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On the Class of the Transverse Divergence

Volume 8, Number 2 (2009), 163 - 172

On the Class of the Transverse Divergence

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The class of the transverse divergence of a transversally oriented foliation F on a smooth manifold M, denoted I(F ) is the obstruction to the existence of a transverse volume form which is invariant by all foliated vector fields. We analyse the connection of this new invariant with more classical invariants such as the Godbillon-Vey invariant and the Reeb class. We also show that the vanishing of this invariant imposes some restrictions on the geometry of the foliation. For instance, we prove that if M is compact and the codimension of the foliation is one, I(F ) is trivial if and only if F is a Lie R- foliation with dense leaves. For transversally parallelizable codimension q foliations F with dense leaves, we prove that the vanishing of I(F ) implies that the q dimensional basic cohomology of M is non-trivial.