The KV-homology theory is a new framework which yields interesting properties of lagrangian foliations. This short note is devoted to relationships between the KV-homology and the KV-cohomology of a lagrangian foliation. Let us denote by ${}_F$ (resp. $V^F$) the KV-algebra (resp. the space of basic functions) of a lagrangian foliation F. We show that there exists a pairing of cohomology and homology to $V^F$. That is to say, there is a bilinear map $H^q{(}_F,V^F)\times H_q{(}_F,V^F)\to V^F$, which is invariant under F-preserving symplectic diffeomorphisms....

We establish a formula for the Schouten-Nijenhuis bracket of linear liftings of skew-symmetric tensor fields to any Weil bundle. As a result we obtain a construction of some liftings of Poisson structures to Weil bundles.

We discuss the existence of closed geodesic on a Riemannian manifold and the existence of periodic solution of second order Hamiltonian systems.

Let $M$ be a smooth manifold. The tangent lift of Dirac structure on $M$ was originally studied by T. Courant in [3]. The tangent lift of higher order of Dirac structure $L$ on $M$ has been studied in [10], where tangent Dirac structure of higher order are described locally. In this paper we give an intrinsic construction of tangent Dirac structure of higher order denoted by $L^r$ and we study some properties of this Dirac structure. In particular, we study the Lie algebroid and the presymplectic foliation...

We discuss the gluing principle in Morse-Floer homology and show that there is a gap in the traditional proof of the converse gluing theorem. We show how this gap can be closed by the use of a uniform tubular neighborhood theorem. The latter result is only stated here. Details are given in the authors' paper, Tubular neighborhoods and the Gluing Principle in Floer homology theory, to appear.

The object of the present paper is to study some types of semisymmetry conditions on two classes of almost Kenmotsu manifolds. It is shown that a $(k,\mu )$-almost Kenmotsu manifold satisfying the curvature condition $Q·R=0$ is locally isometric to the hyperbolic space ${ℍ}^{2n+1}(-1)$. Also in $(k,\mu )$-almost Kenmotsu manifolds the following conditions: (1) local symmetry $(\nabla R=0)$, (2) semisymmetry $(R·R=0)$, (3) $Q(S,R)=0$, (4) $R·R=Q(S,R)$, (5) locally isometric to the hyperbolic space ${ℍ}^{2n+1}(-1)$ are equivalent. Further, it is proved that a ${(k,\mu )}^{\text{'}}$-almost Kenmotsu manifold satisfying...

The notes consist of a study of special Lagrangian linear subspaces. We will give a condition for the graph of a linear symplectomorphism $f:({ℝ}^{2n},\sigma ={\sum }_{i=1}^{n}d{x}_i\wedge d{y}_i)\to ({ℝ}^{2n},\sigma )$ to be a special Lagrangian linear subspace in $({ℝ}^{2n}\times {ℝ}^{2n},\omega =\pi *₂\sigma -\pi *₁\sigma )$. This way a special symplectic subset in the symplectic group is introduced. A stratification of special Lagrangian Grassmannian $S{\Lambda }_{2n}\simeq SU\left(2n\right)/SO\left(2n\right)$ is defined.

We construct a family of Lagrangian submanifolds in the complex sphere which are foliated by $(n-1)$-dimensional spheres. Among them we find those which are special Lagrangian with respect to the Calabi-Yau structure induced by the Stenzel metric.

We study the action of a real-reductive group $G=Kexp\left(𝔭\right)$ on a real-analytic submanifold $X$ of a Kähler manifold. We suppose that the action of $G$ extends holomorphically to an action of the complexified group $G^{ℂ}$ on this Kähler manifold such that the action of a maximal compact subgroup is Hamiltonian. The moment map induces a gradient map ${\mu }_{𝔭}:X\to 𝔭$. We show that ${\mu }_{𝔭}$ almost separates the $K$–orbits if and only if a minimal parabolic subgroup of $G$ has an open orbit. This generalizes Brion’s characterization of spherical...

We give an example of a symplectic manifold with a stable hypersurface such that nearby hypersurfaces are typically unstable.

We study the stability of singular points for smooth Poisson structures as well as general Lie algebroids. We give sufficient conditions for stability lying on the first-order approximation (not necessarily linear) of a given Poisson structure or Lie algebroid at a singular point. The main tools used here are the classical Lichnerowicz-Poisson cohomology and the deformation cohomology for Lie algebroids recently introduced by Crainic and Moerdijk. We also provide several examples of stable singular...

In this paper we firstly define a tangential Lichnerowicz cohomology on foliated manifolds. Next, we define tangential locally conformal symplectic forms on a foliated manifold and we formulate and prove some results concerning their stability.

On a $4$-dimensional compact symplectic manifold, we consider a smooth family of compatible almost-complex structures such that at time zero the induced metric is Hermite-Einstein almost-Kähler metric with zero or negative Hermitian scalar curvature. We prove, under certain hypothesis, the existence of a smooth family of compatible almost-complex structures, diffeomorphic at each time to the initial one, and inducing constant Hermitian scalar curvature metrics.