Publicacions
Ma emá iques,
Vol
34
(1990),
311-321
.
A
bs ac
LIE
ALGEBRAS
OF
VECTOR
FIELDS
AND
CODIMENSION
ONE
FOLIATIONS
ToMASz
RYBICKI
The
main
esul
is
a
Pu sell-Shanks
ype
heo em
o
codimension
one
oli-
a ions
.
This heo em can be iewed
as
a
pa ial
solu ion o
a
hypo he ical
gene al
e sion
o
he
heo em
o
Pu sell-Shanks
.
Se e al
p oposi ions
and
lemmas
on
olia ions
a e
con ained
in
he
p oo
.
1
.
In oduc ion
The
Lie
algeb a
X(M)
consis ing
o
all
ec o
ields
onasmoo h
mani old
M
gi es
an
impo an
example
o
an
in ini e
dimensional
Lie
algeb a
.
A
un-
damen al
heo em
p o ed
by_
L
.E
.
Pu sell
and
M
.E
.
Shanks
in
[10]
s a es
ha
he Lie
algeb a
s uc u e
o
X(M)
comple ely
de e mines
he
unde lying
opo-
logical
and
smoo h
s uc u e
o
M
.
A
whole
se ies
o£
pape s
ollowed, e
.g
.
[1],
[6], [9]
.
Ou
objec
o
in e es
is
he Lie
algeb a
o£
a
codimension
one
olia ion
.
Le
(Ml,
F,)
and
(M2,
F2)
be
smoo h,
second
coun able
and
smoo hly
oli-
a ed
mani olds
.
By
Y(Mi,
F¡),
i
=
1,
2,
we
deno e
he Lie
algeb a
o£
all
lea£
p ese ing
ec o
ields
on
Mi
.
I
.
Amemiya
in
[1]
p o ed
a
heo em
which
can
be
o mula ed
as
ollows
.
Theo em
1
.1
.
I
he e
exis s
a
Lie
algeb a
isomo phism
oP
o
y(M
1
,F
1 )
on o
Y(M2,F2)
hen
he e
exis s
a
olia ion
p ese ing
di eomo phism
cp
o
M
l
on o
M2
such
ha
So
.
=
(P
.
This no e
is
de o ed
mainly
o a
gene aliza ion
o£
Theo em
1 .1
.
Fo
any
olia ed
mani old
(M,
F)
he
symbol
X(M,
F)
s ands
o
he Lie
algeb a
o
all
olia ed
ec o
ields
i
.e
.
ec o
ields
wi h
a
low
ans o ming
each
lea
o£
F
op o
a
lea£
o
F
.
We
ecall
ha
X
is
a
olia ed
ec o
ield
i
[X,
Y] E
Y(M,
F)
o
any
Y
E
y(M,
F)
.
Thus
y(M,
F)
is
an
ideal o
X(M,
F)
.
Now
ou
esul
is
he
ollowing
312

T
.
RYBICKI
Theo em
1 .2
.
Le¡
(Ml,
FI)
and
(M2,
F2)
be
one
codi nensioal
smoo h
non-
i ial
(i
.
e
.
dim
Ml
>_
2,
dim
M2
>_
2)
olia ions
wi h
Ml,
M2
compac
.
I
hese
is
a
Lie
algeb a
isomo phism
<D
o
X(M1,F1)
on o
X(M2,F2)
hen
hese
is
a
olia ion
p ese ing
di eomo phism
cp
o
Ml
on o
M2
such
ha
cp
*
=,¿
.
Fo
simplici y
sake
we
es ic
ou
a en ion
in
he
p oo
o
ans e sely
o ien able
olia ions
.
Le
us
indica e
analogous
esul s
o
Theo ems
1 .1
and
1 .2
.
The
case
o
compac
olia ions
was
sol ed
in
[6]
as well as
he
case o
ans e sely
comple e
olia ions
in
[111
.
Two
e y
special
cases
o
codimension
one
olia ions
we e
also
conside ed
in
[6,
Theo ems
C
and
D]
.
In
he
las
sec ion
o
ou
no e
we
o mula e
b ie ly
a
conjec u e
b oadly
gene alizing
hese
esul s
.
In
his
no e
all
mani olds
a e
second
coun able
and
all
mani olds
and
olia-
ions a e o
caass
C°°
.
Howe e ,
all
conside a ions
in
he
sec ion 2
emain
ue
in
he
case
o
C
2
olia ions
and
C
2
unc ions
.
Finally
I
would
like
o
hank
D
.
Robe
Wolak
o
ui ul
con e sa ions
.
I
alsó
exp ess
my
deep
g a i ude
o he
e e ee
whose
c i ical
ema ks
helped
me
o
imp o e
his
no e
.
2
.
S uc u e
o
open
sa u a ed
se s
Va ious
au ho s
ha e
s udied
a
heo y
o
open
sa u a ed
se s
o
one
codimen-
sional
olia ions
.
Re e en es
o
he
ac s
p esen ed
in
his
sec ion a e
Can well
and
Cónlon
[2],
Hec o
[3]
and
Hec o
and
Hi sch
[4]
.
We
shall
o mula e
and
p o e
he e
some
p epa a o y
esul s
which
will
be
use ul
in
he sequel
.
Le
(M,
F)
be
a
olia ed
compac
mani old wi h
codimension
one
.
The e
is
a
inc easing
sequence
Mk,
k
=
0,
1, 2,
. .
.,
o closed
sa u a ed
subse s
o
M
such
ha
1)
Mo
is
he
union
o
all
minimal
se s
o
F
;
2)
i
Mk_1
is
de ined
o
k
>
0
hen
he
se
Mk Mk_1
is
he
union
o
all
minimal
se s
o
FIM Mk_1
.
Each
lea
con ained
in
Mk Mk_1
is
called
a
lea
a
le el
k
.
In gene al, a
lea
con ained
in
some
Mk
is
said o
be
a
ini e
le el
.
On
he
con a y,
i
hese
a e
any
lea es
in
M
U{Mk,
k
='0,1,
. . .
}
hey
a e said o
be
a
in ini e le el
.
Le
a
lea
L
be
a
in ini e le el
.
The
s uc u e
o
L
is
he
union
o
all
lea es o
L
ha
a e a
ini e
le el
.
Then Theo em
5
.0
in
[2]
s a es
ha
he subs uc u e
o
L
is
dense
in
L
.
Ou
objec
o
in e es
a e
local
minimal
se s
.
A
local
minimal
se
is
a
minimal
se
o
FlU,
whe e
U
is
some
open
sa u a ed
se
.
In pa icula ,
e e y
p ope
lea
is
alocal
minimal
se
.
Each
lea
o a
local
minimal
se
lies
a
ce ain
ini e
le el
.
On
he
o he
hand,
lea es
ha
lie
in
no
local
minimal
se
a e
a
in ini e
le el
.
LIE
ÁLGEBRA
OF
CODIMENSION
ONE
FOLIATION

31
3
We
ecall
ha
minimal
se s
can
be
ei he
closed
lea es,
o
excep ional
se s,
o
he
whole
mani old
M
.
Deno e
by
C(F),
E(F)
and
Z(F)
he
union
o
all
closed
lea es,
he
union
o
all
excep ional
minimal
se s
and
he
union
o
all
minimal
se s
o
(M,
F)
espec i ely
.
Le
U
be
any open
sa u a ed
se
.
Then
Z(F1U)
7É
0,
C(F1U)
is
closed
and
E(F1U)
is
a
ini e
union
(c
.
[3])
.
In
pa icula ,
he
se
Z(F1
U)
is
closed in
U
.
F om
now
on
we
ix
a
olia ion
T
ans e sal
o
F
;
i is
one
dimensional
and
o ien able
.
Following
[3] le
us
conside
a si ua ion
nea
a
p ope
lea
L
.
Ha ing
ixed
T
one can
cons uc
a
pseudog oup
P
o
global
holonomy on an
o bi
C
o
T
.
Le
x
E
C 1L
.
I is
essen ial
ha
he e
is
a
compac
neighbo hood
V
o x
con ained
in
a
T-plaque
wi h
a
amily
n,
n
=
1,
2,
. .
.,
o
unc ions
gene a ing
P
on
V
and
de ined
on
V
.
We
may
iden i y
x
wi h
0
and
V
wi h an
in e al,
say [-1,1]
.
The
ac ion
o
P
.,,
he
iso opy
subpseudog oup
a x,
induces
on
V
an
equi alen e
ela ion
.
Sh inking
V
i
necessa y,
we
ha e
he
ollowing h ee
ypes
o
minimal
subse s
o
[-1,1]
:
(i)
a
single
poin
;
(ii)
a
cyclic
o bi
Le
.
a
sequence
(xj
in
(0,1]
o [-1, 0)
con e ging
o
0
;
(iii)
he
in e als
(0,1]
o
[-1,
0)
.
Deno a ion
.
Le
U
be a
union
o
all
lea es
which
a e
no
locally
dense
and
ha e
an
open
sa u a ed
neighbo hood
U
such
ha
FI
U
is
wi hou
holonomy
.
Then
U
is
open and
sa u a ed
.
O
cou se,
U
need
no
con ain
all
lea es
wi hou
holonomy
.
Lemma
2
.1
.
Le¡
L
be
a
p ope
lea
a
le el
k
such
ha
he e
is
an
open
sa u a ed
neighbo hood
o
L
consás ing o
p ope
lea es
a
le el
k
only
.
Then
LcU
.
P oo
..
Le
V
be
a compac
neighbo hood
o x
E
L
con ained
in a
T-plaque
.
We
may
assume
ha
V
mee s
only
p ope
lea es a
le el
k
.
Hence
he
ypes
(ii)
and
(iii)
abo e
a e no
admissible
on
V
.
Hence
FIV
is
wi hou
holonomy,
whe e
V
is
he
sa u a ion
o
V
.
As
a
consequence
o
Lemma
2
.1
we
ha e
he
ollowing
P oposi ion
2
.2
.

Le
L
be
a
lea a
pi e
le el
such
ha
L
~
U
.
Then
he e
is
an
open
sa u a ed
neighbo hood
U
o
L
wi h
he
ollowing p ope y
:
o
any
GC
U, an
open
sa u a ed
neighbo hood o
L,
he e
exisis
W,
an open
sa u a ed
subse
o
G,
such
ha
L
C
W
and
any smoo h
unc ion
on
G
cons an
along
¡he
lea es
o
FIG
is
cons an
on
each
componen¡
o
W
.
P ooL
The
lea
L
may
be
p ope ,
locally
dense
o
excep ional
.
The
p oo
is
i ial
i
L
is
locally
dense
.
I
L
is
excep ional
hen
L
is
con ained
in
an
excep-
ional
minimal
se
N
.
Then
we
apply
Theo em
4
.1 .1
in
[4]
which
es ab_lishes
he
exis en e
o
an
open
sa u a ed
neighbo hood
U
o
N
such
ha
N
C
L'
o
any
L'
in
U
.
The e o e
any smoo h
unc ion cons an
along
he
lea es
which
is
de ined
on
some
W
C
U,
a
neighbo hood
o
L,
is
a
cons and
unc ion
.
314

T
.
RYBICICI
Le
us
ake
up
he
case
o
L
p ope
a
le el
k
.
Le
U
be
an open
sa u a ed
se
such
ha
L
is
minimal
o
FIU
.
We
se
W
=
U C(F1U)
.
Le
W'
be a
connec ed
componen
o
W
.
I s
bounda y
consis s o
a
ini e
numbe
o lea es
Ll,
. . . ,
L,
.
a
le el
k
.
Each
lea
in
W'
spi als
on o
one
o
he
lea es
Ll,
...
,
L,
.
o
on o
one
o
excep ional
minimal
se s
N,
.
+1
,
. .
.
,
N
.,
con ained
in
W'
.
The e o e
a
unc ion
on
U
cons an
along
he
lea es o
FIU
can
assume
only
a
ini e
numbe
o
alues
on
W'
and
hus
i
is
cons an
on
W'
.
Finally,
by
Lemma
2
.1
and by
he
assump ion,
we
ge
L
C
W
.
No ice
ha
in
he
abo e
a gumen
U
may
be
eplaced
by
any open
sa u a ed
G
C U
such
ha
LC
G
.
This
comple es
he
p oo
o
he
p oposi ion
.
H
.
Imanishi
and
K
.
Yagi
in
[8]
ha e
s udied
a
gene alized
Reeb
componen
.
We
ecall
ha
a compac
olia ed
mani old
(N,
F),
whe e
áN
:~
0,
is
called
a
gene alized
Reeb componen
i
he
ollowing
wo
condi ions a e
sa is ied
:
(1)
all
lea es in
In (N)
a e
non-compac
and
p ope
;
(2)
FlIn (N)
has
i ial
holonomy
.
The e
was
shown
he
ollowing
P oposi ion
2 .3
.
(6,
P oposi ion
2
.1
in
[8])
.
Leí
(N,,F)
be
a
gene alized
Reeb
componen¡
.
Then
he e
is
a
ans e se
ec o
ield
X
on
In (N)
such
ha
X
has an
o bií
C
and
C
l
L
=
{one
poin }
o
any
lea
L
in
In (N)
.
Such
a
ec o
ield
is
said
o
be
nice
and
C
is
a nice o bi
.
Le
U
bean open
sa u a ed
se
.
By
U`
we
deno e
he
comple ion
o
U
(c
.
[2]
o
[4])
.
Le
i°
:
U`

+
M
be
a
canonical ex ension
o
he
inclusion
i :
U
-->
M
.
Recall
ha
U`
is
a
mani old wi h
bounda y
and
i`
maps
di eomo phically
each
componen
o
U°
on o
a
pe iphe al
lea
o
U
.
Now
i
U
is
a
connec ed
componen
o
he
se
U
we
ha e
he
ollowing
wo
possibili ies
:
(i)
U`
=
L
x
[0, 1] is
olia ed
as a
p oduc
by
he
olia ions
F
and
T
;
(ii)
U`
is
a
mani old wi h
bounda y,
he e
is
a
ini e
numbe
o lea es
which
a e
he
connec ed
componen s
o
aU`
and
he
es ic ion
FlIn (U`)
is
a
ib a on o e
S'
.
Thenwe
ha e
he
ollowing
co olla y
.
Co olla y 2
.4
.

The e
is
a
nice
ec o
ield
on
U
.
3
.
Maximal
ideals
o
X'(M,F)
By
X'(M,
F)we
deno e
he
de i ed
ideal
o
X(M,
F)
Le
.
he
ideal
gene a ed
by
all
b acke s
[X, Y],
whe e X,
Y
E
X(M,
F)
.
We
begin
wi h
he
ollowing
p oposi ion
which
is
ue
o
an
a bi a y
olia-
ion
(N,
F)
.
LIE
ALGEBRA
OF
CODIMENSION
ONE
FOLIATION

31
5
P oposi ion
3
.1
.
The
de i ed
ideal
Y(N,
.F)
coincides
wi h
y(N,
.F)
.
The
p oo
is
a
modi ica ion
o ha o
Lemma
14
in [11]
and
is
omi ed
.
Co olla y 3
.2
.
y(N,
.F)
is
an
ideal
o
X'(N,
.F)
.
Co olla y
3
.3
.
Fo any
mani old
N
he
de i ed
ideal
X'(N)
coincides
wi h
X(N)
.
Le
X
E
X(M,
F)
.
We
ha e
he
unique
decomposi ion
X
=
X
l
+
X2,
whe e
X
I
is
angen
o
F
and
X2
is
angen
o he
ans e sal
olia ion
T
.
Be o e
gi ing
a
desc ip ion
o
maximal
ideals
we
shall
show
ha
any
X
E
X'(M,
.F)
is
angen
o
F
on
M U,
ha
is
X2
=
0
on
M U
.
Le
LC
M U
be
an
a bi a y
lea
a
ini e
le el
.
I
he e
does
no
exis
any
X
E
X(M,
F)
such
ha
X
is
no
angen
o
L
hen
we
a e
done
.
Suppose
ha
X
E
X(M,
F)
such
ha
X2
q¿
0
on
L
.
Hence
X2
:~
0
on
an
open
sa u a ed
neighbo hood
G
o
L
.

Le
us
conside
any
b acke
[Y,
Z]
E
X'(M,
F)
such
ha
Y
=
Y1
+
Y2
and
Z
=
Z
1
+Z
2
belong
o
X(M,
F)
.

Then
he e
exis
smoo h
unc ions
and
g
on G,
cons an
along
lea es,
such
ha
Y
2
=
X2
and
Z
2
=
gX2
.
By
P oposi ion
2
.2,
he e
is
an
open
sa u a ed subse
W
o
G
such
ha
LC
W
and
,
g
a e
cons an
on
each
connec ed
componen
o
W
.
Hence
[Y,
Z]
=
[Y1
+
Y
2
,
Z
1
+
Z2]
=
[Y1,
Zl]
+
[Y
1
,
Z
2
]
+
[Y2,
ZI]
+
[Y2,
Z2]
=
[Y1,
Z
I ]
+
g[Y1,
X2]
+
[X2,
ZI]
is
angen
o
F
on
W
and, by
con inui y, also
on
L
.
Nex ,
le
L
be
a
in ini e le el
.
The
subs uc u e
o
L
is
ob iously con ained
in
M U
and
by
he
abo e
a gumen
X'(M,
F)
is
angen
o
F
on
he subs uc u e
.
Hence, by
con inui y,
any
elemen
o
X'(M,
F)
is
angen
o
L
.
Thus
we
ha e
he
p oo
o
he
ollowing
Theo em
3
.4
.
Le¡
(M,
F)
be
an
a bi a y
one codimensional
olia ion
on a
compaci
mani old
M
.
The e
is
an open
sa u a ed
subse
U
o
M
such
ha
FIU
is
wi hou
holonomy
andany
X
E
X'(M,
F)
is
angen
o
F
on
M U
.
Rema k
.
I is
known
ha
any
one
codimensional
olia ion
wi hou
holon-
omy
admi s
an
ex ension
o
any
ec o
a
a
poin
o a olia ed
ec o
ield
.
Ou
heo em
comple es
his
p ope y
:
he
ans e sal
pa
o
X'(M,
F)
simply
anishes
ou side
an
open
se
wi hou
holonomy
.
Le
C
be
a nice
o bi
on
U
o
T
(now
we
conside
T
on
U
only)
.

31
6

T
.
RYBICKI
Co olla y 3
.5
.
The
ans e sal
pa
X2
o
X
E
X'(M,
F)
is
comple ely
de e mined
by
i s
es ic ion
X
2
¡C
.
In
ac ,
he
low
o
X2
P
uniquely
de e mines
he
low
o
X2
.
De lni ion
.
F om
now
on,
by
T ans(M)
we
deno e
he
union
o
all
lea es
L
such
ha
he e
is
X
E
X'(M,
F)
wi h
X2
:~
0
on
L
.
T ans(M)
is
open, sa u a ed
and
con ained
in
U
by
Theo em
3
.4
.
The
ol-
lowing
examples
jus i y
he
de ini ion
.
Examples
:
(3
.6)
Le
M
=
T
2
and
F
be
he
well
known
olia ion
by
a
o a ion
wi h
a ional
slope
.
He e
he
ans e sal
pa
o
X(M,
F)
is
isomo phic
o
X(S'
)
.
Since,
by
Co olla y
3
.3,
X'(S')
=
X(S
1 )
we
ha e
X'(M,
F)
_
X(M,
F)
.
In pa icula ,
T ans(M)
=M
.
(3
.7)
The
si ua ion
is
qui e
di e en
in
he
case
o
he
Reeb
olia ion
o
S
3
.
Ob iously
Ll
=
S3
T',
whe e
T
2
deno es
he
unique
compac
lea
.
A
nice
o bi consis s o
wo
disjoin
ci cles
.
Howe e ,
he
ans e sal
pa
o
X(M,
F)
is
ini e
dimensional
as
i
was
shown
by
K
.
Fukui
in
[5]
.
Consequen ly
T ans(M)
=
0
.
Now
we
shall
y o
desc ibe
maximal
ideals
.
Le
N
be
a
mani old
and
p
E
N
.
By
m
P
we
deno e
he
ideal
o
X(N)
o med
by
all
ec o
ields
anishing
a
p
wi h
all
i s
de i a i es
.
By
abuse
o
no a ion,
m
p
deno es
also
such an
ideal
in
any
subalgeb a
o
X(N)
.
The
ollowing
ac
is
undamen al
.
P oposi ion
3
.8
.
Le
(N,
.F)
be
any
compac
olia ed
mani old
and
dim
.F
>
0
.
The
ideals
m
p
,
p
E
N,
a e ¡he
unique
maximal
ideals
in
y(N,
.F)
.
In
pa ic-
ula ,
m
P
a e
¡he
unique
maximal
ideals
in
X(N)
.
The
second
pa
was
p o ed
in [10]
.
The
p oo
o
he
i s
pa
is
essen ially
he
same
.
Deno a ion
.
Fo any
algeb a
A
we
deno e
by
A*
he
se
o
all
maximal
ideals
o
A
.
Fo
simplici y
sake
we
shall
w i e
A
ins ead
o
X'(M,
F)
.
One
can
easily
see
ha
m
p
E A*
o
p
E
M T ans(M)
.
Le
L
be
a
lea
in
T ans(M)
.
Then
PL
deno es
he
ideal in
A
o
all
ec o
ields
wi h
hei
ans e sal
pa s
in ini ely
la
on
L
.
A
s anda d
a gumen
shows
ha
PL
E
A*
.
Lemma
3
.9
.
Fo any
p
E
T ans(M)
he
ideal
m
p
is
con ained
in
PL,
whe e
pEL
.
In
ac ,
i
he
low
o
a
olia ed
ec o
ield
ixes
p
hen
i
ixes,
o cou se, he
lea
L
.
Now
le
us
conside
he
Lie
algeb a
A
es ic ed
o
T ans(M)
.
By
he
abo e
a gumen s,
he
ans e sal
pa
o
A
may
be
iden i ied
wi h
a
subalgeb a,
say
A,
o
X(C),
whe e
C
is
a
opological
sum
o
a
numbe
o
S
l
o
(0,1)
.
In
pa icula ,
LIE
ALGEBRA
OF
CODIMENSION
ONE
FOLIATION

31
7
he
ideals
PL,
whe e
L
C
T ans(M),
co espond
o
m,,,
x
-
E C
.
On
he
o he
hand,
he e
a e
some
maximal
ideals
in
A
di e en
om
m
z
.
Le
{A
;}
be
he
o alli y
o
such
ideals
.
Fo
any
ideal
A
¡
he e
is
he
unique
co esponding
ideal
A
;
E
A*
.
Such
ideals
exis ,
e en
o
e y
simple
olia ions,
as
he
ollowing
example
shows
.
Example
3
.10
.

Le
a
sequence
(xk)
C
(0, l),
k
=
1,
2,
. .
.,
con e ge
o 0
.
By
A(xk)
we
deno e
he
ideal
o
all
X
E
X((0,1))
such
ha
X
is
ak- la
a
xk o
some
sequence
o posi i e in ege s
%
ending
o
oo
.
A
maximal
ideal
con aining A(xk)
gi es
an
example
o such
A
;
.
P oposi ion
3
.11
.
The
unique
elemen s
o
A*
a e
m
p
,
p
E
M T ans
(M),
PL,
L
C
T ans(M),
and
~i
.
P oo
.
.
Le
I
be
a
maximal
ideal
o
A
such
ha
I
is
nei he
con ained
in
m
p
,
no
in
pL,
no
in
a
i
.
Le
P
=
I
l
y(M,
F)
.
Then,
by
P oposi ion
3
.8
and
Lemma
3
.9,
we
ge
P =
Y(M,
)
and
y(M,
F)
C
I
.
The e o e
he e
is
a
unique
J,
an
ideal
o
A, co esponding
o I
.
Then
J
being
p ope
mus
be
con ained
in
m
x
o
A¡
.
Hence I
is
con ained
in
PL
o Ai
.
This
con adic ion
p o es
he
p oposi ion
.
De ini ion
.
Maximal
ideals
pL
and
al
a e
called
"ho izon al"
;
m
p
a e called
" e ical"
.
We
shall
deno e
E=
he
o alli y
o
/UL,
a=
he
o alli y
o
m
p
,
A=
he
o alli y
o
Ai
.
4
.
S one
opology
o A*
and
he
p oo
o
Theo em
1 .2
In
he
si ua ion
desc ibed
abo e
we
ha e
o ind
an
algeb aic
p ope y
which
cha ac e izes
he
ho izon al
ideals
.
This p ope y
being
p ese ed
by
isomo -
phisms
ensu es
ha
p,L
canno be
mapped
on o
m
p
and
ice
e sa
.
Ou
cha -
ac e iza ion
is
based
on
opological
p ope ies
o
A*
.
We
in oduce
he
S one
opology
on
he
se
A*
in
he
ollowing
way
.
Le
52
be
a
subse
o
A*
.
Then
he
closu e
o
52 is
de ined
by
52=
{mEA*
:n{wE52}Cp}
.
In
pa icula ,
~
=
0
.
Rema k
.
Le
(N,
F)
be
any
olia ed
mani old
wi h
dim
.F
>
0
.
I is
an
easy
consequence
om
P oposi ion
3
.8
ha
he
mapping
N
Dp
->
m
pE
y(N,
F)*
is
a
homeomo phism
.
Speci ically,
N
and
X(N)*
a e
homeomo phic
.
Now
we
conside
he
pa i ion
o
A*
by
connec ed
componen s
in
he
S one
opology
.
We
wan
o
show
ha
such
a componen mus be
" e ical"
Le
.
o med
by
e ical
ideals,
o
"ho izon al"
Le
.
o med by
ho izon al
ideals
.
Fi s
we
s eng hen
Theo em
3
.4
.
31
8

T
.
RYBICKI
Theo em
4
.1
.
The
ans e sal
pa
X
o
a
ec o
aeld
belonging
o
A
an-
ishes
wi h
all
i s
de i a i es
on
M T ans(M)
.
P oo
. .
Le
pEL and
L
9~-
T ans(M)
.
Fi s case
:
L
C
U
.
The e
is
a
olia ed
ec o
ield
Y
on
M
angen
o
T,
Y(p)
:~
0
.
Le
(U, xl,
..
.
,
x
n )
be
a
bidis inguished cha
a
p
(wi h
espec
o
F
and T)
such
ha
Y=
a1ax
n
nea
p
.
Suppose
ha
he e
is
k
>
0
such
ha
he
k- h
de i a i e
a
k
XIaxñ
~
0
a
p
.
We
ha e
[Y,
X]
E
X(M,
F)
and
[Y,
X]
=
OXIax
n
on
a
neighbo hood
o
p
.
Mo eo e ,
X
E
A
implies
[Y,
X]
E
A
.
Con inuing
his
p ocedu e
k
imes
we
ge
Xo
E
A
such
ha
X
o
(p)
:~
0
.
This
con adic s
he
de ini ion
o
T ans(M)
.
Second
case
:
L
9~-
U
.
Since
T ans(M)
C
U
is
an open
sa u a ed
se
wi h-
ou
holonomy
one
can
cons uc
a
non- anishing
olia ed
ec o
ield
Y
on
T ans(M)
angen
o
T
IT ans(M)
.
Now,
i
X
=
Y
on
T ans(M)
hen
is
a
smoo h
unc ion
on
T ans(M)
cons an
along
lea es
.
Fi s
we
assume
L
o
be
a ini e
le el
.
I
L
is
locally
dense
o
excep ional
hen
X
=
0
on
a
neighbo hood
o
p
(see
he
p oo
o
P op
.
2
.2)
.
Nex ,
le
L
be
p ope
and
U
be an open
sa u a ed
neighbo hood
o
L
such
ha
L
is
a
minima
l
se
o
U
.
Then
W
=
U C(F1
U)
sa is ies
LC
LV
.
Le
L
9~-
W
n
T ans(M)
.
The e
is
G, an
open neighbo hood
o L,
such
ha
G
l
W
1
T ans(M)
=
0
and
G 1W
:~OsinceL~U
.
ThenX=OonGnWáx=OonGnWandso
on
.
Thus
X
is
in ini ely
la
on
G
l
W
and
a
p
.
Nex ,
le
L
C
W
l
T ans(M)
.
As
in
he
p oo
o 2
.2,
he
unc ion
is
cons an
on
each
connec ed
componen
o
W
l
T ans(M)
.
The e o e
'
=
a
=
0
on
W
1
T ans(M)
and
aX
=
0
on
W
1
T ans(M)
and
so
on
.
In
ac ,
locally
Y
=
a1ax,,
in
some
dis inguished
cha
.
Hence,
by
con inui y,
X
is
in ini ely
la
a
p
.
Finally,
le
us
conside
L
a
in ini e le el
.
I
is
isible
ha
o
any
L
a o
he
subs uc u e
o
L
he e
is
an open
se
W
a
such
ha
X
is
in ini ely
la
on
W
1
,
and
L
a
C
W
a
.
Hence
X
is
in ini ely
la
on
he
union
W
=
U
W
a
and
L
C
U
L
a
C
W
.
Thus
X
is
in ini ely
la
a
p
.
This
comple es
he
p oo
.
Co olla y
4
.2
.
We
ha e
E
l
° _

and
A
n

_
~
.
In
pa icula ,
he
componen s
o
A*
a e pu ely
" e ical"
o
pu ely "ho izon al"
.
P oo
..
E
U
A
=
EUA
ollows
om
he
e iden
p ope y
ha
he anishing
wi h
de i a i es
o
ans e sal
pa s
o
olia ed
ec o
ie1ds
canno
yield
he
anishing
o
he
angen
pa s
.
On
he
o he hand,
he
abo e
heo em
implies
"
=
E
.
In
ac ,
le
A
be
any
ho izon al
ideal
.
I
X
is
a
ans e sal
o
an
elemen
o
A
which
does
no
belong
o A hen,
by
he
heo em,
X
E
n
E
.
Then
he
de ini ion
o
he
opology
implies
ha
A
1
"
.
De lni ion
.
pE A*
is
said o
be
dis inguished
i
p
belongs
o
an
elemen
o
a
unique
minimal
amily
R
o
connec ed
componen s
o
A*
chaxac e ized
by
he
equali y
whe e
R
=
U
R
.
LIE
ALGEBRA
OF
CODIMENSION
ONE
FOLIATION

31
9
n{,,E(nñ)*}=6,
Lemma
4
.3
.
The e
does
no
exis
any
connec ed
componen¡
o
A*
consis ing
o
elemen s
o
A
only
.
In
ac ,
i is
immedia e om
he
de ini ion
ha
E
=
E U
A
.
P oposi ion
4
.4
.
The
ho izon al
ideals
a e
uniquely
cha ac e ized
by
he
p ope y
" o
be
dis inguished"
.
P oo
::
In
ac ,
he
in e sec ion o
all
ho izon al
ideals gi es
y(M,
F)
.
I
is
clea
ha
y(M,
F)*
sa is ies
he
de ini ion
by
P oposi ion
3
.8
.
Suppose
now
ha
R
does
no
con ain
some
ho izon al
componen¡,
say
E'
.
By
Lemma
4
.3,
he e
a e
a
connec ed
componen¡
U
o
T ans(M)
and a
lea
LCU
such
ha
JIL
0
R
=
U
R
.
Now,
i
ollows
om
he
de ini ion
o
he
opology
ha
p
V
E
E'
o
e e y L'
C
U
.
Consequen ly,
n
{ i
E
(n
R)*
}
con ains
all
ec o
ields
angen
o
FIU
wi h
suppo s
in
U
and
hus
i
is
non-ze o
.
This
and
he
minimali y
o
R
implies
he
p oposi ion
.
Co olla y 4
.5
.
Y(M,
F)
is
cha ac e ized
in
A
as
¡he
in e sec ion o
all
dis inguished
ideals
.
Now
we
ha e
o
e o mula e
sligh ly
Theo em
1 .1
.
Co olla y
4
.6
.
I
4
is
a
Lie
algeb a
isomo phism
o
X(M
I
,F
I
)
on o
X(M2,
F2) such
ha
$(Y(M1,
Fl))
=
Y(M2,
F2),
hen
he e
is
a
olia ion
p ese ing
di eomo phism
cp
o
Ml
on o
M2
such
ha
cp
*
=
~¿
on
X(MI,FI)
.
The
p oo
is
in [11]
.
The
p oo
o
Theo em
1
.2
is
now
a
consequence
o
Co olla ies
4
.5
and
4
.6
and
o
he
ac
ha
an
isomo phism
o
Lie
algeb as
induces
an
isomo phism
o
hei
de i ed
ideals
.
Rema k
.
Theo em
1
.2
can
be
o mula e
in
sligh ly
mo e
gene al
e sion
.
Namely,
X(MI,
FI)
and
X(M2,
F2)
can
be
eplaced
by any
hei
subalgeb as
A1
and
A2
espec i ely
such
ha
Y(MI,
F
I )
C
A1
and
Y(M2,
F
2
)
C
A2
.
The
p oo
is
he
same
.
5
.
Conjec u e
A
.
Lichne owicz
in oduced
a
no ion
o
Jacobi
mani old
(see
e
.g
.
[7])
which
co eponds
o a
no ion
o
local
Lie
algeb a
o e
R
.
In pa icula ,
his
no ion