Losik classes for codimension-one foliations

Following Losik's approach to Gelfand's formal geometry, certain characteristic classes for codimension-one foliations coming from the Gelfand-Fuchs cohomology are considered. Sufficient conditions for non-triviality in terms of dynamical properties of generators of the holonomy groups are found. The non-triviality for the Reeb foliations is shown; this is in contrast with some classical theorems on the Godbillon-Vey class, e.g, the Mizutani-Morita-Tsuboi Theorem about triviality of the Godbillon-Vey class of foliations almost without holonomy is not true for the classes under consideration. It is shown that the considered classes are trivial for a large class of foliations without holonomy. The question of triviality is related to ergodic theory of dynamical systems on the circle and to the problem of smooth conjugacy of local diffeomorphisms. Certain classes are obstructions for the existence of transverse affine and projective connections.Comment: The final version accepted to Journal of the Institute of Mathematics of Jussie

Non-diffeomorphic Reeb foliations and modified Godbillon-Vey class

The paper deals with a modified Godbillon-Vey class defined by Losik for codimension-one foliations. This characteristic class takes values in the cohomology of the second order frame bundle over the leaf space of the foliation. The definition of the Reeb foliation depends upon two real functions satisfying certain conditions. All these foliations are pairwise homeomorphic and have trivial Godbillon-Vey class. We show that the modified Godbillon-Vey is non-trivial for some Reeb foliations and it is trivial for some other Reeb foliations. In particular, the modified Godbillon-Vey class can distinguish non-diffeomorphic foliations and it provides more information than the classical Godbillon-Vey class. We also show that this class is non-trivial for some foliations on the two-dimensional surfaces.Comment: a corrected versio

The Spin(7)Spin(7)-structures on complex line bundles and explicit Riemannian metrics with SU(4)-holonomy

We completely explore the system of ODE's which is equivalent to the existence of a parallel $Spin(7)$-structure on the cone over a 7-dimensional 3-Sasakian manifold. The one-dimensional family of solutions of this system is constructed. The solutions of this family correspond to metrics with holonomy SU(4) which generalize the Calabi metrics.Comment: 11 page