Six-dimensional nearly Kähler manifolds of cohomogeneity one

We consider six-dimensional strict nearly Kähler manifolds acted on by a compact, cohomogeneity one automorphism group G$G$. We classify the compact manifolds of this class up to G$G$-diffeomorphisms. We also prove that the manifold has constant sectional curvature whenever the group G$G$ is simple.     

Ruled CR-submanifolds of locally conformal Kähler manifolds

The purpose of this paper is to study the canonical totally real foliations of CR-submanifolds in a locally conformal Kähler manifold.     

HyperKähler and quaternionic Kähler manifolds with S -symmetries

We study relations between quaternionic Riemannian manifolds admitting different types of symmetries. We show that any hyperKähler manifold admitting hyperKähler potential and triholomorphic action of S¹$S^1$ can be constructed from another hyperKähler manifold (of lower dimension) with an action of S¹$S^1$ that fixes one complex structure and rotates the other two and vice versa. We also study the corresponding quaternionic Kähler manifolds equipped with a quaternionic Kähler action of the circle. In particular we show that any positive quaternionic Kähler manifolds with S¹$S^1$-symmetry admits a Kähler metric on an open everywhere dense subset.     

Some remarks on special Kähler geometry

Given a special Kähler manifold M$M$, we give a new, direct proof of the relationship between the quaternionic structure on T^∗M$T^{\ast }M$ and the variation of Hodge structures on T^CM$T^{\mathbb{C}}M$.     

Kähler metrics generated by functions of the time-like distance in the flat Kähler–Lorentz space

We prove that every Kähler metric, whose potential is a function of the time-like distance in the flat Kähler–Lorentz space, is of quasi-constant holomorphic sectional curvatures, satisfying certain conditions. This gives a local classification of the Kähler manifolds with the above-mentioned metrics. New examples of Sasakian space forms are obtained as real hypersurfaces of a Kähler space form with special invariant distribution. We introduce three types of even dimensional rotational hypersurfaces in flat spaces and endow them with locally conformal Kähler structures. We prove that these rotational hypersurfaces carry Kähler metrics of quasi-constant holomorphic sectional curvatures satisfying some conditions, corresponding to the type of the hypersurfaces. The meridians of those rotational hypersurfaces, whose Kähler metrics are Bochner–Kähler (especially of constant holomorphic sectional curvatures), are also described.     

Deformation quantization of nonholonomic almost Kähler models and Einstein gravity

Nonholonomic distributions and adapted frame structures on (pseudo) Riemannian manifolds of even dimension are employed to build structures equivalent to almost Kähler geometry and which allows to perform a Fedosov-like quantization of gravity. The nonlinear connection formalism that was formally elaborated for Lagrange and Finsler geometry is implemented in classical and quantum Einstein gravity.     

Random Kähler metrics

The purpose of this article is to propose a new method to define and calculate path integrals over metrics on a Kähler manifold. The main idea is to use finite dimensional spaces of Bergman metrics, as an approximation to the full space of Kähler metrics. We use the theory of large deviations to decide when a sequence of probability measures on the spaces of Bergman metrics tends to a limit measure on the space of all Kähler metrics. Several examples are considered.     

Extrinsic curvatures of distributions of arbitrary codimension

In this article, using the generalized Newton transformation, we define higher order mean curvatures of distributions of arbitrary codimension and we show that they agree with the ones from Brito and Naveira [F. Brito, A.M. Naveira, Total extrinsic curvature of certain distributions on closed spaces of constant curvature, Ann. Global Anal. Geom., 18 (2000) 371–383]. We also introduce higher order mean curvature vector fields and we compute their divergence for certain distributions and using this we obtain total extrinsic mean curvatures.     

Some remarks on the converse of Weyl’s conformal theorem

One of Weyl’s classical theorems states that a certain tensor, the Weyl tensor, is unchanged when the metric from which it is constructed is replaced by another metric conformally related to it. This paper explores the converse of this theorem.     

Symmetries of parabolic contact structures

We generalize the concept of locally symmetric spaces to parabolic contact structures. We show that symmetric normal parabolic contact structures are torsion-free and some types of them have to be locally flat. We prove that each symmetry given at a point with non-zero harmonic curvature is involutive. Finally we give restrictions on the number of different symmetries which can exist at such a point.     

Quantum ball as an example of quantum Kähler manifold

Using the notion of the coherent state map we present a construction of the C^∗$C^{\ast }$-algebra which is considered as a quantum version of the unit ball in C^N${\mathbb{C}}^N$. The structure of the quantum ball as well as its physical interpretation is investigated. Applying the reduction procedure to the quantum ball we obtain the Heisenberg–Weyl algebra.     

Hyper-Kähler induction and self-duality on Riemann surfaces

Universal hyper-Kähler spaces are constructed from Lie groups acting on flat Kähler manifolds. These spaces are used to describe the moduli space of solutions of Hitchin's equation — self-duality equations on a Riemann surface — as the contangent bundle of the moduli space of flat connections on a Riemann surface.     

Essential points of conformal vector fields

An essential point of a conformal vector field ξ$\xi$ on a conformal manifold (M,c)$(M,c)$ is a point around which the local flow of ξ$\xi$ preserves no metric in the conformal class c$c$. It is well-known that a conformal vector field vanishes at each essential point. In this note we show that essential points are isolated. This is a generalization to higher dimensions of the fact that the zeros of a holomorphic function are isolated. As an application, we show that every connected component of the zero set of a conformal vector field is totally umbilical.     

Killing initial data on totally umbilical & compact hypersurfaces

In this note, we give a geometric characterization of the compact and totally umbilical hypersurfaces that carry non-trivial locally static Killing Initial Data (KID). More precisely, such compact hypersurfaces (Mⁿ,g,cg)$(M^n,g,cg)$ endowed with a Riemannian metric g$g$ and a second fundamental form cg$cg$ (where c∈C^∞(M)$c\in C^{\infty }\left(M\right)$ a priori) have constant mean curvature and are isometric to one of the following manifolds:

(i)

Sⁿ${\mathbb{S}}^n$ the standard sphere,     

Holomorphic vector fields of compact pseudo-Kähler manifolds

It is well known that a pseudo-Kähler structure is a natural generalization of the Kähler structure. In this paper, we consider holomorphic vector fields of a compact pseudo-Kähler manifold from the viewpoint of Kähler manifolds.     

Ricci flatness of certain compact pseudo-Kähler solvmanifolds

It is well known that a pseudo-Kähler structure is one of the natural generalizations of a Kähler structure. In this paper, we consider Ricci curvature tensor of certain compact pseudo-Kähler solvmanifolds.     

Dolbeault cohomology groups of compact pseudo-Kähler homogeneous manifolds

It is well known that a pseudo-Kähler structure is one of the natural generalizations of a Kähler structure. In this paper, we consider the Dolbeault cohomology groups of compact pseudo-Kähler homogeneous manifolds.