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GEN - lCHl OSHIKIRI

THEOREM SI (Sullivan [10]). OeC^^{F), i.e., F is taut if and only if each compact leaf of F is cut out by a closed transversal.

THEOREM W (Walczak [12], Oshikiri [6]). F is taut if and only if Cj^^{F) = C^u{0}.

Now recall the set-up of Sullivan [9]. Let D^ be the space of smooth /7-forms on M and D* be the dual space of D^,, i.e., the space of/?-currents. Then we have:

THEOREM SW (Schwartz [8]). (D;)* = D^.

Let X e M and 5 = {^i,..., ^„} be an oriented basis of T^F. We define a Dirac current oj,ß by

охАФ ) = Фх{е\ л • • • л e„ } for 0 G D„,

and set

Cp = the closed convex cone in D* spanned by Dirac currents ö^ß for x e M.

PROPOSITION S (Sullivan [9]). Cp is a compact convex cone in D„. Here ''compact'' means that there is a continuous linear functional L :D* -^R so that the set L~\\) nCp is compact non-empty.

We shall call a compact set L~'(l) nQ of the cone Cp the base of Cp and denote it by С Let ^ : D^, -►Dp^ i be the exterior differentiation and д : D*+1 ->D* be the dual of d, i.e., (#, c) = (ф, de) for ф eD^, С e D*^ i. Here (, ) means the natural coupling D^, x D^ -^ R. Set В = д{Щ+ , ) and Z = Ker 3 : DJ -^ D*_, i.

THEOREM S2 (Sullivan [9]). There is a canonical one to one correspondence between invariant transversal measures and elements in Z nCp.

The main result in Oshikiri [6] is the following:

THEOREM O. For fe C^(M), the following three conditions are equivalent.

( 1 ) / бСд , ( П

( 2 ) There are an n-form œ and an oriented volume form dV on M so that dco =

fdV and 0) is positive on F. Here ''positive'' means that co^(ei л • • л ^„) > 0

for all oriented bases {ci,... ,e„} of T^F and x e M.