In this paper, we compute the adiabatic limit of the scalar curvature and prove several vanishing theorems by taking adiabatic limits. As an application, we give a Kastler-Kalau-Walze type theorem for foliations.

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The notion of a generalized CRF-structure on a smooth manifold was recently introduced and studied by Vaisman (2008)[6]. An important class of generalized CRF-structures on an odd dimensional manifold M consists of CRF-structures having complementary frames of the form , where is a vector field and is a 1-form on M with ()= 1. It turns out that these kinds of CRF-structures give rise to a special class of what we called strong generalized contact structures in Poon and Wade [5]. More precisely, we show that to any CRF-structures with complementary frames of the form , there corresponds a canonical Lie bialgebroid. Finally, we explain the relationship between generalized contact structures and another generalization of the notion of a CauchyRiemann structure on a manifold.

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We prove some structure results for transversely reducible Sasaki manifolds. In particular, we show a Sasaki manifold with positive Ricci curvature is transversely irreducible, and so join (product) construction cannot be generalized to irregular SasakiEinstein manifolds, as opposed to the quasi-regular case done by WangZiller and BoyerGalicki. As an application, we classify compact Sasaki manifolds with nonnegative transverse bisectional curvature, which can be viewed as the generalized Frankel conjecture in Sasaki geometry.