para-Kahler Structures Research Articles

Para-Kahler and Para-Hermitian Structures on Six-Dimensional Unsolvable Lie Algebras

In this paper, we investigate into the matter of the existence of para-Kahlerian and para-Hermitian structures on six-dimensional unsolvable Lie algebras that are semidirect products. According to the classification results, there are four Lie algebras that are semidirect products of the Lie algebras so(3), sl(2, R) and three soluble Lie algebras A3.1=R3, A3.3 and A3.5. We show that only g=A3.5⋉sl(2, R) has a symplectic structure, and it admits a para-Kahlerian structure of zero Ricci curvature. The paper presents calculated curvature characteristics and the method to find other para-Kahler structures based on deformations of some initial para-Kahler structure. Other Lie algebras admit para-Hermitian structures, i.e. integrable paracomplex structures consistent with the natural non-degenerate 2-form. It follows from the results of the paper that the sixdimensional symplectic Lie algebra must be solvable except for one case when g=A3.5⋉sl(2, R). It complements the well-known result of Chu Bon-Yao that a four-dimensional symplectic Lie algebra must be solvable.

A para-Kähler structure in the space of oriented geodesics in a real space form

In this article, we construct a new para-Kahler structure $$({{\mathcal {G}}},{{\mathcal {J}}},\Omega )$$ on the space of oriented geodesics of a n-dimensional (non-flat) real space form. We first show that the para-Kahler metric $${{\mathcal {G}}}$$ is scalar flat and when $$n=3$$ , it is locally conformally flat. Furthermore, we prove that the space of oriented geodesics of hyperbolic n-space equipped with the constructed metric $${{\mathcal {G}}}$$ is minimally isometrically embedded in the tangent bundle of hyperbolic n-space. We then study submanifold theory, and show that $${{\mathcal {G}}}$$ -geodesics correspond to minimal ruled surfaces in the real space form. Lagrangian submanifolds (with respect to the symplectic structure $$\Omega $$ ) play an important role in the geometry of the space of oriented geodesics as they come from the Gauss map of hypersurfaces in the corresponding space form. We demonstrate that the Gauss map of a non-flat hypersurface of constant Gauss curvature is a minimal Lagrangian submanifold. Finally, we show that a Hamiltonian minimal submanifold is locally the Gauss map of a hypersurface $$\Sigma $$ , which is a critical point of the functional $$\mathcal {F}(\Sigma )=\int _{\Sigma }\sqrt{|K|}\,dV$$ , K denoting the Gaussian curvature of $$\Sigma $$ .

Anti-abelian nearly Kähler structures on nilpotent Lie algebras

Morozov (Izv Vyss Ucebn Zaved Mat 4(5):161–171, 1958), classified six-dimensional nilpotent Lie algebras (see also Di Scala and Vezzoni in Ann Global Anal Geom 40:21–45, 2011 for more details). In this paper, we classify anti-abelian almost product structures for these Lie algebras. Then we determine anti-abelian almost para-complex structures on 6-dimensional nilpotent Lie algebras. Finally, it is shown that only one of them admits nearly para-Kahler structures.

Conical Ricci-flat nearly para-Kähler manifolds

In this paper, we classify conical Ricci-flat nearly para-Kahler manifolds having isotropic Nijenhuis tensor. More precisely, we give a bijective correspondence between this class of nearly para-Kahler manifolds and local cones \(M_1 \times (a,b)\) over para-Sasaki-Einstein manifolds \((M_1,g_1,T)\) carrying a parallel 3-form with isotropic support. Moreover, we show that the cone over a para-Sasaki-Einstein five-manifold \((M_1,g_1,T)\) admits a family of parallel 3-forms with isotropic support. As an application our result yields first examples of Ricci-flat (non-flat) nearly para-Kahler structures.

Local rigidity and nonrigidity of symplectic pairs

It is shown that indefinite strictly almost Kahler and opposite Kahler structures (J, J′) on a four-dimensional manifold with J-invariant Ricci operator are rigid, thus extending a previous result of Apostolov, Armstrong and Drăghici from the positive definite case to the indefinite one. In contrast to this, examples of nonhomogeneous four-dimensional manifolds which admit strictly almost paraKahler and opposite paraKahler structures \({(\mathfrak{J},\mathfrak{J}^{\prime})}\) with \({\mathfrak{J}}\) -invariant Ricci operator are shown.

Mean curvature flow of space-like Lagrangian submanifolds in almost para-Kähler manifolds

Given an almost para-Kahler manifold equipped with a metric and para-complex connection, we define a generalized second fundamental form and generalized mean curvature vector of space-like Lagrangian submanifolds. We then show that the deformation induced by this variant of the mean curvature vector field preserves the Lagrangian condition, if the connection satisfies also some Einstein condition. In case the almost para-Kahler structure is integrable, the flow coincides with the classical mean curvature flow in the pseudo-Riemannian context.