2 H. BLAIN E LAWSON , JR.
liation o n M , on e ca n introduc e a secon d topology , calle d th e leaf topology, whos e b
sis consist s o f set s o f th e form : { / eC: y ft) = constant , , y (t) = constant! , wher
(x, y): U R^ x R ^ i s a distinguishe d coordinat e an d 0 i s a n ope n subse t o f (J. I
this topolog y th e connecte d component s o f M ar e th e leaves , an d M carrie s th e struc
ture o f a n (uncountable ) p-dimensiona l submanifol d o f clas s C r.
Observe tha t whil e th e leave s ar e no t necessaril y properl y embedde d i n M , the y
"continue forever" . Thus , fo r example , i f r 1 an d M carrie s a complet e Riemannia
metric, the n eac h lea f i s als o complet e i n th e induce d metric .
As w e shal l see , foliation s aris e naturall y i n a numbe r o f differentia l geometri c
settings. Whe n the y do , on e woul d usuall y lik e t o understan d thei r interna l structure ,
that is , th e structur e o f th e leave s an d th e wa y the y fi t together . Thi s i s th e essentia
concern i n th e stud y o f th e "topolog y o f foliations" .
Let u s continu e wit h ou r sequenc e o f definitions . Suppos e M an d 3 " ar e a s abov
and assum e r 1. I f (x , y) an d (x' , y') ar e distinguishe d loca l coordinate s o n a n op
set ( j i n M , the n th e coordinat e chang e satisfie s th e equation s
(1.1) dy'./dx. = 0
for i = 1 , . . . , q an d / ' = 1 , . . . , p. Le t § r denot e th e pseudogrou p (cf . [KN] ) o f loca
CT diffeomorphism s o f R ^ x R ^ whos e Jacobian s satisf y conditio n (1.1). The n w e h
Definition 1.1' . A foliation of class C r ( r 1) and codimension q on a manifold M
is a Q r -structure on M , i.e., a maximal atlas whose coordinate changes lie in § r .
Thus, a foliatio n ca n b e viewe d a s a certai n pseudogrou p structur e o n a manifol d
and on e ca n brin g t o bea r th e concept s o f deformatio n theory , characteristi c classes ,
etc., develope d t o stud y suc h structure s i n th e mor e classica l cases . Thi s notio n o f a
foliation a s a structur e o n a manifol d i s als o inheren t i n th e nex t definition .
Observe tha t th e simples t exampl e o f a foliation i s give n b y a submersion (i.e. , a map
whose differential i s everywher e surjective) . Le t Q b e a manifold o f dimensio n q m an d
class C r, r 1, an d let / : M Q b e a C T submersion . The n / induce s a natural foliatio n o
codimension q an d class C r o n M by defining th e leave s t o be th e connecte d component s o f
f~Kx) fo r x e Q. (Thi s i s a direct consequenc e o f th e Implicit Functio n Theorem. )
In fact, ever y foliatio n 3 " o n M i s give n locall y i n thi s way . Le t (x , y): U —* R p
Rq b e a distinguishe d syste m o f loca l coordinate s fo r 3" . The n th e projectio n y: (J—
is a submersio n whic h induce s th e foliatio n 3 i n U. I f y' : U 1 —+ R* i s anothe r suc h
submersion an d (J O V 1 4 0 , the n b y (1.1) th e coordinat e chang e i n U O {]' ha s th e
form
(1.2) (*' , / ) = bff(x, y), £()/) )
where rank(^ ) = q. Thi s lead s t o th e followin g
Definition 1.1" . A foliation of class C r ( r 1) and codimension q on M is a maxi
mal family of C
r
submersions f
a
: £/
a
~~"*R* where \U
a
\aeA is an open cover of M and
where for each a, f$ A and each x 6 U
a
C\ VQ there exists a local diffeomorphism cf^
of R q such that fg = f\ ° fa in a neighborhood V
x
of x.
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