6 H . BLAIN E LAWSON , JR.
that eac h elemen t X i n the Lie algebr a 3 of CJ , induces a vector field X o n M . Fi x x e M
and choos e X. , . . . , X $ suc h tha t X. , . . . , X ar e linearl y independen t an d spa
the tangen t spac e t o th e orbi t C{x) a t x . The n X. , . . . , X ar e linearl y independen t
in a neighborhoo d ( 7 o f x i n M , an d sinc e th e dimensio n o f th e orbit s i s constan t
(= dim(C7 ) - dim( G )) , the y spa n th e tangen t space s i n U t o al l th e orbit s throug h U.
This plan e fiel d i s clearl y integrable , s o Theore m 1.5 ca n b e applie d t o produc e distin
guished coordinate s fo r th e orbi t decompositio n nea r x.
Note tha t eve n i n th e relativel y simpl e cas e wher e G = R an d dim( G ) = 0 fo r al l
x ( a nonsingula r dynamica l system) , th e resultin g foliatio n ca n b e quit e complicated .
We shal l no w conside r som e mor e specifi c examples .
Example 1.4. Le t M = 5 I x S 1 = R 2 /Z 2 . The n M admit s severa l interestin g folia
tions.
(i) Linear foliations. Conside r th e idea l o f differentia l form s o n M generated b y
a) = a dx
1
+ ^2^* 2 w n e r e (*i x o^ a r e c o o r dinates o n R 2 an d a
v
&
2
e ^ ' Clearl y do
- 0 , an d s o co defines a foliatio n whic h lift s t o a foliatio n o f R b y paralle l straigh t
lines. Not e tha t i f *
l
/a2 e Q ( t n e rationa l numbers) , the n ever y lea f i s closed . I f a^a
4 Q , the n ever y lea f i f dens e i n M .
(ii) The suspension of a diffeomorphism. Le t / : S 1 —• 51 b e a diffeomorphis m o f
the circl e an d conside r th e actio n o f Z o n R x S , generate d b y th e ma p
(*
p
*
2
) - » Uj - f 1, f{x
2
)).
The foliatio n 3 " = { R x 5 x
2
^x es 1 iS P r e s e r v e ^ ^ y tn ^s action , an d thu s descend s t o a
foliation5 " o n M . Not e tha t b y followin g alon g th e leave s o f 5 " i n M , on e ca n recaptu
the diffeomorphis m / . Not e als o tha t i f / i s a rotation, on e obtain s a linea r foliation .
(iii) A general dynamical system. Le t V b e an y nowhere-vanishin g vecto r fiel d o
M, an d le t J b e th e resultin g foliation . The n afte r changin g basi s i n Z x Z , th e folia
tion wil l hav e th e followin g aspec t i n a fundamenta l domain :
Example 1.5. Le t M = D 2 x S l wher e D 2 = {(x , y) 6 R2: x 2 + y 2 l} . Agai n M a
mits severa l interestin g foliations .
(i) The Reeb foliation. Le t (r , 6, z) b e cylindrica l coordinate s o n M , an d le t f
0 r l b e a C ° ° functio n suc h tha t 0 f(r) 1 f or 0 r 1 an d suc h tha t f ext
in a ° manne r b y definin g f t o b e = 0 fo r r 0 an d = 1 fo r r 1. The n
co = f(r)dr + {l -f(r))dz
satisfies conditio n (1.5) an d s o define s a codimension-1, ° foliatio n o f M with th
property tha t dM i s a leaf .
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