Geometric natur e o f integrabilit y
1.1. Foliations , contac t structure s an d confoliation s
1.1.1. Definitions . Unlik e a lin e field, a two-dimensiona l plan e field £ o n a
3-dimensional manifol d M i s not necessaril y integrable , that is , there are not neces -
sarily surfaces , eve n locally , whos e tangen t plane s ar e i n £. 1 Whe n £ is integrable ,
then it s integral surfaces for m a foliation o f the manifold M , s o that locall y it look s
as a fibration o f the cylinde r D 2 x / b y the horizonta l disc s D 2 x t, t G /. Usually ,
we will no t distinguis h betwee n a foliatio n an d th e plan e field £ tangent t o it , an d
often cal l £ itself a foliation , i f it i s integrable .
To understand bette r th e geometr y o f a 2-dimensional plan e field and th e geo -
metric meanin g o f integrability, i t i s helpful t o conside r th e trace s whic h th e plan e
field cut s o n 2-dimensiona l surfaces . Suppos e F i s a 2-dimensiona l surfac e i n th e
3-manifold M. I f F i s transverse t o £ then th e lin e field T(F) D £ on F integrate s
into a 1-dimensional foliatio n F% whic h i s called characteristic. Generically , F an d
£ have isolate d point s o f tangency, an d thu s th e characteristi c foliatio n F^ still ca n
be define d a s a foliation wit h isolate d singularities .
In mos t o f this pape r w e assume fo r simplicit y tha t £ is co-orientable o r trans-
versely orientable, i.e. tha t th e norma l bundl e t o th e plan e field i s orientable .
(The non-co-orien t able cas e i s discussed briefl y i n Sectio n 2.9. 1 below) . A non-co -
orientable tangen t plan e field admit s a co-orientable doubl e cover , s o that man y of
the result s prove n i n th e co-orientabl e cas e ca n b e automaticall y re-prove n i n th e
non-co-orientable one . O f course , fo r loca l consideration s thi s i s irrelevant .
We wil l als o assum e tha t th e manifol d M i s compact , althoug h i n mos t case s
the result s remai n tru e i n th e non-compac t case .
Suppose w e are give n a triangulation o f an oriente d manifol d M b y smal l sim -
plices suc h tha t eac h 1- an d 2-dimensiona l simple x o f th e triangulatio n i s trans -
verse t o £ (compar e [55]) . Thu s th e characteristi c foliatio n o n eac h 2-simple x i s
non-singular. Eac h 3-simple x a o f the triangulatio n ha s exactl y tw o vertices p an d
q such th e £
an d £
ar e supportin g plane s fo r th e simplex . Le t T b e the edg e con -
necting p an d q. Le t u s pick an y co-orientatio n o f £. Thi s co-orientatio n define s a n
orientation o f T , an d togethe r wit h th e orientatio n o f M i t define s a n orientatio n
of £ , whic h give s a n orientatio n o f th e characteristi c foliatio n (dcr)^. Th e holo -
nomy alon g the leave s of the characteristi c foliatio n (da)^ define s a diffeomorphis m
ha :T -T (se e Figure 1.1).
1 This usag e o f "integrable " i s th e sam e a s i n multivariabl e calculu s an d differentia l equa -
tions. I n suitabl e loca l coordinates , integra l surface s ar e graph s (x,y, f(x,y)) o f function s /
that satisf y ("integrate" ) a syste m o f first-orde r partia l differentia l equations , namel y f x(x,y) =
F(x,y,f(x,y)) an d f y(x,y) = G(x,y, f(x,y)).
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