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By Leonard Lovering Barrett

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Example text

Two i-forms wO,wi o~ t on M are isotopic, if there is a family ~ 1, with ~O = id and Wo * = ~iwi' ~t of diffeomorphisms, For two closed i-forms wO,wi a [wO] = [wi]' since ~t* = id on cohomology. The following result is a partial answer, using the technique of Moser [MR]. 7 THEOREM. closed i-forms with Let M be a closed manifold, and wO,wi nonsingular [wO] = [wi] HtR(M). If there is a family wt of nonsingular closed i-forms coinciding with wO,w1 for t = 0,1 and such that [wt ] E HtR(M) E is independent of t, then wO,w1 are isotopic.

Wq defined by v = O. that locally v is of the form v the tangent bundle L E TM is This is obvious in a distinguished chart. and The proof of the global statement follows from a partition of unity argument. Note that the transition function of the line bundle AqQ * is det gap' * transition function of Q. * E rQ, if gap is the The transversal orientability assumption on 1 means that det gap can be assumed positive throughout. 10) dv = a A v. 4 generalizes then as follows. 11 THEOREM (Godbillon-Vey).

Was initiated by Reinhart [RE 2]. transition functions on ~q. ). 7) invariantly defined because of the isometric property of For this metric it follows then that 8(X)gQ 0 for all X E rL. This condition is called the holonomy invariance of gQ' It is the infinitesimal equivalent of the invariance under the holonomy transformations sketched in Chapter 4 on transversal manifolds, and serves as the technical definition of the Riemannian property. A metric gM on M is bundle-like, if the induced metric gQ on Q is holonomy invariant.

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