AND H. KARCHER
IJ(s) 1
(H)
<
IJ(o) 1 0 C
(Hi)
IJ(t) 1
<
"f(S')
+
2 (s)
IJI'(0)05
II
Rauch's
Equality
Proof.
Recall
that
J
(o,t)
<
2(s)
/:,r
(
E
s
f(t)
IJ(s) 1 ' s €
[o,tJ
fir
lower
bound ") .
holds
if and only if
L
is assumed
c'
= /:, and
K
6.3.3
after
IJI' (0)
iJ'
(0) 1
.
.
therefore
Now
o
lim (1JI f/:,- IJlof~) s + 0
gives
the first
inequality.
Next
IJI
>
(-)
0
and
f/:,
IJI lim -f- (s) s+o /:,
6.3.6
Lemma.
the second.
(Comparison
between
K < /:, , J (0) = 0 ,
Assume on
proves
1
c'
1
Rauch's
J = r
and
lower bound
is a special
J'
. Let
t
be such that
<
IJ(s) -soJ'(s)1
sharp
is sharp
if the curvature
positive;
proofs
S
2
is increasing
/:,r
(o,t) . Then
This inequality
case.
only
IJ(t)
1
0 } maxlKI
in the flat case;
is constant;
a weaker
2 2
o
s r
a more
complicated
one is
one holds
as long as
S
/:,r
are similar.)
96
2
is
CURVATURE
Proof.
For every
CONTROLLED
unit parallel
I(J(s) - s'J'(s),
field
p(s»'1
(with 6.3.5:)
CONSTRUCTIONS
P
and
s E (o,t)
=
IS"(R(J,C')C',p)(S)I
<
IJ(t) I
'Llr (s) SLlr
Use the crude
6.3.7
Lemma.
Assume Let
K
K
2(s) Llr
(Vectorvalued
twosided
2(t) Llr
0:K:
bounds
and put
for larger
s
S
and integrate
J .....
ds .
o
K:
A
=
LI and for simplicity
max(LI-K,K-O)
Ic'l
. (The inequalities
}(LI+o) , but they hold on a larger
=
1
will
interval
if
is smaller.)
Let
A
be the vectorfield
D
along
D
o ,
ds ds A + K"A
Let
a
be the solution
an
Let
t
+ (K-A)
a
be such that
6.3.8 Assume
Corollary.
SK
that
which
A(o)
A"IAI
satisfies
J(o)
, A' (0)
<
on
a(s)
initial J(o)
a' (0)
, a(o)
is positive
(Special
in addition
c
J' (0)
•
of
IJ-AI (s)
t
:
(t)
estimates)
curvature
be a parameter
be sharper
S
estimate
holds
and
(o,t) . Then
on
data,
o .
[o,t] .
lower
curvature
J'(o)
are linearly
S
is positive
bounds) dependent
and that
is such that
f
K
IA(o) I "C
K
+
IAI' (0) "
97
K
on
(o,t)
.
.... P. BUSER AND H. KARCHER
Then:
f
(i)
<
IJ-AI
(ii)
1
2"(LHo)
on
[Olt]
field,
this gives "Rauch's upper bound":
,
< -
fo(S)
=
IJ(o) 10Co(s) + IJI
with
=
0 , A2
=
A
K
a
;
IJ(s) I
c (iii)
=
K
is a parallel
(f _ - f ) K A K
-
with
, IAI-10A
K
=
maxlKI
one has a good comparison
IJ(s) - Ps (J(o)+soJ'
I
(0»
Proof. For every unit parallel
S
and
P
(0)
05 (s)
0
parallel
s
;
translation
with the flat case: IJ(o) I(cosh As-l) + IJI' (0)
*
along
(sinh As - As) •
field we have with 6.1.1
I(J-A,P)" + KO(J-A,P)I
If
b
satisfies
b"+Kb
vanishes
at
0 and is nonincreasing,
}"
This gives
IJ-AI ~ b
o
AOIJI ' b(o)=b'(o)
0
its derivative
SK - {
as long as
SK
then
}
0
S") K
<
is nonnegative
being
o. and implies
b" + Kb < Ab + AlAI •
As
before
;
0
{b-a}
vanishes
at
0
and is nonincreasing,
which proves
K
b < a
and therefore
the leIiuna.All claims in the corollary
98
are then obvious.
CURVATURE
Applications
to geodesic
6.4 Proposition. Assume exp
p
CONTROLLED CONSTRUCTIONS
constructions.
(Rauch's and other
curvature
bounds
: TpM + M
is a radial
exp rv
to the radial
(Ivl=l)
S~(r)
v
vi
0
=
one has the Rauch
The bounds
are sharp
inequality
holds
as long as the lower bound
inequality
holds
as long as
holds
Proof.
rv
We saw
P
r
r
SK
6.4.3
Corollary.
along
cases.
is nonnegative,
along
I
<
is nonnegative.
a different
can be compared
(Ivl = 1 , w.l. v)
Iwl
(K
The left
the right
(or with
The differential
exp rv
ow)
r
curvature
is positive
point).
= J(r)
(i), (ii) with
with these
(Angle comparisons
curvature
T = pqr c M
bounds
with
:
0
r arbitrary,
~ > 0
also
J(o) = 0 , J'(o) initial
A
max(~-K,K-O),
u
-~
Mo
' M~
T
("Aleksandrow
of constant
a geodesic
less than the radius
curvature
triangles").
99
•
o,~
in
6.3.2
triangles) triangle of injectivity
u < 2n ~ 1 . Then we have triangles
lengths
= w
conditions.
for small geodesic
0 ~ K ~ ~ • Consider
circumference
the planes as
constant
f(~+O)
SK(r) (-
(d exp)rvorw
6.3.5,6.3.8
and if
estimates
.)
Apply
Assume
(w) - P
as long as
S
conjugate
translation
Id exp
see 6.3.8
M
in the respective
up to the first
which
map
<
I(d exp) owl p rv
r
6.4.2
) p exponential
Ivl
direction
<
0_-
with parallel
exp
:
6.4.1
proof:
on
The radial
isometry:
I (d exp)
perpendicular
estimates
0 ~ K ~ ~ . Then:
which
have
For corresponding
To
' T~
of in
the same edgeangles
one has
•
...
I P. BUSER AND H. KARCHER
Proof.
S
To prove
Tt
S
a a a~ consider also triangles T{, o and the two adjacent edge lengths the same as T -1 use the distance decreasing maps exp 0 exp (resp. exp p Po p~ (6.4.1) to map the third joining
those
(resp.
p~)
vertices . Since
for the third
geodesic
of
T
edge lengths
of
shows
that the third
6.4.5
Remark.
without proof
this
consequence
Proposition. r
r <
1
2"
"IT ~
bound
-1
To
(resp.
T~);
this decreases
(resp. increases
the angle
that
u
is less than twice
a :
("Toponogow's
If more positive is more
[7
estimates
holds
theorem").
curvature
complicated
in fact The above
occurs,
in
and not just a
].
injectivity
ball
a
o
and derivatives
radius
Br(p)
of
exp
p
~ > 0
, and if
of
; let
exp) be an upper
~
assume
•
its second
B
f
r
derivative
the geodesic = X
p we have
has to be decreased
The lower bound
than the injectivity
grad
J(1)
SO.
field
from
than this curve
T~~) To
of
on the triangle
on the compact
(i) The function
along
(resp.
we only used
exp.
(Convexity,
be smaller
curvature
T;'
0 , then the proof
of Jacobi
are different
T~).
a~
of
0 + ~
if
0 >
if
S
a
To prove
any restrictions
shows
6.4.6
a~ of
of injectivity
particular
edge of
to the angle
0
p
a~
edge of
to the angle
the radius
>
a
ao
a
T~
which
T"
a
of
T~)
Then -1
exp
(resp. T ) onto a curve
edge is not longer
to give the third
the angle
of
T:
0
To*' T , T~~ the inequalities
>
(resp. increased)
Let
(resp.
the geodesic
6.4.4
This
edge of
have
which
a
the angle
e
1 2 R , f(q) ._ .- 2" d(p,q)
->-
-1
eXPq
f(q)
is expressible
from
p
to
q
T M
q
100
has the first
derivative
of the Jacobi
field
P
in terms determined
by
J(o) = 0 ,
J(s)
CURVATURE
CONTROLLED
CONSTRUCTIONS
J' (1)
DX grad f(q)
In particular 6.3.5 (i) and 6.3.6 imply
I D2X,X
<
r .
fl
(1 +
<
I
1
2
"2 maxiKI •
I xl 2
r ).
S'
f
so that
(ii) Any two points in
Br
in
(p)
-s6.
(r'
<
2r;
-
>
(r)
0)
•
6.
r
have in
connection of length q € Br
B
is convex on
Br
a unique and shortest goedesic
in other words,
a diffeomorphism onto
exp
q
is for each
Br
from a suitable preimage of
in
Br
Br
T M
q
Proof. (i) _____
For any geodesic
of geodesics from
p
to
q(t)
with
q(o)
=
x
define the family
q(t)
c(s,t)
exp
f(c;r(t))
1 2
p
s·
exp
-1
p
-1
q(t)
eXPq(t)(l-s) • eXPq(t) p.
Then 1
3
f < as c
3 , 3s c )
ds ,
o
3 < 3t3 c , as c)
d
dt f(q(t))
since
D 3 3s 3t
--c
-1
eXPq(t) P To the family
c(s,t)
and
3 3t c (o,t)
3 as c
o .
(l,t) ,
Now
grad f (q(t)) .
(1,t)
of geodesics corresponds the family of Jacobi fields
along these geodesics ._ -
3 3t
c (s,t)
which are determined by the boundary data
101
---------~!P;!'"...-... P. BUSER AND
H. KARCHER
.
q(t) D - dt
Clearly
The bounds 6.3.6 .
(ii) r1
-1 eXPq(t)
of the second
First
let
:= d(ql'P)
Because
exp
of 6.3.5
p
q(t)
exp
points
< 2r
of length
The exponential so that these (qo,ql) Br
E
minimal d(qo,q1)
Br x Br
form a closed
nections
from
and
cannot
c2
excludes
that
Br(p)
go
to
with metric
from
<
space
qo
to the following
:=
of
v(t)
Br(p)
• r
to obtain
any geodesic
at least
but have to
1
"2
«
r
o
to
IT
Br(p)
we
ql
11
-1
curve , hence
/ max(r
we
) q(t)
which
is
a shortest
,/v(t) /) 0
a shortest
of length:
unique.
geodesic
qo
two geodesic
connection
connections
rank
(6.4.1)
the set of pairs of length
take a pair
~ 2r
in
(qo,q1)
with
geodesics
c ,c in Br ' 1 2 (c ) : r . Now, if c and c would not 2 1 2 an angle < IT at, say, ql ' then we could than
ql
Ci1 ' contradicting goedesic
has a maximum
2r maximal
In particular,
set were not empty we could
form a closed f
(i) and
.
(c1) ~ length
closer
from 6.3.5
2r :
v(t)
are locally
with
geodesic
Ci1
q(t)
/v(t) /
q(t)
and two joining
length
find points
with
v(t)
If this
distance
=
v(t)
map has along
is compact.
curve
J~ (1)
which
<
geodesics
points
(l,t)
c
directly
< r . In the compact
a shortest
we take limits
in
now follow
but in the interior
p
D a as at
(1,t)
be interior
we can change
not longer than q(t) geodesic of length <
For boundary
Br(p)
=: ro
~ d(qo'p)
have (by Arzela-Ascoli) can write as
q(t)
Clt
derivative
E
qo,ql
asa c
D
(p)
and still
with
the choice
of
either,
since
on it. This proves
102
two geodesic (qo,q1)
the lower (ii)
con-
. But
bound
of
c 1
D2f
I'
CURVATURE
The following theorem power
example
CONTROLLED CONSTRUCTIONS
is not used
but in the application
of the results
components
6.5 Example. Because
fields
(The normal
special
2.5.3 which
in section
of Jacobi
initial
in the proof
Gromov
(6.3.3) need
exponential
conditions
special
were
curvature
goemetries
(of the same dimension).
Assume:
M
Let
[0,1]
a:
geodesic
is complete + M
give bounds
where
the
the tangential
which
for sharp
Jacobi
can be interpreted For certain
field
in constant
applications
it is
bound.
and simply
connected
be a geodesic
rays perpendicular
to illustrate
attention.
important
always
some explicit
flat manifold
map of a geodesic)
one cannot
to have
used
2 . It is also an example
estimates
enough
of the almost
to
segment
a
with
with
and
curvature
bound
[0,00) + M
S,y:
S(o) ; a(o)
, y(o)
K:
unit
; a(l)
_A2 <
speed
.
Then:
cosh Ad(S(t),y(t»
>
(cosh Ad(S(o),y(o»
;.
cosh Af(t)
- 1) "cosh
2
(At) + 1
,
hence
6.5.1
d(S(t) ,y(t»
with
f
strictly
>
increasing
f(t)
>
and
d(S(o) ,Y(o»
,
lim f (t) t+oo
Proof.
Let
where
v (r)
introduce sponding
s + J
From
r
c(r)
; eXPa(r)v(r)
the family Jacobi
along
a
s + c(r,s)
of geodesics
with
; 0
we have
J
ii~.«
from
; eXPa(r)S"v(r)
S(t)
to
y(t)
c' (r) and the corre-
we have J
r
(0)
;
D
(0)
r
c'(r),
0;
Jnorm(o) r
103
iiiiid
geodesic
and.J..a' • To estimate
fields
(s) ; ~ c(r,s) or
J (o)-L v(r) r give
be the shortest
is a vectorfield
-=..
_.&III!
...
---...,.".....,~-
ds Jr(o)
D
dr v(r)
D
<J,J'>(o)
(6.3.3) so that 6.3.5
since
can be used
to
o.
P. BUSER AND H. KARCHER
IJnorm (1) I r
>
-
Ia' (r) I • cosh A Iv (r) I •
Since D
Jtan(o)
o
r
and
~ Jtan(o) ds r
• v(r)
we find I:r Iv(r) I I, hence 1
f
ic'(r)ldr
>
o
f1 o
2 2 d 21(la' (r) I ·cosh A Iv(r) I + I Iv(r) II )~ dr dr
The right hand side of this inequality bound can be obtained by interpreting hyperbolic
plane of curvature
depends only on that expression
Iv(r) I and a lower in the 2-dimensional
- A2 ; in the 2-dimensional
case the inequality
is sharp since
~ Jtan(o) ds r
la' (r) I • cosh A Iv(r) I •
f(t)
1i(t)
The minimum is obtained
from the hyperbolic
quadrilateral
in the figure and
the formulas
cosh Ah
=
cosh At • cosh AR
sin
<jJ
=
sinh At/sinh Ah
cosh Af
=
cosh At • cosh Ah - sinh At sinh Ah • sin cosh AR • cosh2 At - sinh2 At
104
<jJ
CURVATURE
The next
result
the commutator polation
does not appear estimates
in section
6.6 Proposition.
Let
CONTROLLED
in section
M
of riemannian
translation.
)
which
agree
tangent
spaces,
<
G(w»
If we use parallel
the equivariant
inter-
und euclidean
parallel
T M let w(t) be the parallel vectorfield p q . We compare the following two maps from
=
in the flat case
: W + exp
G : w + exp
d(F(w),
We use it to prove
wE
F
6.6.1
manuscript.
2 and to construct
(Comparison
M be fixed; for p c(t) = exp tv ; c(l)
T Minto p
in Gromov's
5.
veT
along
CONSTRUCTIONS
t
p q
(!):
(v+w) (w(l»
Ivl • Iwl • A sinh A(!vI
translations
along
then we also control
the geodesics
+
Iwl) • sin ~ (v,w) .
involved
to identify
the differentials:
IdF ·X - xl w
<
-
Ixl
.
(sinh Ap _ 1)
IdG ·X - xl w
<
Ixl
.
(sinh Ap _ 1)
Ap
,
where
p
=
Iv+wl
,
where
p
=
Iwl
6.6.2
Proof.
The control
applied
to geodesics
consider
the family
Then
s + ~~ (s,t)
initial
-
of the differentials
p
of speed
is given by 6.4.2
rather
than
s + c(s,t)
of geodesics
is a family
Ap
of Jacobi
(K=O)
. To estimate
=
when
the distance,
eXPc(t) (s·(w(t)+(l-t)·
fields
with
linearly
~(t»).
dependent
conditions:
.
-c(t)
Since
the curve
t + c(l,t)
Jt(l)
we can apply
Jacobi
fields
6.3.8
involved
joins
G(w)
and has
(iii) to get the desired
estimate.
clearly
F(w)
to
are not perpendicular
105
the tangent
to their
vectors
In this case geodesics,
the but
P. B USER AND H. KARCH~R
the initial
conditions
are such that
cosh x - sinh x
we obtain
from 6.3.8
1. x
<
x
-
3
applied
• sinh x
<
G(w»
f
0
<
f1
norm
IJ (l) t
l
o
and
to geodesics
1
d(F(w),
J~an(l)
p
t.
With
max (Iwl, Iv+wl)
of speed
dt
dC
norm IJt(o)
for all
1
I· lasl·
3"
.
A • sl.nh II.pdt •
0
Finally
- <
dC ~) dt ' dS
2
I
s=o
2 12 - 2 Ivi • Iw+(l-t)v 2
2
Ivl
which
- 2
ends the proof.
The following of certain Note
Iwl
area estimate
homotopies
that this usually
result
is rather
lower
pinching
6.7 Proposition.
6.2,
from above
curvature
sharp,
We use it because known)
(compare
estimate
whereas
of A. D. Aleksandrow
sharper
is used
2.2) in the proof depends
bounds
than worthwhile
it is geometrically
of Gromov's
only on upper
are employed
elegant
to bound
curvature
the area theorem. bounds,
for such a purpose.
for our present and a typical
The
application.
(though little
result.
(A. D. Aleksandrow's
area estimate
for geodesic
triangles.) Let
T
6.4.3
=
pqr
be a small
geodesic
triangle,
i.e. make
• We span a 2-dimensional "ruled" surface 3 C -differentiable monotone maps f:
choosing p
to
q),
minimizing
g : [o,lJ + pr geodesic.
and by joining
The intrinsic
curvature
106
S
of
into the triangle
by
[o,lJ + pq
f(t) of
the assumptions
to S
g(t) is also
(= segment
from
by the unique < ~ , while
CURVATURE
a lower
curvature
bound
CONTROLLED CONSTRUCTIONS
depends
The area of the ruled the area of
Proof.
The curvature
or by observing on
M
and
surface
bound
for
S
S
g.
Then:
is not larger
follows
that the goedesics
as on
f
than
(6.4.3) • Tl::,.
S ; differentiation as well
on
gives
from
either
f(t)
vectorfields
J
to
from the GauB equation get)
which
are also geodesics
are Jacobi
fields
on
S , therefore
2(:S
J, :s
J) -
2
S
I EJI ds
>
and
Now ist suffices curvature lower
bound
curvature
vanish
Step
1.
Let
T
T~ V T~
=
to consider T as a convex triangle on the surface S . It is technically simpler to work also with K < l::,. bound,
somewhere.
the lower
bound
say
0,
Aleksandrow
although handles
and omit a final
T' V
Tn
is a
(usually)
some spherical
imply
be a division nonconvex
or hyperbolic
area
the general
limit
of
it may not exist
into
subtriangles.
quadrilateral
(T~ v T~ )
107
f
[2];
or
with
a g
we assume
argument.
T
geometry,
case
if
S
we get
<
6.4.3
in the plane
shows
that
Ml::,. and, with
P. BUSER AND H. KARCHER
Therefore, T' U
if the proposition
Repetition
of this argument
very
triangles.
small
dividing
segment
subdivision,
step
in
2. It suffices
T
T'
and
Tn
with
it follows
for
to prove
By repetition
Step 3. Use -----
area
exp
<
(T)
in
T
to prove
with
} u(T)
small,
plane
under
MI'J..
circumference
the midpoints
roughly
for
than the
become
curvature
(TI'J.) • (1 + const·
area
the claim
is not longer
the triangles
triangles
by joining
isometry
also areas
To complete area
(T)
0
<
(T)
exp
PI'J.
but not more
which,
area
2 area -1 P
u
2 u ) ,
of its edges,
. Application
are decreased
the proof
(TI'J.) • (1 +
1
"4'
into four
of step
2
const·
u )
(TI'J.)follows.
to map the triangle
in a ball of radius
than by a factor
T
} u(T)
1 1 -1 SI'J.(2" u) • So (2"u)
but not by more
into
MI'J. ; the map
, decreases
is
lengths
2 • u ). Consequently 2 (l-const • u ) !
= (l-const
than a factor
we show that the area of the image
of
T
in
MI'J.
(TI'J.) ; this then gives
area
(TI'J.)::: area
was needed
We describe obtain
for small
circumferences
area
which
segment
(*) to
improves
a radial
that it is enough
as fast as in the constant
can be subdivided,
triangles
2
for
TI'J.(6.4.4); therefore
area
since
shows
The dividing
at least
(~)
is
is true
Tn = T .
::: area
(T) • (1-const.
u2)
in step 2 .
the image
a division
(image T)
of
T
T = Tl V
as follows:
Subdivide
the edge
V Tn • The trigonometry
<
108
area
qr
of step
(TI'J.) •
of
T
1 gives
to
CURVATURE
As
the subdivision
exp
0
PD.
exp
-1
(T)
is made
CONTROLLED
finer
CONSTRUCTIONS
the polygon
since
TiD.
(6.4.3)
u ... and
P
V TnL!. converges Cl.iL!. --converges Cl. io
to
(by trigonometry).
6.7.1 ~. sphere
Among
the maximal
all triangles
with
area is realized
two edgelengths
by the
angle
"Thales
a,b
triangle",
between a and b equals the sum of the two others. . 1 1 TI 2 arc sw(tg "2 a' tg "2 b) • If a,b ~ 3" then area ~ 1.25'
109
on the unit where The area ab
T
the is
to
7. Lie groups
In this section we treat metric properties
of Lie groups in a fairly self-
contained way. In the compact case we emphasize Finsler metrics since they offer advantages
in applications
The results are needed for connection
the specialization
simplify proofs. (i)
from compact groups to
In the noncompact
SO(n)
SO(n)
does not
Rn
which follow from properties
and
results on left invariant metrics on nilpotent
Lie groups, some of
which we can prove only with detailed estimates of remainder the Campbell-Hausdorff
terms in
formula. Almost flat metrics are given in 7.7.2.
7.1 Basic notions. The tangent space e
proofs.
case we need
results on the group of motions of of
(ii)
without requiring more complicated
SO(n) , but if one works with the left invariant
T G e
of a Lie group
G
at the identity
is made into a Lie algebra as follows:
Extend tangent vectors
X,Y € TeG
via left translations
L
a
G+G,
La (g) = a· g, to left invariant vectorfields
X*(a)
;=
(dL)
a e
•X ,
observe that the bracket of left invariant vector fields is left invariant, [x*,Yt] (a) = dL •[x~,Y"J (e) , and put a
The differential
of the conjugation
K
a
G
+
G , Ka(g)
a g a
-1
is the
adjoint representation
Ad
G + GL(TeG)
,
Ad(a) .- d(K) a e
The 1-parameter subgroup with initial tangent X is denoted exp tX tX (= e for brevity); for example if S is a skewsymmetric matrix then ts e = id + 2:=1 (t~;k is a 1-parameter subgroup in SO(n)
111
;:~
'~''''·''I!tnhiH'''.,li_.
P. BUSER AND H. KARCHER
It is fundamental
that the differential
of
Ad
d(Adl oX
7.1.1
e
ad X(Yl
Immediate consequences
7.1. 2
at
e
is the bracket:
ad X
=:
[X,YJ .
are
Exp(t ad Xl
:=
id +
L ~, k=l
[dK oX,
dKO[X,yJ a
a
(t ad Xl
k
.
dK oyJ a
aoexp tX
exp «Exp
[exp X, exp yJG 7.2 Lie bracket and curvature. connection
DL
ad Xl oyl oexp (-Yl .
On each Lie group we have the left invariant
for which left invariant vectorfields
curvature tensorfield
vanishes,
The torsion field
is obtained by left translating
7.2.1
T (Xli',
T
the bracket
ylfl
Riemannian
The
since on left invariant vector fields obviously
from
T G
e
-[X,yJ*" .
The following torsion free connection of biinvariant
are parallel.
metrics;
7.2.2 is the Levi-Civita
connection
it is also useful for other metrics. 1
2"
7.2.2
T(X,Yl
Its left invariant curvature tensor field is expressible and also D-parallel:
112
by the Lie bracket
LIE GROUPS
7.2.3
R (X~, y"-) Z 1<
o
DR
Both
connections
or right)
have
7.2.4 Let Using
the
as geodesics;
by differentiating
(from
i-parameter
therefore
families
J(t)
The Jacobi
equation equation
k(t)
7.2.5 If on~works these
solutions,
we describe
into formulas
J(t)
=
closely
=
Exp(t
(i)
= 0
0
the biinvariant D-parallel
dLc(t)
fields
c(t)
aoexp
is then reduced
with
the explicit
gave
,where
tX
.
solution
connection
t
2 ad X)
6.3.2
0
[12
X,J(o)
or 6.6.3
D,
along
c
then one can translate can be written
case:
t (2 ad X)
]
0
D) t dt J(o)
t + cosh (2 ad X)
0
1. a:
;
La
carryover:
J(t)
-~. ~ -
-.-----
,
D) J(o) dt
1
t
y
as
,
sinh J(o) + ~
0
f(z)
z
the Riemannian
0
(d exp\x
.&
obtained
to the ordinary
113
& !
(left
R->-TG e
translation
Exp(-
0
resembling
which
the geodesic
+ f(-t ad X)o~(o)
t Pt (cosh (2 ad X)
The arguments
translates
•
ad X)o~(o)
D J(t) = Pt (.s~nh(2t ad X) dt 7.2.6
0)
it as
~,~J =
with
and their
the same Jacobi
dLC (t) ok (t) , k
k(o)
since
Pt
along
D~(D~ J + T(J,c» c c .. k +
=
~(t)
subgroups
they have
field
J(t)
differential
+ [v, [w,uJ ] + [w, [u,v] ]
of geodesics.
be a Jacobi
left translations
J
[u, [v,w]
.. -
•
P. BUSER AND
with
JC a Jacobi
field
along
with
J
a Jacobi
t
D dt J(o)
-1 (exPq(t) p)
d dt
(ii)
the geodesic
= o ,
J(o)
H. KARCHER
field
exp tX
DL dt J(o)
= =
t+
=
satisfying
y
:s Jt (1)
-1
along
s + eXPq(t)
«l-s)
- eXPq(t)
satisfying
p)
-
q(t)
This explicit studies immediate
7.2.7
knowledge
of derivatives
of the relation application
Proposition.
between
which
of
exp
a Lie group
is the key
for more
and its Lie algebra.
careful
One
we need is the
(Differential
equation
for the Campbell-Hausdorff-
formula)
Define
H
e
exp H(t)
where
X
and
Yare
by
[O,lJ + T G
exp X - exp t y
such that we stay in a domain
where
exp
is injective.
Then
d dt H(t)
where
Z
g(Z)
The solution formula;
H(X,ty)
Proof.
Define
;s ct(o)
=
Z-(e -1)
as a power
its first
:=
series
few terms
H(t)
a family
H(t)
g(-ad
H(t»-y -1
l-.!.Z+ 2
in
X
and
ty
is the Campbell-Hausdorff-
are
t
X+ty +
2"
of geodesics
. The induced
X
H(o)
Jacobi
114
2
t
[x,yJ +
ct(s) fields
IT =
t
[X, [x,yJ] +
exp(s-H(t» Jt(s)
=
IT
[y, [y,xJJ .••
with
;t ct(s)
have
the
LIE GROUPS
initial
=
Jt(l)
data
dLCt(l)'Y
f(- ad H)
0
we obtain
with
few terms
= ~
~s Jt(o)
I
is the inverse
can be solved first
=
Jt(o)
H(t)
from 7.2.4
of
=
Y
in thiS case
d f(- ad H(t)) • dt H(t)
g(- ad H) . The analytic
the Picard-Linde16f
are easily
. Since
obtained
iteration
starting
where
I
differential
equation
in any Lie subalgebra;
the
from
t
x+tY
f
+
X
I
g(- ad H
o We now turn to metric
considerations.
7 • 3 Norms.
II
tangent
Any norm
spaces
in particular curvature well
7.3.1
.G
of
lion
a Finsler
and affine
of parallel
(t))Ydt
•
can be left translated
e
and thus gives
for the linear
control
T G
o
on
G . We .use this
group
of
~n
isometry
in 6.2 the following
translations
to all other
metric
T M ;
p norms
to our are
adapted:
Example.
S € so (n)
For
lis II
.=
max{
put
Isvl
v
n
n
E
lR
IvI
I
l} .
IR One has
II [S,T] II The corresponding
(biinvariant)
d(A,B)
Diameter
lisT - TSII ~
:=
and injectivity
From this metric of motions
of
on
A.
1.
(ll
exp
of
we derive
1.
1.
n
v E IR
are
=
I
O(n))
Ivl
(A. E O(n) 1.
115
&",_ ..11
I
metric
a. c IRn) 1.
is
l} .
n.
as
and define
.....
(and
a left invariant
motions
A. 'v + a.
IIT II· IIS II .
SO(n)
I
We write
(v)
on
max{1 ~(AVIBV)
radius
O(n)
T M P
metric
2
on the group
P. BUSER AND
H. KARCHER
7.3.2
- --1 _ d(A oA 2 1
where
c
is an adjustable
influence factor
parameter,
of the translational
A
distance
is a curvature on the group
' id)
parts
bound
the decrease
of
T M
factors)
of the Riemannian
metric
decreases
of trivial
which
the
The additional
; its inclusion
independent
P (by constant
of which
on the definition.
M(IKI ~ A2)
for
of motions
,
makes
the
changes
is desirable
in pinching
situations. If we describe
the tangent
then the corresponding
II (S,a) II
Abbreviate
IIA II
Gromov's following metrics.
7.2.2
work
estimates
for Finsler-
(biinvariant)
hence
in the proof
of the dimension.
of 7.5.1,
namely
Finsler
in the Riemannian
notations.
the Levi-Civita
The
as for Riemannian that the
case distance
case,
of
since
mini-
the connection
connection
of any
case conjugation
is a
(Finsler)
isometry,
it follows
series
is a norm-isometry: in the Riemannian
We also make maps;
many
proved
X)
T G
....
=
II y II.
e
linear
x ~n
metric.)
Exp(ad
7.3.5
so(n)
, abusing
independent
is the proof
free and metric,
In the biinvariant
d(A,id)
become
are in the
Riemannian
that the power
by
I I) .
simplifies
bounds
(This is trivial
is torsion
of motions
cA a
are just as easily
subgroups
curves.
II S II,
,IIA II
metrics
various
estimates
biinvariant
7.3.4
d(A,id)
The only extra
i-parameter mizing
max (
Finsler
theorem;
of the group
norm is
7.3.3
The use of these
space
given
II Exp (ad case;
X) oy
II
7.4.1 and 7.4.2
the convention, any norm
II
T G
e
This makes
many
results
are Rauch
type estimates.
that we always
use the maximum
lion
116
T G e
this
leads
to the
look as
norm
for
(semi-)
LIE GROUPS
norm
(which is biinvariant
Illxlll
Ilad X II
:=
Jacobi-field
seperately. convex ball
7.3.4
2 Illxl 11·lllylII
estimates.
=
Br
or
IIIad
We treat the biinvariant
{x E TeG ; II ad X II ~ r}
the Riemannia~
Lemma.
invariant
max{ Ilad x·y II
II y II = 1} .
implies a dimension
I II ~
imply
Ilk(dl
=
J(t)
solutions.
we will have uniform
On the
estimates;
estimates.
(Upper bounds, compare 6.3.8
norm and
2
and the general case
Proofs are immediate because of the explicit
they resemble
7.4.1
:=
II is biinvariant)
bound for the bracket:
III [x,yJ III ~
7.4
II
ad [x,yJ = ad X· ad y - ad y • ad X
It is useful since independent
if
dLc(t) ·k(t)
(ii)). Let
II
a Jacobi field along
II be a bic . Then 7.2.4,
Ilk(o)11 , hence
=
DL IIJ(t) - dLc(t) J(o) II :::t ·11 dt J(o) II .
In particul&r
7.2.6
(i) gives
II(d exp\x·y
i.e.
exp
7.4.2
Lemma.
invariant (7.2.5)
does not increase lengths.
(Lower bounds, compare 6.3.5
norm and along
c(t)
t i.e.
II :::Ily II ,
(d exp)tX
J(t)
=
=
Pt(S~~h
(iii)). Let
II •
(t ad X) • (t :t J(o)))
exp tX • Then, if
II be a bia Jacobi field
lit ad X II < 2')f ,
IIJ(t)11 :: si~n ( lit ad X II)· II:t J(o)11 decreases
lengths, but not more than by a factor sin .t id (112"ad X II) .
117
P. BUSER AND
Z
Proof. The power series
sinh Z powers with radius of convergence
II(st~
(~ad
H. KARCHER
is an alternating
II id sinh
X)) -1 II =
t ('2
power series in even
S. 121]. Therefore
[21
11
ad X) II
proves the estimate.
7.4.3
Lemma.
(Comparison of
-----
assumptions
J
dD J , compare 6.3.6).
and
t
Under the
of 7.4.2 we have with a similar proof
IIJ(t) - t :t J(t) II <
(11!:.2
(1 - ~
tan
ad X II)) • IIJ(t) II •
In the left invariant case one has to decide whether it is more convenient to work with the exponential map of the group or the metric. The metric exponential is always surjective while the other one may not; on the other hand, the relations between the metric exponential so easily exploited. group exponential
and the group structure are not
If the group is simply connected and nilpotent then the
is a diffeomorphism
(7.7.4) while the metric
exponential
may not. In this case the choice seems clear and the following estimates will be needed.
7,4.4
Lemma.
(Lipschitz estimates for the group exponential
left invariant norm
II
II•)
(d exp)tX·Y
where
f(z)
z
Recall from 7. 2 . 4,
dL
exp tX
• f (-t ad X) • Y
'
(ez_1) • Therefore
II(d exp'tx·Y - dLexp tx·Y II :::(f( lit ad X II)-1)·
(2
-
in terms of a
7. 2 • 6 :
f( lit ad X II))· Ily II :::II(d exp)tx·Y
Proof. Since the power series
f
II ::: f( lit ad
has positive coefficients
IIf (-t ad X)- id II < f ( IIt ad X II) - 1 •
118
Ily II,
X II)· IlyII •
LIE GROUPS
7.4.5 ---Lemma.
(Estimates for
q
-1
exp q
+
Assume that on some neighborhood
v
B(p)
B + TB
p,
compare 6.4.6, 7.2.6, 7.4.3)
the vectorfield -1
:=
v(q)
eXP
q
is defined.
Let
D
be the biinvariant,
abbreviate
r(q)
:=
Ilad v(q) II (after left translation
torsion free connection
identity and using any left invariant
norm
(1 _ ~
1 (2
tan
Proof. Because of 7.2.6
of
for which we use again explicite D
formulas
id P ((-stanh
ds J (1) - J (1)
v(q)
7.2.2 and to the
II • II). Then
r(q»)
·llx II .
we only have to estimate
(ii)
-----
P
II~ J(l) - J(l)11 oS
(7.2.5):
. sinh - ~d) • -. ~d
1 -1 D ) (- ad exp q). -- J (0) 2 P ds
'
hence
II.E.... J(l) - J(1) II < ~
-
1 exp-1 q)11 • IIJ(1)11 . II( idnh - id) (-2ad ~ p
As in 7.4.2, if we change the coefficients
of the power series
(inhd - id) ta (id - -~-- J , this and tan 'd
to their absolute values we obtain the power series Ilad exp-1 q II = Ilad exp-1 p II complete the proof. p
q
7.5 Metric results in the biinvariant
7.5.1 Proposition. translates
Proof.
----
(Shortest curves) The one-parameter
locally minimize
Finsler distances
subgroups
and their
in. the biinvariant
(!) case.
(i) For any (left) invariant metric one has 1
(t + exp tX) 10
=
(ii) Arbitrary
differentiable
geodesic
case
d(id, exp X) < length -
Ilx II • curves can be approximated
by piecewise
curves such that also ~he length is approximated.
(iii) Therefore of geodesic
it suffices to prove the triangle inequality
connections
(not for distances!),
7.2.7)
119
for the lengths
that is we have to prove
(see
P. BUSER AND
7.5.2
We use
the
curve
H :
[o,lJ
into
subdivide
IIH(X,y)
II <
[o,lJ
..,. T G e
H. KARCHER
+
Ilx II
given
intervalls
Ily II
by
exp H(t)
[ti,tHl]
(i=o,
= exp X
.•.
,n)
exp ty
0
of
length
, l.
and
n
use
tHl H(t.)
+
J
g(-
ad H(t»
y dt
0
t.
1.
1.
n
+ 1 g(-
H(ti)
To have
the
tEO [o,lJ
remainder since
may multiply II [x,y]
II
those
the ~
The set is
A:=
clearly if
B
{(X,y)
for
G
or
which
one needs
convergence
Ilad X II ~ IIH (t
)11
<
IIad H (t)
of
g
and assume
y + O(n
0
is
We first
holds,
then
for
TI
To prove
loss
Ilx II.
TI
II <
TI.
without
-2 )
of
prove
all
7.5.2
we have
Ilx II +
T G x T G; e e and open, and if E
7.5.2
on
A,
(X, y) € A
then
Ilx II +
Te G x Te G;
Ily II <
7.5.2
we remove
A
is
for
this
'IT}
•
IIH(t)
(X, ty) €
then
closed
Ily II <
and
TI
in
the
Hence
II <
A
convex
A = B •
n-l II -
I
Ilx II =
i=o IIH(t.)
1.
If
we choose
V. 1.
II =
II Exp(ln 1
- "2
ad V.) 1.
(I y
B
k=l
2k
0
H(t.)
1.
II
(ad H (t ) i (2k) !
(7.3.4)
) 2k-l )
lo
0
y
Y
(7.2.7),
then
hence
n
O(n
-2
)
n-l IIH(1)
II -
Ilx II :::
.I
1.=0
<
IIH(tH1)
Ily II
120
- EXP(~ ad Vi)
0
H(t ) i
'IT }
(o
set
Finally
IIH(l)
we
generality
follows: {(X,y)
e:
(n -2)
0
of
norm by a constant
nonempty
also, =
e
as
as
radius
Ilx II oily II
X, YET
restriction
term
the
ad H(t » i
II
all
LIE GROUPS
7.5.3 Corollary.
(Comparison of distances
be any biinvariant that
exp
norm or seminorm
is injective
on
B
2r
in
and G). Let II • II e Ilad X II ~ Ilx II. Assume
T G
such that
= {x
E T G ;
IIx II ~ 2r}
e
and assume
r <
'IT.
Then (i)
exp(B ) r For
(H)
Proof.
is geodesically
geodesic
from
exp X
then
exp(-Z)-exp
X
exp(Exp(ad
e
=
to
X,Y E Br
If
exp Y
is midpoint
(exp(-Z)-exp
2Z
=
or exp
H(Exp(ad
112z II :: IIExp (ad
Z) -
it suffices to
exp Br . Let
exp 2z
X
=
and
of a minimizing
exp Z
be this
exp(-Z)-exp
Y , i.e.
exp Z exp Y exp(-z) exp X
is injective on
Z)-Y,X)
Y II +
(i)
then the midpoint
exp(-Z)-exp
-1
y)
(i) . For
is also in of
Z)-Y)-exp X . Since
7.2.7, 7.5.2, 7.3.4
IIX-Y II.
r
(ii) follows from 7.4.1, 7.4.2 and
midpoint,
G.
2 sin r/2 -IIX-Y II
holds
show (because of continuity):
=
convex in
B2r
we get from
,
IIX II = IIY II +
IIX II, i. e.
Z E B
7.5.4 Proposition. (i) (H)
(iii)
(Commutators in the group and in its Lie algebra) d(eXeY,eYeX) ~ II[x,yJ II ' X c!(eXeY,e +Y)
t
<
Under the assumptions Cambell-Hausdorff
I/H(X,y) - (X+Y)
(iv)
II[x,YJ II '
If
lIy 1/ ::
r
of 7.5.3 the first remainder
power series satisfies
1/
::
t1_c,~
~!')
-
II[x,YJ
1/.
(as in 7.5.3) then
Proof. (i) The curve t + c(t) = exp(Exp(t ad Y)-X) Y X -Y e e e has length::: II[X,YJ II since
II! Exp(t ad Y) -X
II
from
IIExp(t ad Y) - [y,x] II
121
-
of the
(compare 6.6)
eX
to
II[y,x] II
r
-- --
P. BUSER AND H. KARCHER
and since
exp
does not increase distances,
(ii)
n (n-1) d ( X/n yin yin x/n) --2--e e ,e e
<
< }
(iii) follows from
(ii)
_I')
~,~
Remarks. Biinvariance
,
n· d( e
<
n'IIH(X/n,Y/n)
<
O(1/n)
- (X+Y)/n II (7.4.1)
(Campbell-Hausdorff).
-Y
we have from 7.4.1 and
(iii)
< IIH(Exp(ad X)'Y,-Y) - [x,YJ II ad
IIH(Exp(ad X)'Y,-Y)
A
(i)
H(X/n,Y/n) ,e (X+Y)/n)
<
e
d( e Xe Ye -X e -Y , e [x,YJ)
~
II[x,YJ II
and 7.5.3 •
(iv) Since
<
7.4.1
- (e
X'Y_Y)II + II(ead X,Y_Y) - [x,YJ II
II[x,YJ II( Ilad X 11+ Ilad Y II) • f( Ilad X II)
was used heavily
in the proofs of 7.4.1 to 7.5.4
the results are close to what one would expect from the first terms of the Campbell-Hausdorff
formula.
If one has only left invariance
contain additional
exponental
do not need these in general, metrics on simply connected assumptions.
factors
nilpotent
to
In the following
two propositions
Gromov's dimension
Lie groups with rather special additional
and to the motions
theorem we illustrate dependence
(compare 7.4.4 to 7.4.1 and 7.4.2); we
instead we will come back to left invariant
7.6 Applications
O(n)
then all estimates
of results
of
n
JR.
which are important
tools in the proof of
how the use of Finsler mectrics
improves the
(without other changes of the arguments).
122
LIE GROUPS
7.6.1
Proposition.
(i)
¢e
For given in
O(n)
(Pairwise
distances
(o,TI) there
with pairwise
in
O(n)
)
are at most
distance
>
2 • (2TI/~)dim SO(n)
~
For given A € O(n) there are at :ost m = 2. j A (j = O"",m - 1) with pairwise distance> k exists k < m such that d(A ,id) < ¢ .
(ii)
Proof.
Since
distance So(n)
B~/2
open balls
~ ~
are disjoint
(use the metric
~/2
of radius
and since
7.3.1!)
exp
we have
elements
(2TI/~)[n/2J
around
maps
¢ ; in
the volume
particular
elements
a ball
iterates there
of pairwise
of radius
TI onto
ratio
2 ·vol (SO (n)) / vol (B~/2)
as an obvious
bound for the number of elements with pairwise distance j A are not arbitrary in O(n) but lie on two tori of
The iterates dimension of
O(n)
S
[n/2]
• Since
which
are flat and totally
the exponential
map restricted
tori is length-
(and therefore
explicit
2·(2TI/~)dim torus.
put
k
=
bound
exp)tX
Jacobi
fields which start 2 . Because of DR
(ad X)
volume-) If
IJI-1.J
by using
pairwise
is parallel
0
to the tangent
d(Aj1,Aj2)
< ~
with
(i)
we estimate
bound
orthogonal
in
(in the standard of the nonpositive -1 2 = 4 (ad X)·J
R(J,X)X
and
II ad
0
S
X
II ::: 2 II X II
dim SO(n)-l eXP)tX
IT""
i = 1
hence
123
~NIii!i
iSli
:..
.•..•. -
the
<
j1
Riemannian symmetric
j2 ~ m
metric) operator
(7.2.3) we have:
<
21f
is eigenvalue
of
-(ad X)2
(with 7.3.1).
Therefore
det(d
of these
immediately
(where
A.~ Ai :::2TI since
spaces
we have
in eigenspaces
=
in the two components
preserving
j2 - j1 . - To get the explicit
det(d
geodesic
~ ~ .
7'"
sin
id
decreases
with
t
I
I
-
--------~--
P. BUSER AND
The following
modification
3 to establish
a crucial
7.6.2 Proposition. Let
A. ~ id
to a noncompact
Assume that the
1, ..• ,k)
A.
d(A~lA.,id) ~J
>
max(d(A. lid)-Sd(A. lid)
By definition map of
is used in section
(7.3.2)
SO(n)
~J
I
d(A. lid)-sd(l\..,id)) J
~
cl.(A. ~lid)
=
Since the
max ( lis. ~ II,cAla·l) ~ distances
(7.4.1) we also have
~
I
J
W
=
~
J
so(n) x ~n
we find at least as many
with
w.
satisfying
~
the system
of inequalities
IIw. - w. II > max ( IIw. II-s IIw. II ~
as we had motions homogeneity
J
A ..
~
-
~
J
For a fixed pair
of the inequalities
i,j
on the vector
I
i
IIw. il-s Iw. II) J
~
we may assume space)
1
=
(using the
IIw. II < ~
IIw. II•
-
J
Then
II(Ilw. J
>
11-1.w.J
- w.) II > ~
-
Ilw. - w. II -II (Ilw. J
Ilw. II-s Ilw. II - (11w. 11-1) J
~
1111w. J
11-1•
J
w. II - s Ilw. II • J
i.e.
(3-S)n + dim SO (n) l-s
<
This implies that on the vector space I
SO(n)
max ( lis. - S. II cAla. - a·l) .
-
IIw II:= max ( IIS II cA Ia I)
I
J
does not increase
<
a.(A.,A.)
parts in
satisfy
~
k
exponential
with rotational
pairwise
-
situation
in the group of motions)
be motions
~
then
Proof.
(l)
a priori bound.
(Pairwise distances
=
(i
~
H. KARCHER
~
124
~
J
1I-1.w.J
- w.) II J
LIE GROUPS
In other
words,
the unit
unit vectors around
0;
vectors
Therefore
are disjoint
w./ Ilw. II l
still
l
the open balls and contained
satisfy
of radius
in the ball
around
of radius
ratio (3 -S)dim W is then a bound 1 -S also on the number of motions A.
, hence
the same system
(1-s)/2
the volume
w. l
these
1 + (1-s)/2
on the number
of
l
7.7 Left invariant
7.7.1
vectors
--
of inequalities.
Proposition.
metrics.
(Lie bracket
and curvature
of the Levi-Civita
connection.)
Let n~ Y
+
U(X,Y)
connectipn of a left invariant scalar product <, > L G. (n is the left invariant connection from 7.2 whose
be the Levi-Civita on a Lie group torsion tensor
field
T
is essentially
the Lie bracket;
is a left invariant
field.)
Then
=
<,U(X,Y),z)
RU.(X,y)Z
(This curvature
Proof.
Since
=
U
{-+
U(X,U(Y,Z)
bound
n
t
)-U(y,U(X,Z)
is 24 times
is torsion
the bound
free and
U(X,Y)
nL
U(Y,X)
n
the scalar
and
nL
product,
are metric
connections,
,
)-U(T(X,Y)
,Z)
in the biinvariant
has torsion
-T(X,Y)
U Since
U
i.e.
,z)
+
125
we have
•
product
we have
(U(X,Y)
T
case.)
=
0
differentiations
for
P. BUSER AND
This determines is immediate invariant
7.7.2
U
of
from the definition
vectorfields.
Proposition.
On any nilpotent (7.7.3
in terms
(i)
Lie algebra
L
the following
(E) The unit balls exhaustion
group
Proof. that
there
Choose
E-flat
Li = span (Xl ,..• ,Xi)
The formula
= U(y~,Z*)
RU
for
for left
is then trivial.
are almost
flat)
exist
7.7.1
families
also
quotient
metrics
a "triangular"
as above.
D~AZ·
bound
IIRµ IIµ
D := {x € L ; µ L .
of
on any compact
exist
if we use
there
>
<,
and
of metrics
II
"µ
properties:
- and with IIad II µ of µ .
In particular,
T
The curvature
(Nilmanifolds
(ii)) with
H. KARCHER
IIx II < 1} µ -
for any
{Xl""
is for each
E >
,X } n i
for
an ideal
independent
give an expanding
of the corresponding
(1.3)
basis
- is bounded
nilpotent
Lie
°. L, of
i.e. a basis Land
such
J
[L,L + i 1
c:
Let
[X. ,X.] Define
the length
7.7.3
(i)
y
J
l
of
:= maxly ijkl
.
X n
where
IIxII2 µ
the
are such that
. 1 er con d"ltlon slmp q E (0,1)
4 µi+1
:=
L i=l
Ii-1
2 < 4 < 4 k=l µk - µi - µi+1
>_ µi2 + µi4 ; exp l'lClt . examples
is implied
are obtained
with
(E)
No matter
' which
1, ... ,n)
how
small
q
is taken
y2.
we conclude
L k « min(i,j)
126
~
from
<
•
by the
for any
L • i
LIE GROUPS
<
that the norm of the Lie bracket is
II [l:a.x. , ~ ~
l:b.X.J J J
yn:
IIµ < yn
7.7.4 Proposition.
Let
The group exponential
N
be the nilpotent
exp:
is a diffeomorphism The difference
Lie group with Lie algebra
L+N
which, because of 7.2.7, maps subalgebras
between left translation
Proof.
exp
Y
has maximal rank on
L
(7.7.3). This implies that e E N
by a curve
y(t)
L
is surjective: =
e , y(l)
Finally,
7.8 Remainder
L
=
·11
Y II.
y £ N
can be
y ); then choose with
IIx II < 2· length(y)
can be lifted to a curve
eXPe
is injective,
y
=:
21
implies
in the compact ball
giving the contradiction
estimates
The following estimates
to zero; this homotopy
of two different
for the Campbell-Hausdorff
rays with the
formula
are nee~ed in 5.3 to bound the curvature
left invariant metric on a nilpotent estimates 3.5 .
127
to
since the closed curve formed
by two geodesics with the same endpoint is homotopic can be lifted to same endpoint.
z-1(ez_1»
such that_ exp y(t) = y(t) • (Any lift of yl [o,t) 1 ~ (2-1.4) • 1 < 21 , so that the lift exists on a
that ball has length larger interval.)
y
exp
(y(O)
7.7.2 a left invariant metric such that lIad X II :: log 1.4 ; now
can be
for which the bracket has arbitrarily
small norm
in
=
exp
because of the lower bound 7.4.4 and the
joined to
21
(f(z)
of
dLexp tX • Y II :: (f( IIt ad X II)-1)
existence of left invariant metrics
of radius
to subgroups.
and the differential
bounded with 7.4.4 using any left invariant norm
II (d exp)tx·
L.
map
of a certain
Lie group in terms of the commutator
----
P. B USER AND
7.8.1 Lemma.
(Bound for the Campbell-Hausdorff
group with Lie algebra II • lion if
H. KARCHER
L.
If
L
Ilad
X
formula).
and a left invariant II +
Ilad
y
metric
II < 10g(1+s)
Let coming
< log (2_e-
be a Lie
N
from a norm
If
)
~O.67
and
exp H = exp X • exp Y , then Ilad H II < -log(2-e Ilad -
II +
X
IIad X II + IIad Y II Ilad H II < e • ( Ilad - 2-e IIad X II + IIad Y II (Compare to 7.5.2 and 7.3.5
exp H = exp
X
subgroups
exp
•
X
Y
11+
II) < log(l + _E_ ) < 1-E
+ Ilad y II) < ~ ( Ilad - -E l+E 1-E
; the loss of the factor
proof that the i-parameter
Proof.
Ilad
are distance
implies
Y
If
,
11+
X
destrDYs
Ilad y II).
the
minimizing.)
Exp(ad H) = Exp(ad
0 Exp(ad
X)
y)
From
IIExp(ad H) - id II
II(Exp (ad
X)
-id) + (Exp (ad
-id) + (Exp (ad
Y)
:: e Ilad X II -1 + e Ilad Y II -1 + (e Ilad X II
e Ilad
X
11+
Ilad
we have the convergent
ad
H
II -1
Y
power
<
X)
-id) 0 (Exp (ad
-1)
y)
-id) II
(e Ilad Y II -1)
•
<
s
series (Exp(ad H)_id)k
L k=l
Log (Exp (ad H»
k
hence k
Ilad H II:,:
7.8.2 Lemma. Let >
N
k=l
k
(Remainder
be a nilpotent
{X1, .•• ,xn}
L
IIExp(ad H)-id II
estimates
L. = span{x1, ... ,x.} ~ ~
-log(l-IIExp(ad
for the Campbell-Hausdorff
Lie group with normed
be a triangular
=
(II.
basis (as in 7.7.2)
the descending
sequence
128
for
H)-id II)
formula.)
II) Lie algebra
L.
L,
of ideals
([L,L. lJ c L.). ~+ ~
Let
LIE GROUPS
Let
X E L , Y E L + i 1
Determine
R(t)
(i)
exp H(t)
Li
= g(-ad H(t))·y
IIH(t)-X-ty
Ilad xI
11+
= exp X • exp tY
II [H(t)-X,y]
(ii)
'
II
II
,
Ilad YI
II Li
with
H(o) = X,
<
TI
log (2_e- ) ~
7.2.7 and 7.7.4
g(z)
= z·(e
z
-1)
<
t Ilad YI II II [x,yJ II • (e Li -1)
<
II [X, tYJ II • f ( IIt ad YIL. II)
-1
0.67
from
1 = fez)
Then
,
1.
lit IIH(t)
II
<
ad X • y,-y) II H(e
(iii)
IltYll
-
II [x,yJ II • (~llad
+
<
11+
xiL.
Ilad YIL. II·
1.
Remark. Note that ad YI·L.
1.
, ad XI L.
Y£
for
ad YIL II i ·llxll
e
f( Ilad YIL. II)) • f( Ilad XIL. II)
1.
L+ i 1
1.
1.
the estimates are in terms of
(compare7.5.4
(iv)).
This is
the particular
feature
1.
the nilpotent
case which allows an induction;
ad XIL
= 0 • 1
Proof.
Weestimate the higher order terms in the power series:
Ii
= g(-ad H)·Y = Y + ~ [y,H] 2k
00
where
I
+ g2(-ad H).Y ,
k=l
Z
_z~_
B
2k
2k r (2k)!
(2k)!
=
-
-
-1 + ~ • e +1 2 z--e -1
'
g2(ir)
(if r < From 7.8. 1 we have IIad HI
L.
TI)
II < TI , hence with
1.
IIIi - Y - ~ [y,H] II =
IIg2(-ad H)· Y II
~ ~
Ilad HIL. II 1.
(a)
Consequently
IIIi - Y
II [H,y]·
II
II ~ ~
II [H,y] II·
Ilad YIL. II • II [H,y] II ' 1.
129
·11
[H,y] II
of
P. BUSER AND
hence II [H-X,y](t)
II f II
~
a
0
II
t Ilad YIL II II [x,Y] II (e i-i)
~
II f II
H. KARCHER
, which is
implies 2
t
Ilf(t)
-
f(o)
II
:::
f
allf(T)lldT
:::llf(o)lloat+a
o
<
Ilf(o)
110
The bound II [H,yJ II::: II [x,yJ 110 gration For
gives
(iii)
(eat_i)
II read Xoy,_y] II ~
XOY_Y)-
ad
IIH(e
o
0
Ilfll
by induction.)
t Ilad YI II e Li
is
power series
inserted
in
(a) , inte-
Ilad YIL. 110ft
[x,yJ II ~
tllad
estimates
Ilad X
. II)
IL
xiL.
inequality
Xoy,_y) -
and
ad
(e
(ii)
0
II [x,yJ II ,
1.
11 ft
Ilad X
0
. II)
IL
1.
the triangle
T
f f
,
1.
ad
t
(ii).
we use the trivial
II (e
(i).
oil [x,Y] II ,
1.
:
Xoy _y) II
~ (U)
130
ad
II [e
Xoy,_y] 110
f(
Ilad YIL. II) 1.
8. Nonlinear averages
In section 3.6.4 we need the result 8.2 on almost homomorphisms. 5.4 we need the result 8.3 on the interpolation Both tools are based on a generalization [ 12
J, [
In section
of locally defined maps.
of the euclidean center of mass
J.
18
8.1 The nonlinear center of mass 8.1.1 Mass distribution. specifically
let
A
Let
A
be a measure space of volume
subset of a Riemannian manifold, with volume normalized a convex set diffeomorphic D • The center of mass we call
f
to a ball in a manifold
Lf
to
B
1 •
n
M
R
in
:=
fA f(a) 'Definition
(i)
definition
(ii)
immediately
fA
f:
1 2
Ix - f(a)1
(x - f(a»
A
+
B ;
B
then one can define the center
or as the unique zero of the vectorfield
grad Pf(x)
denotes
with affine connection
either as the unique minimum point of the convex function on
B + IR
(ii)
M
is to be defined for a measurable map
a normalized mass distribution
8.1.2 The euclidean case. If (i)
1 - more
either be a finite set of weighted points or a compact
on
2
~f
B
da,
B
da ,
da
generalizes
to the Riemannian case
(8.1.3);
shows that the center of mass is an affine rather than a
metric notion, it can be made to work under more general circumstances
(8.1.4)
than (8.1.3), for example on a simply connected nilpotent Lie group without a metric.
131
~ ..
P. BUSER AND
8.1.3
Definition.
Let
Let
functions)
be a ball in a Riemannian
manifold
p
<
injectivity
exp
P
<
4"1 • rrl1- i ,
A + B
f:
(Using convex
H. KARCHER
p
radius
if
11
on
B
of
and if
be a normalized
and assume
p
is an upper
p
M
mass
curvature
bound
11 > 0
distribution
(8.1.1).
Then,
the
function
1
is because
of 6.4.6,
(i)
6.3.6
an average
grad Pf(x)
of convex
da
(compare
functions
8.1. 2)
with
-1
JA
-
2
d(x,f(a»
2"
expx
f(a) da
,
(ii)
Its unique f : A + B
L
minimum
point is called f (adapted to the Levi-Civita
p
The next definition as an affine
(generalized
notion.
is used
to prove
of the auxilliary
and 8.1.8,
8.1.4
Definition. D
exp
and let
B
(Using indices
For a normalized
(compare
8.1.2
succeed
See 8.1.6,
of the massdistribution
aims to define
since
some metric
However
the center
application
information is inde-
to compact
Lie groups,
of vectorfields) on a manifold
set for the geodesics mass distribution
f:
M of
with
exponential
D,
A + B
B
vf(x)
define
132
-1 - A expx f(a) da
J
map
diffeomorphic
(ii»
B + TB
the center
Lie groups.
connection
be a convex
(ii»
of the center.
metric.
to nilpotent
be a (symmetric)
to a ball. field
the uniqueness
application
Let
from 8.1.2
It does not fully
pendent
the center connection).
.
the vector-
NONLINEAR
The convexity
of
ClB
. therefore
in
B
Assume
B
that one can prove
The unique
distribution The center denoted
adapted
of finitely
'e {f. ,~.}
Let
A + A
in case 8.1.3 all
x E B
f
in
B
zero of
is called
DV f has index
v
f
t
the center
as in 8.1.3 1 .
of the mass f
to the connection
many points
D.
fi E B
with weigths
~i
will
also be
1.
Proposition. ~:
for the derivative
that every
v
v points outward along f and consequently at least one zero
•
As in the euclidean
8.1.5
1
index
an estimate
implies
zero of
f
1.
that the vectorfield
has total
(ii) or 7.4.5 which
Then:
implies
AVERAGES
case we have
(Compatibility
be a volume resp.
directly
properties
preserving
an affine
of the center)
map and let
diffeomorphism
M + M
U:
in case
8.1.4,
be an isometry
then we have
for
from the definitions
PU 0 f 0 ~ (Ux)
(in case 8.1.3),
(in case 8.1.4),
hence
i.e. the center
commutes
and is unaffected
with
by volume
8.1.6
Example.
Given
any biinvariant
(Center on compact Finsler
normalized
so that
connection
(7.2.2) and have
(i)
isometries
preserving
Ilad X II ~
Convexity
(resp. connection "permutations"
preserving
Lie groups)
metric
on a compact
Lie group
G
Ilx II) , we use the biinvariant
of Finsler
maps)
of the masses.
balls
133
B
p
of radius
P
<
(assumed (and metric)
IT
(7.5.3) .
P. BUSER AND
(ii)
<
Therefore
8.1.4
connection)
gives
a center
for any normalized
H. KARCHER
(1 - ta~ p)-
II X II
Ilx II
p < !!. 2
if
C
of mass
(adapted f distribution
mass
the present
balls which
satisfy
It may be helpful
to the biinvariant
p < !!.
f
Note that
(7.4.5)
2
balls
B
are in general much p for Riemannian metrics on
8.1.3
to see in a simple
application
larger
convex
sets than
G .
how easily
the center
can
be used:
8.1.7
Proposition.
Let
Hand
G
with
biinvariant
and
H" C. G
(Conjugation
be compact Finsler
metric
be two homomorphic
d(h' ,h")
Define
f
of isomorphic
Lie groups,
H
normalized
to
images
H
of
'IT
<
volume
II ad
X
normalized
to
II ~ II X II.
Let
1 ,
G
H' C G
and assume
hE
for all
2
H -+-B'Tf/2(e)C G , f(h)
subgroups)
with
:= (h,)-l - h"
H •
•
Then '1"'-1 l.,f
. r:
• h'
i.e. the center
Proof.
Fix
isometries therefore
h"
f
of
for all
f
conjugates
k ~ H • Left translation of 8.1.5
G,
right
by
translation
by
k'
H'
and
and right 1
k-
is volume
=
r
h-+-k'-(h')
-1 h -+-((h - k
134
-1
-1 )')
by
k"
preserving
on
H ,
-h"
-1 - (h-k
•
translation
implies
r
H"
)" - k"
are
NONLINEAR
Corollary.
For a finite
maps
H
form
there
into
G
are at most
Next we apply as an affine
N
8.1.4
group
nonconjugate
to nilpotent
be a compact
let
D
be the biinvariant subgroups
Any normalized
where
H
into
the center
G
can be defined
mass
Choose
connection
distribution
in the D-convex have
a
nilpotent
(7.2.2) which
Lie group
has the
Nand
(left) translates
A
f:
+ C
(8.1.1) has a center
of mass
lemma
unit ball
2
µ-length
some metric
II such that C is II µ B :~ exp Dµ • Geodesics, which join two 1 9 1+e / (7.8.2 (ii)), therefore we have
from 7.4.5
<
The assumptions
8.1.9
Lemma.
Normalize
of 8.1.4
<
to define
(Convex exhaustion
the almost
µ
< -
the center
of nilpotent
flat metrics
Ilad X II
7.7.3
1 -9 Ilx II
µ
of
fare
1 90
satistied.
Lie groups)
so that
by putting
µ.
L
:~ µ. (q) .9yn L
•
Then (i)
The compact
unit balls
are an exhaustion
(ii)
The balls are a convex
B
of the Lle algebra
'I'';ili"IIiIIQIII.
"'III'"""" ..",.,...."""'
L
(as
q + 0).
:~ exp (r • D) are D-convex for µ,r µ exhaustion of the Lie group N.
135
'I
~f
D.
in the following
B
connected
as its geodesics.
to the connection
of
from
(V01(G)/vOl(B / )IHI rr 2 8.1.7 (1f) • Therefore
Lie groups)
set in a simply
of l-parameter
points
Lie groups
N:~
satisfy
homomorphisms
(Center on nilpotent
C
contained
are at most
such that no two of them
Let
Proof.
there
notion.
8. 1 .8 Example.
adapted
H
AVERAGES
.......
--------
r < 2
the balls
P. BUSER AND
H. KARCHER
Proof. (i) restates 7.7.3. To prove ----diffeomorphism (7.7.4) and define
h:N+JR
-1 [ 1 2 = h (0'-2 r )
J
Clearly
B
Consider
the set
µ,r
A
A
Obviously
the endpoints This proves
that B
µ,
A 2
sublevels
B
B
~ =
(d exp)x.
µ,r
•• l<x,x>1
µ,r
and
h
dL • x
µ
+
differentiation
in
I.(X,2t)
µ
I
from
=
x
y
to
is in
, 0 < t < 1 , be
2
--
> 0 ; then,
t
geodesic
µ,
on
dd 2 h(x(t»
2
is a
µ
exp X(t)
show
is a convex
is also
hence
= B µ,2
A
function
in
on
B
µ,
2
B
~h
o x
dt
µ,
2
(with
< x,x > , µ
(x,·x) µ .
•• f (-ad X) • X
+
p.
(2 - f(p/9»
- 9
•
2 µ
and 7.4.4
give
X
II '
::: (2 - f ( IIad X II>)-1 .
1 Ilad II < -
I
f
o .
::: f' ( Ilad X II) • Ilad
with
0.93 xl
series
(if
136
IXI := p < 4
and
-1
• f' (p/9)
p/9
::: ~)
1
• 9"
if
µ,r
x B
we have
µ
of the power
II f (- ad X) -1 II
is used
µ,
>
is convex
x(t)
2 x B
L + N
e
• 2 d2 From :::0.93Ixl ,implying dt2hox>0 µ • µ DL • f(-ad X). X (7.2.4, 7.2.6) and dt x = 0 we get
II d~ f(- ad X(t»11
This
B
). By definition
d • dt f (-ad Xl) • X
(:jt)
Termwise
Let
We shall
~ <. X(t) ,X(t»)
X
h
µ, (r < 4) , the whole
<. x,x) prove
4'
exp
exp -1 x e
x
we show that
is also open in
h(x(t»
We shall
-1 (exPe
and closed. in
is convex B
that
µ, 2 x B µ, 2 ; the geodesic
geodesic are in
words
convex
B
is nonempty
any nonconstant
other
E
{(x,y)
1
"2
:=
h(x)
(ii) recall
to end the proof
In
NONLINEAR
In many
applications
expected
to be close
such a guess
8.1.10 Let (i)
A + B
Under
5.4) one knows
to the center;
by estimating
Proposition.
f:
(e.g. 8.2.7,
AVERAGES
with
d(x,t ) f
(Length of
be a normalized
the assumption
in terms
Vf(x)
of
distribution
which
can be
one can exploit
to the center) (8.1.1).
B = B
with
x
result
Ivf(x) I .
and distance
mass
8.1.3
a point
the following
(and recalling
P
6.3.4)
holds
S' d (x, rfl • 2p S 11_
>
(2pl
11 if in addition
then
2 d(x,l"'fl' 2 (lip)
<
(ii)
In the case 8.1.6 metric
(iii)
holds
In the case invariant put
2r:=
of a compact
(recall
8.1.8
Finsler
7.5.1,
Lie group with
If
C C exp D
r(l + er tlad II) , then
In all cases
For the first
in
(i)
where
id tan
~f
we have
For the other
inequalities
Finsler
d2 -2 o dt
J
we have with
Dr = {x ; Ilx II ~ r}
(r'
by a geodesic
(p f 0 yl dt
». i
IIad II
to the point
>
• 2 S~ Iyl • 2p ,t,
we have
137
••
any left ,
IIad II
x •
from 6.4.6
1
>
(Ell.
r
we join the center
inequality
biinvariant
N
Lie group
<
Proof.
(see 8.1.2
7.5.3)
of a nilpotent metric:
also
_
m
•...
"'~ ..,
(2pl .
~=~=~~~--.
........ --------------------------.~.-.-
P. BUSER AND
Iyl
• 2(Ap)2
H. KARCIJER
incase
Ily II • (1 - -t p) an P
(i)
from 6.4.6,
in case
(ii)
from 7.4.5,
in case
(iii)
{ Ily II • (1 - t~~ (r: Ilad II»
for
(iii)
by 7.8.2
observe that any geodesic which joins two points of (ii)
Finally,
from 7.4.5;
<
a length
in cases
(i),
exp D
r (1 + er IIad II) = 2r
(ii)
use
II -
IIX(y(o) III ::
has
r
1
IIIX(y(1»
f
Ild~ X
0
Y Iidt
o
for any differentiable
vectorfield
and any metric connection;
this inequality holds for the components relative so that 7.2.5 gives another factor er Ilad II I.
Almost homomorphisms
8.2
of compact groups
morphisms
[ 13
deckgroup
of the finite covering
Let G Let
H
and
G
+
G
(Almost homomorphisms
with volume normalized
Finsler metric normalized
to
<
q
<
d(wo(h),w(h»
G
near
w
o
<
138
i.e.
IT
6
for all
II·
1
and
Ily II
i.e. assume for all
Then there exists a (continuous) homomorphism
+
to
II[x,yJ II :: IIX
be a (continuous) q-almost homomorphism,
H
O(n)
are near homomorphisms)
h,k " H
w
of the
(3.6.4)
in Gromov's theorem with a subgroup of
be compact Lie groups, H
with biinvariant Wo : H
can be improved to homo-
on this result depends the identification
Proposition.
8.2.1
(iii)
to parallel basis fields,
In this section we prove that almost homomorphisms
J;
in case
h E H
NONLINEAR
Proof. ---
Assume
defined.
that a
Then define
q.-almost
AVERAGES
homomorphism
~
k E H
for each
-
11k(h)
w.
H + G
~
has already
been
the map
-1
w.(koh)ow.lh)
~
~
€B
qi
(w.(k»
~
and define
H + G
8.2.2
We plan
8.2.3
to prove
h,k E H
-1 d (wi+1 (k oh)
This shows
0
wi+1 (h)
that the sequence
to a limit,
8.2.4
for all
by
Lemma.
H1 (k)
r
of almost
~
homomorphisms
converges
commutes
with
inversion
(an isometry
~
w. (k
~
0
0
h)
-1
'Ch
-1 h)
+ w. (k
~
0
w. (h)
is volume
Lemma.
H1 (k
For all
G).
-1
~
(since left multiplication
8.2.5
of
-1
h + w. (h)
w
rapidly
w.
> 0)
(i
w
Proof.
{w.}
the homomorphism
<
,wi+1 (k»
-1
)
k,k',h
€ H
holds
139
preserving
in
bei H)
k-1
P. BUSER AND
H. KARCHER
Proof.
w.(k'k
-1
l
The fact that
w.
fore the center
homomorphic gives the first statement; there-1 -1 T]k'• T]k exists in B (w. (k' • k )) . Now the qi l from
l
of
statement
follows
-1 -1 w. (k 'k • kh) • w. (kh) l
by
k
is volume
Next we translate
T]k' and
e S G
our notation:
and shorten
T] -1 (k' h) k' • k
l
(left mUltiplication
preserving
T]k so that their
T]'
These
definitions
imply
= -1
T]"T]
8.2.6
r =t
-1 T]
T]
= wi+1 (k')
(8.1.5,
-1 )0 -1 -1 -1 ·(....(T]k'· T] ) • wi+1 (k )
(8.1.5)
k
k
-1 ) • wi+1 (k)
(8.2.5)
the
Lemma.
Lemma.
goes to the identity
T](h) € B (e) , qi
is
qi+1-almost
max k ',k
8.2.7
center
H).
h € H)
(for all
-1 wi+1 (k') • wi+1 (k"
This shows
in
T]
T]'(h) ~ B (e) qi
r:
l
is q.-almost
l
seeond
-1
·kh)'w.(kh)
d (e,
rT]/'T]-1)
<
If
homomorphic
e
d(e,t H -1
exp
H
140
-1) T]'• T] 1
I •
(n (h) • T]- (h))dh I
e
for
tan 2q i 2q. l
8.2.4)
NONLINEAR
Proof. Since for all
h E H
AVERAGES
11' (h) 0 11
-1
(h) € B2
(e),
the lemma follows
qi directly
from 8.1.10
II
8.2.8 Lemma.
(ii).
exp-l(l1'(h) o 11-1 (h»dhl
Proof. Define maps
X,Y:
X(h)
IH
Then of
11'
=
X(h)dh
and of
Next let
<
e
H
0 -1
.-
=
H
+
-1
eXPe
IH
TeG
by
11' (h) , Y(h)
Y(h)dh
-1
-
eXPe
-1 (11 (h) )
because of 8.1.4 and since
e
is the center
11
H( ,
be the Campbell-Hausdorff -1
H(X(h) ,Y(h»
Finally we have from 7.5.4
eXPe
map
(11' (h) 011
(7.2.7, 7.5), then -1
(h».
(iii) together with
Ilx II + Ily II <
IIH(X,y) II <
IIH(X(h) ,Y(h})
II <
(X(h)+Y(h»
1
_
q. l
2 sin q.
0
II[X(h) ,Y(h)] II
l
IIX(h) IIoIIY(h) II <
1 2
which proves 8.2.8.
This sequence of lemmas proves
<
Since therefore
C
d (e,
-1) ~
t
qi
IT
6
we can in the proof of 8. 2 .7, which
11'011
was based on 8.1.10, apply the estimate of length
2.5qi
instead of
4qi ' which gives the improvement
tan 1.25q. -,-=-::-----: __
7.4.5 to Jacobi fields along geodesics
l_
1.25 sin qi
<
1
2
2 qi
141
/ cos 1.25qi
P. BUSER AND
and completes
the inductive
One also checks estimate
for
8.3 Averages
In this
8.3.1
1 2 ~~ q. < q +2 q L~=o ~(h),w(h» in 8.2.1.
o
of differentiable
F :
we prove
M_
N
~ r > 0
aiscrete
Let
M
r/l0-dense
:
differentiable
maps
B
r
in
B
satisfies
x
be defined
for mass
....
~
into another
connection
~
M
(m.)
D).
Assume
, which
the maximal
proves
rank proof
x
the assumptions distributions
(2.1.4).
5.4 for
B
{m
with
eM; i
i
injectivity
e~}
1jJ : IR+ .... [0, 1]
bea
manifold
N
for each
x EM
d(x,m.)
(Riemannian
or at
r ,
or 8.1.4
so that
a center
x
C -cut off function
o ,
1 ,
and define
.=
cp.~ (x) In this
cp.~ (x)
weights
situation
1jJ(
for the points
10d(x,m.) ~ r
)
•
(I.J
we can interpolate
1jJ(
such that
<
-1
1jJ'
F i (x) € Bx 10d(x,m.) J r
the local maps
»-
<
0 ,
by 1
to a map
F
M""N
via the
Definition.
F(x)
:=
t' {F.~(x) ,cp.~ (x)}
142
be a
Let
00
Let
the
N
of 8.1.3
in
manifold
A2 • Let
~ <-
if
F. (x) E BeN
where
1f
6")
(~f q ::
Riemannian
IKI::
bounds
~
affine
4.
on which
be a complete
and curvature
F.
least with
q
maps
the result
set of points,
be local
+
depends.
Assumptions.
radius
step 8.2.3.
that
d(w
section
the map
H. KARCHER
(8.1.3 or 8.1.4).
can
NONLINEAR
The following equivariance
r
acts isometrically
on
AVERAGES
result will be used in the case where a group M
resp. connection preserving
the local maps are compatible with these actions, i.e. with
defined by
8.3.2 Proposition.
yo m
i
and where
i
Yi
(Equivariance of
F) m ' U(m ) = ~i i i assume for the local maps
U
M
+
M
be an isometry which permutes the points
let
V
N
+
N
be an affine diffeomorphism;
0
N
= m
Let
V
on
YOF (x)
F. (x)
~
=
FUi (Ux)
("local equi variance" )
F(x)
=
F(Ux)
("equivariance" ).
F(x)
=
V( r{F. (x),•. (x)})
(8.3.1 )
e{v
~
(8.1.5)
(UX)}
(local equivariance)
then Vo
Proof.
V
0
~
~
F. (x),•. (x)}
0
~
C {FUi (Ux)'.Ui
C {F.J (Ux),
(8.1.5)
F(Ux)
(8.3.1 )
Of course one wants 8.3.1 to define a differentiable
8.3.3 Proposition. The interpolation
(Differentiability (8.3.1)
(see 8.1.4)
(i)
F:
e.A/
Define in a neighborhood
M
v :vf!+ TN , v(x,n)
v(x,F(x))
=
+
N
of graph :=-
F)
is differentiable. F
o
D2V(X,F(X»)
0
dF
x
are defined by
143
vector field
o ;
is atlinvertible linear map;
where the above partial derivatives
More precisely:
the differentiable
\' -1 exp F. (x) ~ n ~
L'
0
(ii) (iii)
of
map:
P. BUSER AND H. KARCHER
(iv)
d1V(
T M .... TN, x n
d1 v (x,n)
(t .... v(x(t) ,n)
).
d dt v(x(t),n)
;(0)
X,TI
is a curve
in the vector
space
TN)
,
n
D
dt v(x,n(t»
(t .... v(x,n(t»
t .... n(t»
Proof. L{F.
(i) is the definition
~ (x),~.~ (x)}
on the set
known.
8.1.6
(8.3.1).
But
of
(ii), resp.
8.1.8
F,
since
as the unique
from
(iii) - which
along
the curve
•
(iv) and
x by differentiation
follows ready
is defined
B
is a vectorfield
for each
x
the center
zero of the vectorfield
(v) are the definitions
(i) if the differentiability is only a restatement
(*) - is the crucial
of
of 8.1.3
assumption
n"" v(x,n)
such that F
(ii) were
al-
(ii), resp.
in the implicit
function
theorem:
If one computes
in local
coordinates
d dt v(x,n(t»
then we have
the so called
d2V( x,n horizontal
d
:=
(dt v(x,n(t»)hor
d (dt v(x,n(t»)ver
Now,
if
=
v(x,n(O»
of the zero section
in
TN
to the zero
of the zero
section
,F(x »TN
o v(x,G(x»
TN
the directional
;'(0) € T
).
and vertical
d dt
(IT
0
N
while
and the image xo)'
( )TN, v x,n
components
v(x,n(t»)
of this derivative:
d dt net)
D
section.
(for each
derivative
dt v(x,n(t»
0 , then the horizontal
transversal
Tv(x
for
components
the vertical Therefore of
d2V(X
components
(iii) ,F(x»
o
span the tangent - if
#
space
0 - are
says that the tangent span the tangent
space
space
0
so that by the implicit
function
theorem
0
=
0 , G(x ) o
=
F(X ) o
locally
144
defines
a differentiable
map
G,
but
NONLINEAR
locally
=
G
F
since
the differentiability
B.3.4
Remark.
(Maximal
We see from 8.3.3 dF
x d v
of
rank of
has maximal
enough
has only the zero
n
=
F(x)
dF
(iii): rank
(8.3.3
if and only
(iii»
information
if the euclidean
has maximal about
d v
1 (x,n)
d1V
has maximal
rank,
145 Tn:·
:will
a
derivative
rank.
the local maps
F. 1
that
. This proves
F.
(ii) and
1 (x,F(x»
In 5.4 we have
n + v{x,n)
AVERAGES
to prove
from
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ERE
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