arXiv research

The paper defines and characterizes para-Kahler structures on hom-Lie algebras.

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

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

Complete classification of para-Kähler structures on Lie groups.

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…

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

The paper classifies para-Kähler structures on Lie groups.

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…

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…

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

Study para-Kähler Lie algebroids and generalize pseudo-Hessian structures.

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…

Study on statistical manifolds with product structures and their properties.

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…

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…

Introduces Legendre bundle for dually flat manifolds and quantum field theories.

New para-Kähler structure found in geodesic space.

Let L\subset V=\bR^{k,l} be a maximally isotropic subspace. It is shown that any simply connected Lie group with a bi-invariant flat pseudo-Riemannian metric of signature (k,l) is 2-step nilpotent and is defined by an element η\in Λ^3L\subset Λ^3V. If ηis of type (3,0)+(0,3) with respect to a skew-symmetric endomorphis…

Study geodesics and F-geodesics on tangent bundles over para-Kähler-Norden manifolds.

Study on new warped product submanifolds in para-Kähler manifolds.

Study on new metrics on para-Kähler-Norden manifolds with conformal deformation.

It is a classical fact that the cotangent bundle $T^* \M$ of a differentiable manifold $\M$ enjoys a canonical symplectic form $Ω^*$. If $(\M,\j,g,ω)$ is a pseudo-Kähler or para-Kähler $2n$-dimensional manifold, we prove that the tangent bundle $T\M$ also enjoys a natural pseudo-Kähler or para-Kähler structure $(\J,\G,…

In this paper, we initiate the study of $\p R$-warped products in para-Kähler manifolds and prove some fundamental results on such submanifolds. In particular, we establish a general optimal inequality for $\p R$-warped products in para-Kähler manifolds involving only the warping function and the second fundamental for…

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…

We describe natural Kähler or para-Kähler structures of the spaces of geodesics of pseudo-Riemannian space forms and relate the local geometry of hypersurfaces of space forms to that of their normal congruences, or Gauss maps, which are Lagrangian submanifolds. The space of geodesics L(S^{n+1}_{p,1}) of a pseudo-Rieman…

Survey para-Hermitian geometry and its applications in physics.

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…

Study properties of para-Kähler manifolds with conformal Einstein soliton metrics.

The paper explores new structures on cotangent bundles induced by natural Riemann extensions.

Method finds MAEs on contactified para-Kähler manifolds.

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…

Study on biharmonic and interpolating sesqui-harmonic vector fields on para-Kähler--Norden manifolds.

Let L be a Lagrangian submanifold of a pseudo- or para-Kähler manifold which is H-minimal, i.e. a critical point of the volume functional restricted to Hamiltonian variations. We derive the second variation of the volume of L with respect to Hamiltonian variations. We apply this formula to several cases. In particular …

We address the study of some curvature equations for distinguished submanifolds in para-Kähler geometry. We first observe that a para-complex submanifold of a para-Kähler manifold is minimal. Next we describe the extrinsic geometry of Lagrangian submanifolds in the para-complex Euclidean space D^n and discuss a number …

The aim of this note is the study of Einstein condition for para-holomorphic Riemannian metrics in the para-complex geometry framework. Firstly, we make some general considerations about para-complex Riemannian manifolds (not necessarily para-holomorphic). Next, using an one-to-one correspondence between para-holomorph…

Let $(M,g)$ be an n-dimensional Riemannian manifold and $T^{*}M$ be its cotangent bundle equipped with a Riemannian metric of Cheeger Gromoll type which rescale the horizontal part by a nonzero differentiable function. The main purpose of the present paper is to discuss curvature properties of $T^{*}M$ and construct al…

Researchers identify bi-Lagrangian structures on specific nilpotent Lie algebras.

We show that any dimension $6$ nearly Kähler (or nearly para-Kähler) geometry arises as a projective manifold equipped with a $\textrm{G}_2^{(*)}$ holonomy reduction. In the converse direction we show that if a projective manifold is equipped with a parallel $7$-dimensional cross product on its standard tractor bundle …

Study para-Kähler-Einstein metrics and their non-integrable twistor distributions.

We show that the pseudo-Riemannian geometry of submanifolds can be formulated in terms of higher order multi-linear maps. In particular, we obtain a Poisson bracket formulation of almost (para-)Kähler geometry.

Survey of linking information geometry and optimal transport.

On a smooth manifold M, generalized complex (generalized paracomplex) structures provide a notion of interpolation between complex (paracomplex) and symplectic structures on M. Given a complex manifold (M,j), we define six families of distinguished generalized complex or paracomplex structures on M. Each one of them in…

The paper explores bi-Lagrangian structures in Teichmüller theory.

6D manifolds with special symmetries have limited automorphism groups.

In a previous paper, we have shown that the geometry of double field theory has a natural interpretation on flat para-Kähler manifolds. In this paper, we show that the same geometric constructions can be made on any para-Hermitian manifold. The field is interpreted as a compatible (pseudo-)Riemannian metric. The tangen…

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

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

Let $(M,g)$ be an $n-$dimensional Riemannian manifold and $T_{1}^{1}(M)$ be its $(1,1)-$tensor bundle equipped with the rescaled Sasaki type metric $$ which rescale the horizontal part by a nonzero differentiable function $f$. In the present paper, we discuss curvature properties of the Levi-Civita connectio…