arXiv research

Introduces pqc structures, generalizing para 3-Sasakian geometry.

Local flatness theorem for paraquaternionic contact structures.

We investigate an obstruction for hypersymplectic manifolds equipped with a free, isometric action of SU(1,1). When the obstruction vanishes, we show that the manifold is a metric cone over a split 3-Sasakian manifold. Furthermore, if the action of SU(1,1) is also proper, then the hypersymplectic manifold fibres over a…

Spinorially constructs Sasakian and 3-Sasakian structures in arbitrary dimensions.

In this paper we deal with two classes of mixed metric 3-structures, namely the mixed 3-Sasakian structures and the mixed metric 3-contact structures. Firstly we study some properties of the curvature of mixed 3-Sasakian structures, proving that any manifold endowed with such a structure is Einstein. Then we prove the …

Geometric flows study nearly parallel G2-structures on 3-Sasakian 7-manifolds.

Study 3-Sasakian and G2 structures on manifolds.

Study the geometry of weak para-f-structures and subclasses.

Study para-complex structures on specific Lie groups, finding explicit forms and properties.

Develops theory of para-holomorphic algebroids with para-complex connections.

Researchers found all invariant contact structures on tangent sphere bundles of compact symmetric spaces.

We define a generalized almost para-Hermitian structure to be a commuting pair $(\mathcal{F},\mathcal{J})$ of a generalized almost para-complex structure and a generalized almost complex structure with an adequate non-degeneracy condition. If the two structures are integrable the pair is called a generalized para-Kähle…

The paper classifies para-Kähler structures on Lie groups.

We study the problem of existence of geometric structures on compact complex surfaces that are related to split quaternions. These structures, called para-hypercomplex, para-hyperhermitian and para-hyperkähler are analogs of the hypercomplex, hyperhermitian and hyperkähler structures in the definite case. We show that …

Develops theory of para-holomorphic algebroids on Calabi-Yau manifolds.

Defines Lewy curves in para-CR geometry and characterizes their path geometries.

We work in both the complex and in the para-complex categories and examine (para)-Kähler Weyl structures in both the geometric and in the algebraic settings. The higher dimensional setting is quite restrictive. We show that any (para)-Kaehler Weyl algebraic curvature tensor is in fact Riemannian in dimension at least 6…

It is well-known that 7-dimensional 3-Sasakian manifolds carry a one-parametric family of compatible G_2 structures and that they do not admit a characteristic connection. In this note, we show that there is nevertheless a distinguished cocalibrated G_2 structure in this family whose characteristic connection along wit…

We investigate the integrability of natural almost complex structures on the twistor space of an almost para-quaternionic manifold as well as the integrability of natural almost paracomplex structures on the reflector space of an almost para-quaternionic manifold constructed with the help of a para-quaternionic connect…

Study on $G_2^*$ structures and almost para-contact structures in 7D.

A para-Kähler manifold can be defined as a pseudo-Riemannian manifold $(M,g)$ with a parallel skew-symmetric para-complex structures $K$, i.e. a parallel field of skew-symmetric endomorphisms with $K^2 = \mathrm{Id}$ or, equivalently, as a symplectic manifold $(M,ω)$ with a bi-Lagrangian structure $L^\pm$, i.e. two c…

We prove that any simply connected compact 3-Sasakian manifold, of dimension seven, is formal if and only if its second Betti number is $b_2<2$. In the opposite, we show an example of a 7-dimensional Sasaki-Einstein manifold, with second Betti number $b_2\geq 2$, which is formal. Therefore, such an example does not adm…

We give an elementary proof of the fact that any 4-dimensional para-Hermitian manifold admits a unique para-Kaehler--Weyl structure. We then use analytic continuation to pass from the para-complex to the complex setting and thereby show any 4-dimensional pseudo-Hermitian manifold also admits a unique Kaehler--Weyl stru…

We provide a general criteria for the integrability of the almost para-quaternionic structure of an almost para-quaternionic manifold (M,P) of dimension bigger or equal to eight, in terms of the integrability of two or three sections of the defining rank three vector bundle P. We relate it with the integrability of the…

An almost para-CR structure on a manifold $M$ is given by a distribution $HM \subset TM$ together with a field $K \in Γ({\rm End}(HM))$ of involutive endomorphisms of $HM$. If $K$ satisfies an integrability condition, then $(HM,K)$ is called a para-CR structure. The notion of maximally homogeneous para-CR structure of …

Study para-hyperKähler geometry of anti-de Sitter structures.

In this paper, we introduce the notions of pseudo-Riemannian, para-Hermitian and para- Kahler structures on hom-Lie algebras. In addition, we present the characterization of these structures. Also, we provide an example including these structures. We then introduce the phase space of a hom-Lie algebra and using the hom…

We analyze degenerate homogeneous structures of linear type in the pseudo-Kähler and para-Kähler cases. The local form and the holonomy of pseudo-Kähler or para-Kähler manifolds admitting such structure are obtained. In addition the associated homogeneous models are studied exhibiting their relation with the incomplete…

Complete classification of para-Kähler structures on Lie groups.

The paper generalizes para-Kähler Lie algebras to k-para-Kähler Lie algebras and explores their structures.

We study the class of non-degenerate homogeneous structures of linear type in the pseudo-Kähler, para-Kähler, pseudo-quaternion Kähler and para-quaternion Kähler cases. We show that these structures characterize spaces of constant holomorphic, para-holomorphic, quaternion and para-quaternion sectional curvature respect…

The present paper is devoted to quasi-Para-Sasakian manifolds. Basic properties of such manifolds are obtained and general curvature identities are investigated. Next it is proved that if $M$ is quasi-Para-Sasakian manifold of constant curvature $K$. Then $K$ $\leq 0$ and $(i)~$if $K=0$, the manifold is paracosymplecti…

In this note we discuss the problem of existence of para-hyperhermitian structures on compact complex surfaces. We construct examples of para-hypercomplex structures on Inoue surfaces of type $S^{-}$ which do not admit compatible metrics.

We consider 3-webs, hyper-para-complex structures and integrable Segre structures on manifolds of even dimension and generalise the second heavenly Plebański equation in the context of higher-dimensional hyper-para-complex structures. We also characterise the Segre structures admitting a compatible hyper-para-complex s…

On a para-quaternionic Kähler manifold $(\widetilde M^{4n},Q,\widetilde g)$, which is first of all a pseudo-Riemannian manifold, a natural definition of (almost) Kähler and (almost) para-Kähler submanifold $(M^{2m},\mathcal{J},g)$ can be given where $\mathcal{J}=J_1|_M$ is a (para-)complex structure on $M$ which is the…

In this paper we consider a manifold $(M,\nabla )$ with a symmetric linear connection $\nabla$ which induces on the cotangent bundle $T^*M$ of $M$ a semi-Riemannian metric $\overline g$ with a neutral signature. The metric $\overline g$ is called natural Riemann extension and it is a generalization (made by M. Sekizaw…

Research explores Kähler and semi-para-Kähler structures on specific Lie groups.

For a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev-Zelenko-Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet-Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian. As an application, we consider …

Characterizes chains in 3D CR and para-CR structures.

Almost para-Hermitian manifold it is manifold equipped with almost para-complex structure and compatible pseudo-metric of neutral signature. It is considered a class of immersions of almost para-Hermitian manifolds into almost para-Hermitian manifolds. Such immersions are called slant submanifolds. The concept is an an…

The study constructs associative 3-folds in squashed 3-Sasakian manifolds.

The pseudo-Riemannian manifold $M=(M^{4n},g), n \geq 2$ is para-quaternionic K\" ahler if $hol(M) \subset sp(n, \RR) \oplus sp(1, \RR).$ If $hol(M) \subset sp(n, \RR),$ than the manifold $M$ is called para-hyperK\" ahler. The other possible definitions of these manifolds use certain parallel para-quaternionic structure…

We discuss relations between the para-CR structures and differential equations (both ODEs and PDEs of finite type).

We survey physical models which capture the main concepts of double field theory on para-Hermitian manifolds. We show that the geometric theory of Lagrangian and Hamiltonian dynamical systems is an instance of para-Kahler geometry which extends to a natural example of a Born geometry. The corresponding phase space geom…

The paper constructs almost para-Kähler-Einstein metrics on cotangent bundles.

The subject of this paper is six-dimensional nearly (para-)Kähler geometry with pseudo-Riemannian metrics. Firstly, we derive the analogue of the well-known exterior differential system characterising a nearly Kähler manifold and prove applications to the automorphism group of a nearly (para-)Kähler structure. Secondly…

New proof classifies homogeneous 3-Sasakian and quaternionic Kähler manifolds.

We study Lie algebras admitting para-Kähler and hyper-para-Kähler structures. We give new characterizations of these Lie algebras and we develop many methods to build large classes of examples. Bai considered para-Kähler Lie algebras as left symmetric bialgebras. We reconsider this point of view and improve it in order…