F. ,lBrisque
81
Gromov's almost flat manifolds by Peter BUSER and Hermann KARCHER*
*
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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
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
(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)
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
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
(~)
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
8
in 2.3
ROTATIONAL
and 2.4 we have to
(K)
for
the
present
Now the
assume the
0.48
3.1.4
Proposition. a.
hence
ra.,y] ~~
E
YE
we have
rp
Proposition.
a priori
the
p
rp,-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
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
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
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 ..
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
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
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 2wd(M)m4
48
0 -
I[a,S] I -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
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
)-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
~
(2pl
11 if in addition
then
2 d(x,l"'fl' 2 (lip)