MARLIES GERBER
These two foliations have the same singularities x ,. . . ,x
and the same number ofprongs p = p(i) at each x., which
we will call a p-prong singularity of f. Moreover, there
e x i s t a 0 and a system of C°° coordinat e c h a r t s (p.,U.),
^ 1 1
i = 1,...,L, L m, on M such that U U- = M and for
i = 1, (cp.,U.) satisfies
/ (i) cpi(Ui)
(ii) cp.(x.) = 0
l l
(2.2) /
(iii) leaves of 7 get mapped to components of the
sets {Re zp = constant} f ) B
a
(iy) leaves of 7 get mapped to components of the
sets {Im 7T = constant} f l 2
a
u
(v) on U.,
(s
x and [i are given by
p-2 2-2
2 2
|Re z dz| and |Im z dz|, respectively
and for i m, (tp.,U.) satisfies
(i) (p.(U.) = (0,b.) x (0,c.) c (t ,t.)-plane,^21
i i i i
for some b.,c. 0
l l
(ii) leaves of 7 get mapped to segments
{tT = constant} f l cp.(U.)
1 ^i
I
(2 3 )\
(iii) leaves of J get mapped to segments
{t0 = constant} f l p.(U.)
2 ^i i
(iv) on U., \xS and iiU are given by |dt1|
V ^ and |dt2|9 respectively.
For convenience we assume that IL,...,U are pairwise disjoint.
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