para-Kahler Manifolds Research Articles

Locally conformally flat Kähler and para-Kähler manifolds

We complete the classification of locally conformally flat Kahler and para-Kahler manifolds, describing all possible non-flat curvature models for Kahler and para-Kahler surfaces.

Semi-invariant semi-Riemannian submersions

In this paper, we introduce semi-invariant semi-Riemannian submersions from para-Kahler manifolds onto semi-Riemannian manifolds. Wegive some examples, investigate the geometry of foliations that arise fromthe de…nition of a semi-Riemannian submersion and check the harmonicity ofsuch submersions. We also find necessary and su¢ cient conditions for a semiinvariant semi-Riemannian submersion to be totally geodesic. Moreover, weobtain curvature relations between the base manifold and the total manifold

Lagrangian submanifolds in para-complex Euclidean space

We address the study of some curvature equations for distinguished submanifolds in para-Kahler geometry. We first observe that a para-complex submanifold of a para-Kahler manifold is minimal. Next we describe the extrinsic geometry of Lagrangian submanifolds in the para-complex Euclidean space $\mathbb{D}^n$ and discuss a number of examples, such as graphs and normal bundles. We also characterize those Lagrangian surfaces of $\mathbb{D}^2$ which are minimal and have indefinite metric. Finally we describe those Lagrangian self-similar solutions of the Mean Curvature Flow (with respect to the neutral metric of $\mathbb{D}^n$) which are $SO(n)$-equivariant.

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.

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.

Instantons, black holes, and harmonic functions

We find a class of five-dimensional Einstein-Maxwell type Lagrangians which contains the bosonic Lagrangians of vector multiplets as a subclass, and preserves some features of supersymmetry, namely the existence of multi-centered black hole solutions and of attractor equations. Solutions can be expressed in terms of harmonic functions through a set of algebraic equations. The geometry underlying these Lagrangians is characterised by the existence of a Hesse potential and generalizes the very special real geometry of vector multiplets. Our construction proceeds by first obtaining instanton solutions for a class of four-dimensional Euclidean sigma models, which includes those occuring for four-dimensional Euclidean N = 2 vector multiplets as a subclass. For solutions taking values in a completely isotropic submanifold of the target space, we show that the solution can be expressed in terms of harmonic functions if an integrability condition is met. This condition can either be solved by imposing that the solution depends on a single coordinate, or by imposing that the target space is a para-Kähler manifold which can be obtained from a real Hessian manifold by a generalized r-map. In the latter case one obtains multi-centered solutions. Moreover, if the integrability condition is met, the second order equations of motion can always be reduced to first order equations, which become gradient flow equations if the solution is further required to depend on one coordinate only. The dualization of axions into tensor fields and the lifting of four-dimensional instantons to five-dimensional solitons are used to motivate the addition of a boundary term to the action, which accounts for the instanton action. If the sigma model is coupled to gravity, and if the Hesse potential is of a suitable form which we specify, then the four-dimensional Euclidean Lagrangian can be lifted consistently to a five-dimensional Einstein-Maxwell type Lagrangian. Instanton solutions lift to extremal black hole solutions, and the instanton action equals the ADM mass.

Special geometry of Euclidean supersymmetry III: the local r-map, instantons and black holes

We define and study projective special para-Kahler manifolds and show that they appear as target manifolds when reducing five-dimensional vector multiplets coupled to supergravity with respect to time. The dimensional reductions with respect to time and space are carried out in a uniform way using an -complex notation. We explain the relation of our formalism to other formalisms of special geometry used in the literature. In the second part of the paper we investigate instanton solutions and their dimensional lifting to black holes. We show that the instanton action, which can be defined after dualising axions into tensor fields, agrees with the ADM mass of the corresponding black hole. The relation between actions via Wick rotation, Hodge dualisation and analytic continuation of axions is discussed.

Para-tt*-bundles on the tangent bundle of an almost para-complex manifold

In this paper we study para-tt *-bundles (TM, D, S) on the tangent bundle of an almost para-complex manifold (M, τ). We characterise those para-tt *-bundles with \({\nabla=D + S}\) induced by the one-parameter family of connections given by \({\nabla^{\theta}=\exp(\theta \tau) \circ \nabla \circ\exp(-\theta \tau)}\) and prove a uniqueness result for solutions with a para-complex connection D. Flat nearly para-Kahler manifolds and special para-complex manifolds are shown to be such solutions. We analyse which of these solutions admit metric or symplectic para-tt *-bundles. Moreover, we give a generalisation of the notion of a para-pluriharmonic map to maps from almost para-complex manifolds (M, τ) into pseudo-Riemannian manifolds and associate to the above metric and symplectic para-tt *-bundles generalised para-pluriharmonic maps into \({{{\rm Sp}(\mathbb{R}^{2n})/U^{\pi}(C^n)}}\) , respectively, into SO 0(n,n)/U π(C n ), where U π(C n ) is the para-complex analogue of the unitary group.