Let $(M,J,\omega )$ be a manifold with an almost complex structure $J$ tamed by a symplectic form $\omega$. We suppose that $M$ has the complex dimension two, is Levi-convex and with bounded geometry. We prove that a real two-sphere with two elliptic points, embedded into the boundary of $M$ can be foliated by the boundaries of pseudoholomorphic discs.

We consider a compact almost complex manifold $(M,J,\omega )$ with smooth Levi convex boundary $\partial M$ and a symplectic tame form $\omega$. Suppose that $S^2$ is a real two-sphere, containing complex elliptic and hyperbolic points and generically embedded into $\partial M$. We prove a result on filling $S^2$ by holomorphic discs.

We prove that one can obtain natural bundles of Lie algebras on rank two $s$-Kähler manifolds, whose fibres are isomorphic respectively to $\mathbf{so}(s+1,s+1)$, $\mathbf{su}(s+1,s+1)$ and $\mathbf{sl}(2s+2,ℝ)$. These bundles have natural flat connections, whose flat global sections generalize the Lefschetz operators of Kähler geometry and act naturally on cohomology. As a first application, we build an irreducible representation of a rational form of $\mathbf{su}(s+1,s+1)$ on (rational) Hodge classes of Abelian varieties with rational period matrix.

Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operators, i.e. isomorphisms of such Lie algebras are induced by the appropriate class of diffeomorphisms of the underlying manifolds.

The main result is a Pursell-Shanks type theorem describing isomorphism of the Lie algebras of vector fields preserving generalized foliations. The result includes as well smooth as real-analytic and holomorphic cases.

For any Lie-Rinehart algebra $(A,L)$, B(atalin)-V(ilkovisky) algebra structures $\partial$ on the exterior $A$-algebra ${\Lambda }_{A}L$ correspond bijectively to right $(A,L)$-module structures on $A$; likewise, generators for the Gerstenhaber algebra ${\Lambda }_{A}L$ correspond bijectively to right $(A,L)$-connections on $A$. When $L$ is projective as an $A$-module, given a B-V algebra structure $\partial$ on ${\Lambda }_{A}L$, the homology of the B-V algebra $({\Lambda }_{A}L,\partial )$ coincides with the homology of $L$ with coefficients in $A$ with reference to the right $(A,L)$-module structure determined by $\partial$. When...

Let $r,n$ be fixed natural numbers. We prove that for $n$-manifolds the set of all linear natural operators $T^{*}\to T^{*}{T}^{\left(r\right)}$ is a finitely dimensional vector space over $R$. We construct explicitly the bases of the vector spaces. As a corollary we find all linear natural operators $T^{*}\to T^{r*}$.

Let $J^{r}T*M$ be the r-jet prolongation of the cotangent bundle of an n-dimensional manifold M and let $(J^{r}T*M)*$ be the dual vector bundle. For natural numbers r and n, a complete classification of all linear natural operators lifting 1-forms from M to 1-forms on $(J^{r}T*M)*$ is given.

Summary: The article is devoted to the question how to geometrically construct a 1-form on some non product preserving bundles by means of a 1-form on an original manifold $M$. First, we will deal with liftings of 1-forms to higher-order cotangent bundles. Then, we will be concerned with liftings of 1-forms to the bundles which arise as a composition of the cotangent bundle with the tangent or cotangent bundle.