QUANTITATIVE THEOR Y O F FOLIATION S
Note tha t th e plan e fiel d spanne d b y A an d L" i s als o involutive . Thi s foliatio
corresponds i n th e firs t descriptio n t o choosin g ou r trivializatio n T AD) = D 2 x 5 1 b
taking th e geodesi c ra y whic h goe s backward s i n tim e (th e ra y emanatin g fro m -v fo r
v T AD )) . Thes e tw o foliation s ar e transversa l an d intersec t i n curve s whic h ar e th
orbits o f th e geodesi c flo w o n T AD ) .
So far , al l o f ou r examples hav e bee n foliation s o f codimension-1. I n highe r codi -
mensions, lif e ca n b e muc h mor e complicated . Th e followin g exampl e i s du e t o G . Ree
Example 1.6. Le t M = (R 2 - {(0 , 0) , (1, 0) , (-1, 0)! ) x S 1, an d introduc e coordina
(* » 0) wher e (x , y) are linea r coordinate s o n R
2
an d 0 i s a paramete r wit h 0 6
2n fo r S . Conside r th e 1-dimensional, C 60 foliatio n o f M whos e idea l i s generate d b
the form s
rdr an d r 2 dx + (1- r 2)d6
where r 2 = x 2 + y 2. I f w e lif t thi s foliatio n t o M * = (R 2 - j(0 , 0) , (1, 0) , (-1, 0)} ) x R ,
it ca n b e picture d a s follows . Fro m th e firs t for m w e se e tha t eac h curv e o f th e foliatio
is containe d entirel y i n a circula r cylinde r centere d abou t th e #-axis . Th e secon d for m
states tha t i n the cylinde r o f radiu s r , thes e curve s ar e obtaine d b y slicin g th e cylinde
with th e famil y o f paralle l plane s whos e norma l vector s ar e ( r , 0 , 1 - r ) . A s r ap -
proaches 1, th e lengt h o f thes e curve s approache s infinity ; an d when r - 1, th e plane s
cut th e cylinde r i n a famil y o f vertica l lines . O f course , afte r passin g agai n t o th e quo
tient M , thes e vertica l line s becom e close d curves .
This i s a C 03 foliatio n o f a 3-manifol d suc h tha t ever y lea f i s compact . Superfi -
cially, on e migh t expec t suc h foliation s t o b e extremel y regular . However , th e followin
disturbing fact s ar e eas y t o verify . Le t y b e a lea f i n th e "cylinder " r = 1, an d le t K
be any compac t neighborhoo d o f y i n M . Then :
(i) Th e unio n o f th e leave s passin g throug h K i s no t compac t i n Af .
(ii) Th e functio n whic h assign s t o eac h poin t th e lengt h o f th e lea f passin g throug
that poin t i s unbounde d o n K.
Note tha t i n thi s exampl e th e manifol d M is no t compact . I n fact, a dee p theore m
D. Epstei n [EP2 ] say s tha t i f M is a compac t 3-manifol d foliate d b y circles , the n th e
above phenomen a canno t occur . I n fact , th e foliatio n wil l b e th e collectio n o f orbit s fo
some circl e actio n o n M .
For explici t construction s o f codimension- 1 foliation s o n odd-dimensiona l sphere
and othe r manifolds , th e reade r i s referre d t o [L2] ,
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