We give several sufficients conditions for a 2-cycle of $B$ Diff$(S^1{)}_d$ (resp. $B$ Diff${}_K(\mathbf{R}{)}_d$) represented by a foliated $S^1$-(resp. $\mathbf{R}$-) bundle over a 2-torus to be homologous to zero. Such a 2-cycle is determined by two commuting diffeomorphisms $f$, $g$ of $S^1$ (resp. $\mathbf{R}$). If $f$, $g$ have fixed points, we construct decompositions: $f=\pi f_i$, $g=\pi g_i$, where the interiors of Supp$\left(f_i\right)\cup$ Supp$\left(g_i\right)$ are disjoint, and $f_i$ and $g_i$ belong either to $\{h_i^n;n\in \mathbf{Z}\}$ ($h_i\in$ Diff${}^{\infty }$) or to a one-parameter subgroup generated by a $C^1$-vectorfield ${\xi }_i$. Under some conditions on the norms...

We consider the problem of extending the result of J.-P. Jouanolou on the density of singular holomorphic foliations on $\mathbf{C}P\left(2\right)$ without algebraic solutions to the case of foliations by curves on $\mathbf{C}P\left(3\right)$. We give an example of a foliation on $\mathbf{C}P\left(3\right)$ with no invariant algebraic set (curve or surface) and prove that a dense set of foliations admits no invariant algebraic set.

Given a non-singular holomorphic foliation $ℱ$ on a compact manifold $M$ we analyze the relationship between the versal spaces $K$ and $K^{\mathrm{tr}}$ of deformations of $ℱ$ as a holomorphic foliation and as a transversely holomorphic foliation respectively. With this purpose, we prove the existence of a versal unfolding of $ℱ$ parametrized by an analytic space $K^f$ isomorphic to ${\pi }^{-1}\left(0\right)\times \Sigma$ where $\Sigma$ is smooth and $\pi$ : $K\to K^{\mathrm{tr}}$ is the forgetful map. The map $\pi$ is shown to be an epimorphism in two situations: (i) if $H^2(M,{\Theta }_{ℱ}^f)=0$, where ${\Theta }_{ℱ}^f$ is the sheaf of...

The class of locally connected and locally homeomorphically homogeneous topological spaces such that every one-to-one continuous mapping of an open subspace into the space is open has been considered. For a foliation F [3] on a Sikorski differential space M with leaves having the above properties it is proved that for some open sets U in M covering the set of all points of M the connected components of U ∩ L̲ in the topology of M coincide with the connected components in the topology of L for L∈...

Riemannian foliations constitute an important type of foliated structures. In this note we prove two theorems connecting the algebraic structure of Lie algebras of foliated vector fields with the smooth structure of a Riemannian foliation.

For a Riemannian foliation, the topology of the corresponding spectral sequence is used to characterize the existence of a bundle-like metric such that the leaves are minimal submanifolds. When the codimension is $\le 2$, a simple characterization of this geometrical property is proved.

Two significant directions in the development of jet calculus are showed. First, jets are generalized to so-called quasijets. Second, jets of foliated and multifoliated manifold morphisms are presented. Although the paper has mainly a survey character, it also includes new results: jets modulo multifoliations are introduced and their relation to (R,S,Q)-jets is demonstrated.

In this note, a topological version of the results obtained, in connection with the de Rham reducibility theorem (Comment. Math. Helv., 26 ( 1952), 328–344), by S. Kashiwabara (Tôhoku Math. J., 8 (1956), 13–28), (Tôhoku Math. J., 11 (1959), 327–350) and Ia. L. Sapiro (Izv. Bysh. Uceb. Zaved. Mat. no6, (1972), 78–85, Russian), (Izv. Bysh. Uceb. Zaved. Mat. no4, (1974), 104–113, Russian) is given. Thus a characterization of a class of topological spaces covered by a product space is obtained and the...

For a codimension $q$ foliation on a manifold, $\eta \times (d\eta {)}^q$ defines the Godbillon-Vey class. We show that $\eta$ itself defines a certain cohomology class, via the Cech bicomplex.