We investigate the existence of symplectic non-Kählerian structures on compact solvmanifolds and prove some results which give strong necessary conditions for the existence of Kählerian structures in terms of rational homotopy theory. Our results explain known examples and generalize the Benson-Gordon theorem (Benson and Gordon (1990); our method allows us to drop the assumption of the complete solvability of G).

We introduce the new notion of pseudo-$𝔻$-parallel real hypersurfaces in a complex projective space as real hypersurfaces satisfying a condition about the covariant derivative of the structure Jacobi operator in any direction of the maximal holomorphic distribution. This condition generalizes parallelness of the structure Jacobi operator. We classify this type of real hypersurfaces.

We give a classification of Hopf real hypersurfaces in complex hyperbolic two-plane Grassmannians ${\mathrm{SU}}_{2,m}/S(U_2·U_m)$ with commuting conditions between the restricted normal Jacobi operator ${\overline{R}}_N\phi$ and the shape operator $A$ (or the Ricci tensor $S$).

This paper consists of two parts. In the first, we find some geometric conditions derived from the local symmetry of the inverse image by the Hopf fibration of a real hypersurface $M$ in complex space form $M_m\left(4ϵ\right)$. In the second, we give a complete classification of real hypersurfaces in $M_m\left(4ϵ\right)$ which satisfy the above geometric facts.

We prove the non-existence of real hypersurfaces in complex two-plane Grassmannians whose normal Jacobi operator is of Codazzi type.

Lee, Kim and Suh (2012) gave a characterization for real hypersurfaces $M$ of Type $\left(\mathrm{A}\right)$ in complex two plane Grassmannians $G_2\left({ℂ}^{m+2}\right)$ with a commuting condition between the shape operator $A$ and the structure tensors $\phi$ and ${\phi }_1$ for $M$ in $G_2\left({ℂ}^{m+2}\right)$. Motivated by this geometrical notion, in this paper we consider a new commuting condition in relation to the shape operator $A$ and a new operator $\phi {\phi }_1$ induced by two structure tensors $\phi$ and ${\phi }_1$. That is, this commuting shape operator is given by $\phi {\phi }_{1}A=A\phi {\phi }_1$. Using this condition, we prove that...

In this paper we classify real hypersurfaces with constant totally real bisectional curvature in a non flat complex space form $M_m\left(c\right)$, $c\ne 0$ as those which have constant holomorphic sectional curvature given in [6] and [13] or constant totally real sectional curvature given in [11].

We determine parts of the contact homology of certain contact 3-manifolds in the framework of open book decompositions, due to Giroux.We study two cases: when the monodromy map of the compatible open book is periodic and when it is pseudo-Anosov. For an open book with periodic monodromy, we verify the Weinstein conjecture. In the case of an open book with pseudo-Anosov monodromy, suppose the boundary of a page of the open book is connected and the fractional Dehn twist coefficient $c$ equals $k=n$, where...

We study to what extent some structure-preserving properties of the geodesic reflection with respect to a submanifold of an almost contact manifold influence the geometry of the submanifold and of the ambient space.

This paper concerns projectively Anosov flows ${\phi }^t$ with smooth stable and unstable foliations ${ℱ}^s$ and ${ℱ}^u$ on a Seifert manifold $M$. We show that if the foliation ${ℱ}^s$ or ${ℱ}^u$ contains a compact leaf, then the flow ${\phi }^t$ is decomposed into a finite union of models which are defined on $T^2\times I$ and bounded by compact leaves, and therefore the manifold $M$ is homeomorphic to the 3-torus. In the proof, we also obtain a theorem which classifies codimension one foliations on Seifert manifolds with compact leaves which are incompressible...

All natural operations transforming linear connections on the tangent bundle of a fibred manifold to connections on the 1-jet bundle are classified. It is proved that such operators form a 2-parameter family (with real coefficients).

The classical concept of affine locally symmetric spaces allows a generalization for various geometric structures on a smooth manifold. We remind the notion of symmetry for parabolic geometries and we summarize the known facts for $\left|1\right|$-graded parabolic geometries and for almost Grassmannian structures, in particular. As an application of two general constructions with parabolic geometries, we present an example of non-flat Grassmannian symmetric space. Next we observe there is a distinguished torsion-free...

Starting by the famous paper by Kirillov, local Lie algebras of functions over smooth manifolds were studied very intensively by mathematicians and physicists. In the present paper we study local Lie algebras of pairs of functions which generate infinitesimal symmetries of almost-cosymplectic-contact structures of odd dimensional manifolds.