4 H. BLAIN E LAWSON , JR .
manifold carrie s a codimension-^ , C r foliation . Thi s topi c wil l b e discusse d i n Chap
ter IV .
The proo f o f th e equivalenc e o f th e abov e definition s i s straightforwar d an d i s lef
to th e reader .
Suppose no w tha t ? i s a foliatio n o f dimensio n p an d clas s C r ( r 1) o n a mani
fold M . The n th e fiel d o f tangen t space s t o th e leave s for m a p-dimensiona l subbundl
r(5") o f th e tangen t bundle , calle d th e tangent bundle t o ? . Th e quotien t uCf) = 7XVf)A
is calle d th e normal bundle t o j .
A natura l questio n i s whethe r an y give n (differentiable ) p-plan e fiel d o n a manifol
M i s automaticall y th e tangen t bundl e t o a foliatio n o n M . Fo r p = 1, th e answe r i s y
since th e questio n i s easil y reduce d t o solvin g a syste m o f ordinar y differentia l equa -
tions. However , fo r p 1, th e answe r i s i n genera l no , sinc e th e correspondin g syste
of partia l differentia l equation s mus t satisf y certai n compatibilit y condition s whic h co
respond t o th e commutativit y o f secon d partia l derivatives . Specifically , le t r b e a
smooth p-plan e fiel d o n M an d le t T{r) denot e th e spac e o f smoot h section s o f r (i.e.
smooth vecto r fields V wit h V €T fo r eac h x £ M). Suppos e r i s th e tangen t bundl e
to a foliatio n 3" , an d le t (x, y): U —* R ^ x R q b e a distinguishe d coordinat e syste m f
5". The n an y V 6 V{r) whe n restricte d t o U ca n b e expresse d i n th e for m
where th e a. ar e smoot h function s o n [7 . I t follow s tha t th e Li e bracke t [V, W] r(r)
for al l V , W e T(r) sinc e [d/dx., d/dx.] = 0.
The conditio n tha t T(r) b e close d unde r th e operatio n o f takin g Li e bracket s i s
clearly nontrivial . (Conside r r = span(d/9x , d/dy + x(d/dz)) i n R . ) An y smoot h plan e
field whic h satisfie s thi s conditio n i s sai d t o b e integrable.
This conditio n ha s a dua l formulatio n whic h i s ofte n usefu l computationally . Le t
i(r) denot e th e idea l o f differentia l form s whic h vanis h whe n restricte d t o r . The n r i s
integrable i f an d onl y i f d(i(r)) C i{r) wher e d denote s exterio r differentiatio n (cf . [ST]
In particular , i f r ha s codimensio n 1, the n i(r) i s generate d locall y b y a singl e 1-form
co (wher e r = \V T M: co(V) = 0\), whic h i s determine d u p t o multiplicatio n b y a no -
where vanishing , smoot h function . Th e condition , d{$(r)) Ci(r) is the n equivalen t t o th
requirement tha t
(1.5) dco - a A co
for som e 1-form a . Mor e generally , i f r ha s codimensio n q, the n ${r) i s generate d lo -
cally b y ^-independen t 1-forms, co
v
. . . , co . Settin g Q = co ^ A A co , th e integra
bility conditio n i s equivalen t t o th e requiremen t tha t
(1.6) fi=aAQ
for som e 1-form a . (Not e tha t i n bot h cases , th e condition s ar e triviall y satisfie d i f th e
forms ar e closed. )
The classica l theorem , whic h i s usuall y ascribe d t o Frobeniu s but , i n fact , goe s
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