Twistor spaces on foliated manifolds

The theory of twistors on foliated manifolds is developed. We construct the twistor space of the normal bundle of a foliation. It is demonstrated that the classical constructions of the twistor theory lead to foliated objects and permit to formulate and prove foliated versions of some well-known results on holomorphic mappings. Since any orbifold can be understood as the leaf space of a suitably defined Riemannian foliation we obtain orbifold versions of the classical results as a simple consequence of the results on foliated mappings.

Basic Cohomology of Canonical Holomorphic Foliations on Complex Moment-Angle Manifolds

We describe the basic cohomology ring of the canonical holomorphic foliation on a moment-angle manifold, LVMB-manifold, or any complex manifold with a maximal holomorphic torus action. Namely, we show that the basic cohomology has a description similar to the cohomology ring of a complete simplicial toric variety due to Danilov and Jurkiewicz. This settles a question of Battaglia and Zaffran, who previously computed the basic Betti numbers for the canonical holomorphic foliation in the case of a shellable fan. Our proof uses an Eilenberg–Moore spectral sequence argument; the key ingredient is the formality of the Cartan model for the torus action on a moment-angle manifold. We develop the concept of transverse equivalence as an important tool for studying smooth and holomorphic foliated manifolds. For an arbitrary complex manifold with a maximal torus action, we show that it is transverse equivalent to a moment-angle manifold and therefore has the same basic cohomology.

On the group of diffeomorphisms of foliated manifolds

Now the foliations theory is intensively developing branch of modern differential geometry, there are numerous researches on the foliation theory. The purpose of our paper is study the structure of the group DiffF(M) of diffeomorphisms and the group IsoF(M) of isometries of foliated manifold (M,F). It is shown the group DiffF(M) is closed subgroup of the group Diff(M) of diffeomorphisms of the manifold M in compact-open topology and also it is proven the group IsoF(M) is Lie group. It is introduced new topology on DiffF(M) which depends on foliation F and called F- compact open topology. It's proven that some subgroups of the group DiffF(M) are topological groups with F-compact open topology.

Einstein-Hilbert type action on spacetimes

The mixed gravitational field equations have been recently introduced for codimension one foliated manifolds, e.g. stably causal and globally hyperbolic spacetimes. These Euler-Lagrange equations for the total mixed scalar curvature (as analog of Einstein-Hilbert action) involve a new kind of Ricci curvature (called the mixed Ricci curvature). In the work, we derive Euler-Lagrange equations of the action for any spacetime, in fact, for a pseudo-Riemannian manifold endowed with a non-degenerate distribution. The obtained equations are presented in the classical form of Einstein field equation with the new Ricci type curvature instead of Ricci curvature

Harmonic measures in embedded foliated manifolds

We study harmonic and totally invariant measures in a foliated compact Riemannian manifold. We construct explicitly a Stratonovich differential equation for the foliated Brownian motion. We present a characterization of totally invariant measures in terms of the flow of diffeomorphisms associated to this equation. We prove an ergodic formula for the sum of the Lyapunov exponents in terms of the geometry of the leaves.