Chapter I . Basi c Definition s an d Som e Example s

As wit h man y importan t concept s i n mathematics , ther e ar e severa l equivalen t way

of definin g th e notio n o f a foliation , an d i t i s valuabl e t o hav e eac h o f thes e formula -

tions a t han d whe n thinkin g abou t th e subject . Th e simples t an d mos t geometri c i s th e

following. Le t M b e a n rw-dimensiona l manifol d o f clas s C , 0 k co.

Definition 1.1 . A foliation of class C r , 0 r k, and of dimension p (or codimen-

sion q = m - p) is a decomposition of M into disjoint connected subsets {£

a

J .,

called the leaves of the foliation, such that each point of M has a neighborhood {] and

a system of C T coordinates (x , y): {] —+ R p x R q such that for each leaf £

a

, the com-

ponents of U n £ are described by the equations:

y\

= constan t

constant

*y

- • x

We denote th e foliatio n b y J = \^

a

\aeA Th e coordinate s referre d t o i n th e defini -

tion ar e sai d t o b e distinguished b y th e foliation .

Note tha t eac h lea f i s a connecte d C

r

submanifol d o f dimensio n p embedde d i n M

The embedding , however , i s no t necessaril y proper ; tha t is , th e natura l manifol d topol

ogy o n th e lea f i s no t necessaril y th e on e induce d fro m M since th e lea f ma y pas s

through a give n char t {] infinitel y often , an d accumulat e o n itself . I n fact , give n a fo -

1

http://dx.doi.org/10.1090/cbms/027/01