QUANTITATIVE THEOR Y O F FOLIATION S
back t o A . Clebsc h an d F . Deahna , i s th e followin g (cf . [ST] )
Theorem 1.2. A smooth plane field on a manifold M is the tangent bundle to a fo-
liation on M if and only if it is integrable.
Note tha t i f 5 " i s a foliatio n o f clas s C f, the n i n genera l r(3" ) i s o f clas s C r~l.
Unfortunately, i f r i s a n integrabl e plan e fiel d o f clas s C r~l, the n th e resultin g folia -
tion i s onl y o f clas s C r~ , sinc e on e doe s no t increas e differentiabilit y i n norma l dire
tions b y integration . However , whe n r = oo , ther e i s n o problem , an d Theore m 1.2 ca n b
used t o giv e a fourt h equivalen t definitio n o f a foliation .
Since no t ever y plan e fiel d o n a manifol d i s integrable , th e natura l questio n i s whe
ther ever y plan e fiel d ca n b e continuousl y deforme d t o on e whic h is . A deep theore m o
W. Thursto n (cf . Chapte r IV ) state s tha t i n codimension-on e thi s i s alway s possible .
However, i n codimensio n (an d dimension ) greate r tha n 1, i t ma y no t b e possible . Thi s
topic wil l b e discusse d i n Chapte r III .
We sa w abov e tha t an y C r submersio n o f a manifol d M induce s a foliatio n o n M .
More generally , w e hav e th e following . Le t N b e a manifol d wit h a codimension-^ , C r
foliation ? , an d suppos e / : M —• N i s a C r ma p wher e r 1. The n / i s sai d t o b e
transversal t o 3 " (writte n / \) 5" ) i f fo r al l x e M,
Lemma 1.3. Suppose / : M --• N is a C T mapping between manifolds which is trans-
versal to a codimension-q, C r foliation 5 " on N for r 1. Then f induces a codimen-
sion»q, C r foliation, f J , on M by defining the leaves to be the connected components
o / / ^ ( £ ) fort e$.
Proof. Th e loca l submersion s /
a
: U
a
~* R * o n N (cf . Definitio n l.l" ) pul l bac k t o
give loca l submersion s f
a
° f: f~\ll^} —* R * o n M . (Thi s i s equivalen t t o th e transve
sality o f /. ) I f l0^
a
l ar e th e connectin g transformation s fo r th e submersion s / , the n
*l*XQa= 4Ba * o r m ^ e connectin g transformation s fo r th e submersion s f
a
o /. Th e famil
{/ of] ca n b e extende d t o a maxima l family , an d th e proo f i s complete .
This lemm a i s ofte n usefu l i n constructin g example s o f foliate d manifolds . Fo r ex
ample, i f f:M—*N i s a submersion , the n / i s transversa l t o an y foliatio n o n N, I n
particular, i f N admit s a nowher e vanishin g vecto r fiel d V , the n th e integra l curve s o
V pul l bac k t o giv e a foliatio n (o f codimensio n = dim(/V ) - l ) o n M .
Before enterin g a discussio n o f genera l theorems , w e shal l presen t som e example
of foliations , bot h i n orde r t o motivat e th e stud y an d t o giv e th e reade r som e feelin g fo
the subject .
We hav e alread y discusse d th e cas e o f foliation s induce d b y a submersio n (e.g .
products, fibe r bundles , etc.) . Anothe r genera l clas s o f example s i s provide d b y grou p
actions. Le t G b e a Li e grou p actin g differentiabl y o n a manifol d M , an d fo r eac h
x e Ai , le t G - {g G: g(x) = x\ denot e th e isotrop y subgrou p a t x. I f th e dimensio n
of G i s independen t o f x , the n th e orbit s o f G for m a foliatio n o f M . T o se e thi s w
first chec k tha t th e tangen t space s t o th e orbit s for m a smoot h plan e fiel d o n M . Reca
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