QUANTITATIVE THEOR Y OF FOLIATION S 7
By takin g tw o copie s o f thi s foliate d manifol d an d gluin g the m togethe r alon g thei
boundaries, w e produc e a codimension-one, ° foliatio n o f th e 3-spher e S .
(ii) Reinhart foliations. Le t co b e a closed , nowher e vanishin g 1-form o n S L x S
= dM (e.g. , Exampl e l(i ) above) . Pul l o bac k ove r a colla r neighborhoo d o f dM vi a th
projection (r , # , 2 ) * (#, z). Le t £. , f , f be a ° partitio n o f unit y o n th e inte
val [0 , l ] suc h tha t supp(£j ) D supp( £ ) = 0, f
{
= 1 nea r 0 , an d f. = 1 nea r 1.
co = ^,(r)o + fSr)dr + cfAr)dz
satisfies conditio n (1.5) an d defines a codimension-1, ° foliatio n o f M whic h is
transversal t o th e boundar y an d induce s th e foliatio n define d ther e b y co.
Note. Usin g th e simplicit y o f th e grou p Diff^( S ) o f orientatio n preserving , C°°
diffeomorphisms o f 5 , W . Thurston ha s show n tha t every codimens ion-one, C°° folia-
tion of dM, give n b y suspendin g a diffeomorphis m o f S , can be extended to a codimen
sion-ly C°° foliation of M . Th e ide a o f th e proo f i s a s follows . Le t G C Diff^XS1) b e
the se t o f diffeomorphism s whic h hav e th e propert y tha t thei r suspensio n foliation s o n
dM exten d t o M . B y properl y attachin g foliate d copie s o f M alon g circle s i n th e bound
ary whic h ar e transversa l t o th e foliations , on e ca n sho w tha t G i s a norma l subgrou p
of Diff^SM . B y th e example s above , thi s subgrou p i s nontrivial . (I t contain s th e ro -
tations.) B y simplicity , G = Diff~(5 1).
Traversing th e boundar y
of A yield s / .
Traversing th e boundar y
of B yield s g.
Traversing th e boundar y
of th e su m yield s f ° g.
(iii) Anosov foliations. Conside r D a s th e hyperboli c plan e wit h th e Poincar e
metric
ds2 =4\dx\ 2/(l- \x\ 2)2
of constan t Gaussia n curvatur e - 1 . Th e tangen t circl e bundl e T AD2) ca n b e canoni -
cally identifie d wit h D x S wher e S = dD b y sendin g eac h uni t tangen t vecto r v
to th e limi t poin t o n dD 2 o f th e geodesi c ra y emanatin g fro m v.
The isometrie s o f th e Poincar e metri c ar e th e Mobiu s transformation s
eiB((x «)/(! - ax))
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