Para-complex Structure Research Articles

Symplectic structures, product structures and complex structures on Leibniz algebras

In this paper, a symplectic structure on a Leibniz algebra is defined to be a symmetric nondegenerate bilinear form satisfying certain compatibility condition, and a phase space of a Leibniz algebra is defined to be a symplectic Leibniz algebra satisfying certain conditions. We show that a Leibniz algebra has a phase space if and only if there is a compatible Leibniz-dendriform algebra, and phase spaces of Leibniz algebras are one-to-one corresponds to Manin triples of Leibniz-dendriform algebras. Product (paracomplex) structures and complex structures on Leibniz algebras are studied in terms of decompositions of Leibniz algebras. A para-Kähler structure on a Leibniz algebra is defined to be a symplectic structure and a paracomplex structure satisfying a compatibility condition. We show that a symplectic Leibniz algebra admits a para-Kähler structure if and only if the Leibniz algebra is the direct sum of two Lagrangian subalgebras as vector spaces. A complex product structure on a Leibniz algebra consists of a complex structure and a product structure satisfying a compatibility condition. A pseudo-Kähler structure on a Leibniz algebra is defined to be a symplectic structure and a complex structure satisfying a compatibility condition. Various properties and relations of complex product structures and pseudo-Kähler structures are studied. In particular, Leibniz-dendriform algebras give rise to complex product structures and pseudo-Kähler structures naturally.

Null hypersurfaces in 4-manifolds endowed with a product structure

AbstractIn a 4-manifold, the composition of a Riemannian Einstein metric with an almost paracomplex structure that is isometric and parallel defines a neutral metric that is conformally flat and scalar flat. In this paper, we study hypersurfaces that are null with respect to this neutral metric, and in particular we study their geometric properties with respect to the Einstein metric. Firstly, we show that all totally geodesic null hypersurfaces are scalar flat and their existence implies that the Einstein metric in the ambient manifold must be Ricci-flat. Then, we find a necessary condition for the existence of null hypersurface with equal nontrivial principal curvatures, and finally, we give a necessary condition on the ambient scalar curvature, for the existence of null (non-minimal) hypersurfaces that are of constant mean curvature.

Conformal vector fields and geometric solitons on the tangent bundle with the ciconia metric

Let (X2k, F, g) be an almost anti-paraHermitian manifold with an almost paracomplex structure F and a Riemannian metric g and let TX be its tangent bundle with the ciconia metric g?. The purpose of this paper is divided into two folds. The first one is to examine the curvature properties of the tangent bundle TX with the ciconia metric g?. The second one is to study conformal vector fields and almost Ricci and Yamabe solitons on the tangent bundle TX according to the ciconia metric g?.

Interaction of Codazzi Pairs with Almost Para Norden Manifolds

In this paper, we research some properties of Codazzi pairs on almost para Norden manifolds. Let $(M_{2n},\ \varphi ,\ g,G)$ be an almost para Norden manifold. Firstly, $g$-conjugate connection, $G$-conjugate connection and $\varphi $-conjugate connection of a linear connection $\mathrm{\nabla }$ on $M_{2n}$ denoted by ${\mathrm{\nabla }}^{*\
 },\ {\mathrm{\nabla }}^{\dagger \ }$ and ${\mathrm{\nabla }}^{\varphi \ }$ are defined and it is demonstrated that on the spaces of linear connections, $\left(id,\ *,\dagger ,\varphi \right)$ acts as the four-element Klein group. We also searched some properties of these three types conjugate
 connections. Then, Codazzi pairs $\left(\mathrm{\nabla },\varphi \right)\ ,\left(\mathrm{\nabla },g\right)$ and $\left(\mathrm{\nabla },G\right)$ are introduced and some properties of them are given. Let $R\ ,\ R^{*\ }$and $R^{\dagger \ }$are $(0,4)$-curvature tensors of conjugate connections
 $\mathrm{\nabla }\mathrm{\ ,\ }{\mathrm{\nabla }}^{*\ }$and ${\mathrm{\nabla }}^{\dagger \ }$, respectively. The relationship among the curvature tensors is investigated. The condition of $N_{\varphi }=0$ is obtained, where $N_{\varphi }$ is Nijenhuis tensor field on $M_{2n}$ and it is known
 that the condition of integrability of almost para complex structure $\varphi $ is $N_{\varphi }=0$. In addition, Tachibana operator is applied to the pure metric $g$ and a necessary and sufficient condition $\left(M,\varphi ,\ g,G\right)$ being a para Kahler Norden manifold is found. Finally, we examine $\varphi $-invariant linear connections and statistical manifolds.

Almost paracomplex structures on 4-manifolds

Reflection in a line in Euclidean 3-space defines an almost paracomplex structure on the space of all oriented lines, isometric with respect to the canonical neutral Kaehler metric. Beyond Euclidean 3-space, the space of oriented geodesics of any real 3-dimensional space form admits both isometric and anti-isometric paracomplex structures.This paper considers the existence or otherwise of isometric and anti-isometric almost paracomplex structures j on a pseudo-Riemannian 4-manifold (M,g), such that j is parallel with respect to the Levi-Civita connection of g. It is shown that if an isometric or anti-isometric almost paracomplex structure on a conformally flat manifold is parallel, then the scalar curvature of the metric must be zero. In addition, it is found that j is parallel iff the eigenplanes are tangent to a pair of mutually orthogonal foliations by totally geodesic surfaces.The composition of a Riemannian metric with an isometric almost paracomplex structure j yields a neutral metric g′. It is proven that if j is parallel, then g is Einstein iff g′ is conformally flat and scalar flat.The vanishing of the Hirzebruch signature is found to be a necessary topological condition for a closed 4-manifold to admit an Einstein metric with a parallel isometric paracomplex structure. Thus, while the K3 manifold admits an Einstein metric with an isometric paracomplex structure, it cannot be parallel. The same holds true for certain connected sums of complex projective 2-space and its conjugate.

Para-Ricci-Like Solitons on Riemannian Manifolds with Almost Paracontact Structure and Almost Paracomplex Structure

We introduce and study a new type of soliton with a potential Reeb vector field on Riemannian manifolds with an almost paracontact structure corresponding to an almost paracomplex structure. The special cases of para-Einstein-like, para-Sasaki-like and having a torse-forming Reeb vector field were considered. It was proved a necessary and sufficient condition for the manifold to admit a para-Ricci-like soliton, which is the structure that is para-Einstein-like. Explicit examples are provided in support of the proven statements.

Some Biharmonic Problems on the Tangent Bundle with a Berger-type Deformed Sasaki Metric

Let (M<sub>2k</sub>,Φ,g) be an almost anti-paraKahler manifold and TM its tangent bundle equipped with the Berger type deformed Sasaki metric gBS and the paracomplex structure Φ<sub>˜</sub>. In this paper, we deal with the biharmonicity of canonical projection π : TM →M and a vector field X which is considered as a map X : M → TM.

Infinitesimal Paraholomorphically Projective Transformation On Cotangent Bundle With Riemannian Extension

The main purpose of the present paper is to study some properties of infinitesimal paraholomorphically projective transformation on 𝑇∗𝑀 with respect to the Levi-Civita connection of the Riemannian extension ( 𝑅∇) and adapted almost paracomplex structure 𝐽. Moreover, if 𝑇∗𝑀 be admits a non-affine infinitesimal paraholomorphically projective transformation, than 𝑀 and 𝑇∗𝑀 are locally flat.

Invariant Tensors under the Twin Interchange of the Pairs of the Associated Metrics on Almost Paracomplex Pseudo-Riemannian Manifolds

The object of study is almost paracomplex pseudo-Riemannian manifolds with a pair of metrics associated each other by the almost paracomplex structure. A torsion-free connection and tensors with geometric interpretation are found which are invariant under the twin interchange, i.e. the swap of the counterparts of the pair of associated metrics and the corresponding Levi-Civita connections. A Lie group depending on two real parameters is constructed as an example of a 4-dimensional manifold of the studied type and the mentioned invariant objects are found in an explicit form.

The Bundle of Paracomplex Structures

Under consideration are the bundle of paracomplex structures and the related problems of the existence of a paracomplex structure on a manifold. We obtain some explicit descriptions for the bundle of paracomplex structures for spheres of dimensions 2, 4, and 6. The existence is proved of a nonitegrable almost paracomplex structure on the six-dimensional sphere.

On widetilde{J}-tangent Affine Hyperspheres

In this paper we study widetilde{J}-tangent affine hyperspheres, where widetilde{J} is the canonical para-complex structure on mathbb {R}^{2n+2}. The main purpose of this paper is to give a classification of widetilde{J}-tangent affine hyperspheres of an arbitrary dimension with an involutive distribution mathcal {D}. In particular, we classify all such hyperspheres in the 3-dimensional case. We also show that there is a direct relation between widetilde{J}-tangent affine hyperspheres and Calabi products. As an application we obtain certain classification results. In particular, we show that, with one exception, all odd dimensional proper flat affine hyperspheres are, after a suitable affine transformation, widetilde{J}-tangent. Some examples of widetilde{J}-tangent affine hyperspheres are also given.

Para-Hermitian Geometry and Doubled Aspects of Vaisman Algebroid

The geometry of double field theory (DFT) is described by a para-Hermitian manifold M. A tangent bundle of the para-Hermitian manifold T M is decomposed into two eigenbundles L and ∼L associated with the eigenvalues of the para-complex structure K. We define a Lie algebroid structure on the eigenbundles L, ∼L. The gauge symmetry algebra of DFT is governed by the C-bracket. The algebraic structure based on the C-bracket is not a Courant algebroid, but a metric algebroid proposed by Vaisman (the Vaisman algebroid). We show that the Vaisman algebroid in DFT is naturally defined on T M by an analogue of the Drinfel’d double of L, ∼L. We also find that an algebraic origin of the strong constraint is the condition for (L, ∼L) to become a Lie bialgebroid.

Nijenhuis operators on pre-Lie algebras

First we use a new approach to define a graded Lie algebra whose Maurer–Cartan elements characterize pre-Lie algebra structures. Then using this graded Lie bracket, we define the notion of a Nijenhuis operator on a pre-Lie algebra which generates a trivial deformation of this pre-Lie algebra. There are close relationships between [Formula: see text]-operators, Rota–Baxter operators and Nijenhuis operators on a pre-Lie algebra. In particular, a Nijenhuis operator “connects” two [Formula: see text]-operators on a pre-Lie algebra whose any linear combination is still an [Formula: see text]-operator in certain sense and hence compatible [Formula: see text]-dendriform algebras appear naturally as the induced algebraic structures. For the case of the dual representation of the regular representation of a pre-Lie algebra, there is a geometric interpretation by introducing the notion of a pseudo-Hessian–Nijenhuis structure which gives rise to a sequence of pseudo-Hessian and pseudo-Hessian–Nijenhuis structures. Another application of Nijenhuis operators on pre-Lie algebras in geometry is illustrated by introducing the notion of a para-complex structure on a pre-Lie algebra and then studying para-complex quadratic pre-Lie algebras and para-complex pseudo-Hessian pre-Lie algebras in detail. Finally, we give some examples of Nijenhuis operators on pre-Lie algebras.

(Para-)Holomorphic and Conjugate Connections on (Para-)Hermitian and (Para-)Kähler Manifolds

We investigate how an affine connection $$\nabla $$ that generally admits torsion interacts with both g and L on an almost (para-)Hermitian manifold $$(\mathfrak {M},g,L)$$, where L denotes either an almost complex structure J with $$J^{2}=-\hbox {id}$$ or an almost para-complex structure K with $$K^{2}=\hbox {id}$$. We show that $$\nabla $$ becomes (para-)holomorphic and L becomes integrable if and only if the pair $$ (\nabla ,L)$$ satisfies a torsion coupling condition. We investigate (para-)Hermitian manifolds $$\mathfrak {M}$$ in which this torsion coupling condition is satisfied by the following four connections (all possibly carrying torsion): $$\nabla ,\nabla ^{L},\nabla ^{*},$$ and $$\nabla ^\dagger = \nabla ^{*L}=\nabla ^{L*}$$, where $$\nabla ^{L}$$ and $$\nabla ^{*}$$ are, respectively, L-conjugate and g-conjugate transformations of $$\nabla $$. This leads to the following special cases (where T stands for torsion): (i) the case of $$T = T^*, T^L = T^\dagger $$, for which all four connections are Codazzi-coupled to g, but $$d\omega \ne 0$$, whence $$\mathfrak {M}$$ is called Codazzi-(para-)Hermitian; (ii) the case of $$T = - T^{\dagger }, T^L = - T^{*}$$, for which $$d \omega = 0$$, i.e., the manifold $$\mathfrak {M}$$ becomes (para-)Kahler. In the latter case, quadruples of (para-)holomorphic connections all with non-vanishing torsions may exist in (para-)Kahler manifolds, complementing the result of Fei and Zhang (Results Math 72:2037–2056, 2017) showing the existence of pairs of torsion-free connections, each Codazzi-coupled to both g and L, in Codazzi-(para-)Kahler manifolds.

Parallel almost paracontact structures on affine hypersurfaces

Let $\widetilde{J}$ be the canonical paracomplex structure on $\mathbb{R}^{2n+2}\simeq\widetilde{\mathbb{C}}^{n+1}$. We study real affine hypersurfaces $f\colon M\rightarrow \widetilde{\mathbb{C}}^{n+1}$ with a $\widetilde{J}$-tangent transversal vector f

(Para-) Kähler Structures on $$\rho $$ ρ -Commutative Algebras

Recently, geometrics objects such as $$\rho $$ -forms, metrics, Levi–Civita connection, differential calculus were introduced on $$\rho $$ -commutative algebras. In this paper, we intend to extend another geometric objects such as almost complex structure, almost para-complex structure, Hermitian metric, para-Hermitian metric, Kahler and para-Kahler forms, on it and introduce new concepts which we will call Kahler $$\rho $$ -commutative algebras and para-Kahler $$\rho $$ -commutative algebras.

On 3 -dimensional e J -tangent centro-affine hypersurfaces and e J -tangent affine hyperspheres with some null-directions

Let $\widetilde{J}$ be the canonical para-complex structure on $\mathbb{R}^4$. In this paper we study $3$-dimensional centro-affine hypersurfaces with a $\widetilde{J}$-tangent centro-affine vector field (sometimes called $\widetilde{J}$-tangent centro-affine hypersurfaces) as well as $3$-dimensional $\widetilde{J}$-tangent affine hyperspheres with the property that at least one null-direction of the second fundamental form coincides with either $\mathcal{D}^+$ or $\mathcal{D}^-$. The main purpose of this paper is to give a full local classification of the above mentioned hypersurfaces. In particular, we prove that every nondegenerate centro-affine hypersurface of dimension $3$ with a $\widetilde{J}$-tangent centro-affine vector field which has two null-directions $\mathcal{D}^+$ and $\mathcal{D}^-$ must be both an affine hypersphere and a hyperquadric. Some examples of these hypersurfaces are also given.

Symplectic, product and complex structures on 3-Lie algebras

In this paper, first we introduce the notion of a phase space of a 3-Lie algebra and show that a 3-Lie algebra has a phase space if and only if it is sub-adjacent to a 3-pre-Lie algebra. Then we introduce the notion of a product structure on a 3-Lie algebra using the Nijenhuis condition as the integrability condition. A 3-Lie algebra enjoys a product structure if and only if it is the direct sum (as vector spaces) of two subalgebras. We find that there are four types special integrability conditions, and each of them gives rise to a special decomposition of the original 3-Lie algebra. They are also related to O-operators, Rota–Baxter operators and matched pairs of 3-Lie algebras. Parallelly, we introduce the notion of a complex structure on a 3-Lie algebra and there are also four types special integrability conditions. Finally, we add compatibility conditions between a complex structure and a product structure, between a symplectic structure and a paracomplex structure, between a symplectic structure and a complex structure, to introduce the notions of a complex product structure, a para-Kähler structure and a pseudo-Kähler structure on a 3-Lie algebra. We use 3-pre-Lie algebras to construct these structures. Furthermore, a Levi-Civita product is introduced associated to a pseudo-Riemannian 3-Lie algebra and deeply studied.

LEFT-INVARIANT ALMOST PARA-COMPLEX EINSTEINIAN STRUCTURES ON SIX-DIMENSIONAL NILPOTENT LIE GROUPS

As is well known, there are 34 classes of isomorphic simply connected six-dimensional nilpotent Lie groups. Of these, only 26 classes admit left-invariant symplectic structures and only 18 admit left-invariant complex structures. There are five six-dimensional nilpotent Lie groups G , which do not admit neither symplectic, nor complex structures and, therefore, can be neither almost pseudo- Kӓhlerian, nor almost Hermitian. In this work, these Lie groups are being studied. The aim of the paper is to define new left-invariant geometric structures on the Lie groups under consideration that compensate, in some sense, the absence of symplectic and complex structures. Weakening the closedness requirement of left-invariant 2-forms ω on the Lie groups, non-degenerated 2-forms ω are obtained, whose exterior differential dω is also non-degenerated in Hitchin sense [6]. Therefore, the Hitchin’s operator K dω is defined for the 3-form dω . It is shown that K dω defines an almost complex or almost para-complex structure for G and the couple ( ω, dω ) defines pseudo-Riemannian metrics of signature (2,4) or (3,3), which is Einsteinian for 4 out of 5 considered Lie groups. It gives new examples of multiparametric families of Einstein metrics of signature (3,3) and almost para-complex structures on six-dimensional nilmanifolds, whose structural group is being reduced to SL (3 , R) SO (3 , 3). On each of the Lie groups under consideration, compatible pairs of left-invariant forms (ω, Ω), where Ω = d ω, are obtained. For them the defining properties of half-flat structures are naturally fulfilled: d Ω = 0 and ωΩ = 0. Therefore, the obtained structures are not only almost Einsteinian para-complex, but also pseudo- Riemannian half-flat.

Interaction of Codazzi Couplings with (Para-)Kähler Geometry

We study Codazzi couplings of an affine connection $$\nabla $$ with a pseudo-Riemannian metric g, a nondegenerate 2-form $$\omega $$ , and a tangent bundle isomorphism L on smooth manifolds, as an extension of their parallelism under $$\nabla $$ . In the case that L is an almost complex or an almost para-complex structure and $$(g, \omega , L)$$ form a compatible triple, we show that Codazzi coupling of a torsion-free $$\nabla $$ with any two of the three leads to its coupling with the remainder, which further gives rise to a (para-)Kahler structure on the manifold. This is what we call a Codazzi-(para-)Kahler structure; it is a natural generalization of special (para-)Kahler geometry, without requiring $$\nabla $$ to be flat. In addition, we also prove a general result that g-conjugate, $$\omega $$ -conjugate, and L-gauge transformations of $$\nabla $$ , along with identity, form an involutive Abelian group. Hence a Codazzi-(para-)Kahler manifold admits a pair of torsion-free connections compatible with the $$(g, \omega , L)$$ . Our results imply that any statistical manifold may admit a (para-)Kahler structure as long as one can find an L that is compatible to g and Codazzi coupled with $$\nabla $$ .