Sirin Aktay
We investigate 7 dimensional almost para-contact metric structures induced by the 3-forms of $G_2^*$ Structures. We calculate the projections that determine to which class the almost para-contact structure belongs, by using the properties of $G_2^*$ Structures.
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Almost Paracontact Manifolds
本文引入了11类基本的几乎仿切流形，并构造了一些例子。
Generalized almost paracontact structures
该研究在广义切丛上引入广义近仿切触结构并探究其性质，给出相关场变换下保持该结构的条件，还定义其正规性并刻画了由近仿切触结构诱导的正规情形。
Almost Paracontact Finsler Structures on Vector Bundle
本文定义了向量丛上的几乎仿接触和正规几乎仿接触Finsler结构，并得到了这些结构可积的一些条件。本文定义了仿接触度量、仿Sasakian和K-仿接触Finsler结构，并研究了它们的一些性质。对于一个K-仿接触Finsler结构，我们发现垂直和水平的标志曲率。那么，我们定义垂直？标记曲率，并证明了每个局部对称的para-Sasakian Finsler结构都有负的垂直曲率。标志曲率。最后，我们定义了仿Sasakian Finsler结构的水平和垂直Ricci张量，并研究了它们的曲率性质。
Almost paracontact almost paracomplex Riemannian manifolds of the lowest dimension 3 are considered. Such structures are constructed on a family of Lie groups and the obtained manifolds are studied. Curvature properties of these manifolds are investigated. An example is commented as support of obtained results.
Almost paracontact almost paracomplex Riemannian manifolds of the lowest dimension are studied. Such structures are constructed on hyperspheres in 4-dimensional spaces, Euclidean and pseudo-Euclidean, respectively. The obtained manifolds are studied and characterised in terms of the classification used and their geometric properties.
Para-CR Structures on almost Paracontact Metric Manifolds
几乎仿接触度量流形是几乎仿CR流形的著名例子。我们找到了这类流形是para-CR的充要条件。接下来，我们研究这些条件在某些子类几乎仿接触度量流形。特别地，证明了正规几乎仿接触度量流形是仿CR的。我们建立了仿接触度量流形和几乎仿共辛流形是仿CR的充要条件。我们还发现基本曲率恒等式的para-CR paracontact度量流形和研究其后果。其中，我们证明了任何具有常截面曲率且维数大于树的仿CR仿接触度量流形必是仿Sasakian的且其曲率等于负1。最后一个断言在维度树中不成立。此外，我们还证明了共形平坦的仿Sasakian流形具有等于负一的常截面曲率。建立了一类新的仿CR流形的例子。
Lie groups considered as three-dimensional almost paracontact almost paracomplex Riemannian manifolds are investigated. In each basic class of the classification used for the manifolds under consideration, a correspondence is established between the Lie algebra and the explicit matrix representation of its Lie group.
We study real affine hypersurfaces $f\colon M\rightarrow \mathbb{R}^{2n+2}$ with an almost paracontact structure $(φ,ξ,η)$ induced by a $\widetilde{J}$-tangent transversal vector filed, where $\widetilde{J}$ is the canonical paracomplex structure on $\mathbb{R}^{2n+2}$. We give a classification of hypersurfaces for which an induced almost paracontact structure is metric relative to the second fundamental form. Some other properties of such hypersurfaces are also studied.
Almost paracontact manifolds of an odd dimension having an almost paracomplex structure on the paracontact distribution are studied. The components of the fundamental (0,3)-tensor, derived by the covariant derivative of the structure endomorphism and the metric on the considered manifolds in each of the basic classes, are obtained. Then, the case of the lowest dimension 3 of these manifolds is considered. An associated tensor of the Nijenhuis tensor is introduced and the studied manifolds are characterized with respect to this pair of tensors. Moreover, cases of paracontact and para-Sasakian t…
This paper is a continuation of our previous work, where eleven basic classes of almost paracontact metric manifolds with respect to the covariant derivative of the structure tensor field were obtained. First we decompose one of the eleven classes into two classes and the basic classes of the considered manifolds become twelve. Also, we determine the classes of $α$-para-Sasakian, $α$-para-Kenmotsu, normal, paracontact metric, para-Sasakian, K-paracontact and quasi-para-Sasakian manifolds. Moreover, we study 3-dimensional almost paracontact metric manifolds and show that they belong to four bas…
Slant curves in 3-dimensional normal almost paracontact metric manifolds
研究了三维正规几乎仿接触度量流形中斜Frenet曲线的曲率和挠率。此外，在这类流形中，还得到了非Frenet斜曲线（具有零切线或零法线）的一些性质.通过实例说明了所取得的成果。
Slant and Semi-Slant Submanifolds of a Lorentzian Almost Paracontact Manifold
本文定义并研究了洛伦兹几乎仿接触流形的斜子流形和半斜子流形。得到了Lorentz几乎仿切流形的子流形倾斜的充要条件。给出了洛伦兹几乎仿接触流形的斜子流形和半斜子流形的一些例子。我们还研究了当周围流形分别是Lorentzian仿共辛流形和Lorentzian para-Sasakian流形时，定义半斜子流形所涉及的分布的可积性条件.
Lifts of Lorentzian r- Paracontact Structure: A Geometrical Dynamics Meaning
本文的目的是利用提升理论在几乎Lorentz r-仿接触流形的切丛上构造几乎仿复结构，并赋予几乎Lorentz r-仿接触流形结构.最后，我们讨论了所产生的几何结构对动力学系统的影响。
There are introduced and studied a pair of associated Schouten-van Kampen affine connections adapted to the paracontact distribution and an almost paracontact almost paracomplex Riemannian structure generated by the pair of associated metrics and their Levi-Civita connections. By means of the constructed non-symmetric connections, the basic classes of the manifolds with the considered structure are characterized. Curvature properties of the studied connections are obtained. A family of examples on a Lie group is constructed.
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. Sekizawa and O. Kowalski) of the Riemann extension, introduced by E. K. Patterson and A. G. Walker (1952). We construct two almost para-Hermitian structures on $(T^*M,\overline g)$ which are almost para-Kähler or para-Kähler and prove that the defined almost para-complex structures are …
Almost paracontact almost paracomplex Riemannian manifolds as extensions of 2-dimensional space-forms
本文研究最低维几乎para接触黎曼流形，其构造为实线与二维空间形式的乘积，度量由锥度量或双曲扩张得到，并依据分类与曲率性质加以刻画。
Bi-paracontact structures and Legendre foliations
研究了切触流形上的几乎双仿切触结构。证明了若在切触流形$(M,η)$上定义几乎双仿接触结构，则在某些自然的可积性假设下，$M$具有两个横截双勒让德结构.相反，如果在接触流形上定义两个横截双勒让德结构，则$M$允许几乎双仿接触结构。我们定义了一个几乎双仿接触流形上的典型连接，我们研究了它的曲率性质，这类似于反超复（或复积）流形的Obata连接。进一步，我们证明了任何Reeb向量场属于$(κ,μ)$ -零分布的切触度量流形正则地具有几乎双仿切结构，并将以前的结果应用到切触度量$(κ,μ)$ -空间理论中.
For almost contact metric or almost paracontact metric manifolds there is natural notion of $η$-normality. Manifold is called $η$-normal if is normal along kernel distribution of characteristic form. In the paper it is proved that $η$-normal manifolds are in one-one correspondence with Cauchy-Riemann almost contact metric manifolds or para Cauchy-Riemann in case of almost paracontact metric manifolds. There is provided characterization of $η$-normal manifolds in terms of Levi-Civita covariant derivative of structure tensor. It is established existence a Tanaka-like connection on $η$-normal man…
$η$-Ricci solitons in $(\varepsilon)$-almost paracontact metric manifolds
该文针对$(\u03b5)$-殆仿切触度量流形，研究了两类势向量场情形下的$\u03b7$-Ricci孤立子，分析了相关特殊流形的孤立子容许性，并结合平行对称(0,2)张量场得到了若干研究结果。
Let $\widetilde{J}$ be the canonical para-complex 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 field. Such vector field induces in a natural way an almost paracontact structure ${(φ,ξ,η)}$ on $M$ as well as the affine connection $\nabla$. In this paper we give the classification of hypersurfaces with the property that $φ$ or $η$ is parallel relative to the connection $\nabla$. Moreover, we show that if $\nablaφ=0$ (respectively $\nablaη…
Almost paracontact metric 3-dimensional Walker manifolds
本文在3维Walker流形上构造几乎para切触度量结构，给出其属于各类别（如para切触、正规、para余辛等）的充要条件，并研究η-Einstein流形的曲率性质。
Lifts of Almost r-Contact and r-Paracontact Structures
本文在几乎Lorentz r-切触和r-仿切触流形的基础上，考虑提升理论，得到了切丛上的几乎复和仿复结构.
H-almost Ricci-Yamabe solitons in paracontact geometry
本文在仿切触几何框架下分类了 h-几乎 Ricci-Yamabe 孤立子，系统刻画了 para-Kenmotsu、para-Sasakian 和 para-cosymplectic 流形上的相关结构，并给出两个实例加以说明。
Geometry of δ-almost gradient Yamabe solitons on pseudo-Riemannian manifolds
本文研究伪黎曼流形中仿切触度量流形上的δ-几乎Yamabe孤立子，证明势向量场的Killing性、与ξ的平行关系及数量曲率条件，并构造了两个K-仿切触度量流形实例。
The purpose of the paper is to study Yamabe solitons on three-dimensional para-Sasakian, paracosymplectic and para-Kenmotsu manifolds. Mainly, we proved that *If the semi-Riemannian metric of a three-dimensional para-Sasakian manifold is a Yamabe soliton, then it is of constant scalar curvature, and the flow vector field V is Killing. In the next step, we proved that either manifold has constant curvature -1 and reduces to an Einstein manifold, or V is an infinitesimal automorphism of the paracontact metric structure on the manifold. *If the semi-Riemannian metric of a three-dimensional paraco…
On basic curvature identities for almost (para)contact metric manifolds
在D. E. Blair“Riemann几何的接触和辛流形“和S.在Zamkoek的“仿接触流形上的典范联络”中，证明了切触度量流形和仿接触度量流形的曲率恒等式。在更广的流形类中得到了曲率恒等式，推广了上述文献中的结果。此外，我们还研究了R+与几乎（仿）切触度量流形的一种特殊半积上的几乎（仿）厄米特结构的一些性质。这个半积在证明主要定理时起辅助作用。
It is provided an overview of existed results concerning classification of contact metric, almost cosymplectic and almost Kenmotsu $(κ,μ)$-manifolds. In the case of dimension three it is described in full details structure of contact metric or almost cosymplectic $(κ,μ)$-spaces. The second part of the paper addresses three-dimensional paracontact metric and almost paracosymplectic $(κ,μ)$-spaces. There is obtained local classification of paracontact metric $(κ,μ)$-spaces, and almost paracosymplectic $(κ,μ)$-spaces, for every possible value of $κ$.
Indefinite almost paracontact metric manifolds
本文引入了$(\varepsilon)$ -几乎仿切流形的概念，特别是$(\varepsilon)$ -仿Sasakian流形的概念.提出了几个例子。得到了$(\varepsilon)$ -paraSasakian流形的曲率张量和Ricci张量的一些典型恒等式.证明了如果半黎曼流形是平坦的、真循环的或真Ricci-循环的，则它不允许有$(\varepsilon)$ -paraSasakian结构。证明了对于$(\varepsilon)$ -paraSasakian流形，对称、半对称和常截面曲率的条件都是相同的.它表明，一个对称的类空（分别。类时）$(\varepsilon)$ -paraSasakian流形$M^{n}$局部等距于伪双曲空间$H_ν^{n}(1)$（分别为.伪球$S_ν^{n}(1)$）。最后证明了对于$(\varepsilon)$ -paraSasakian流形，Ricci-半对称、Ricci-对称和Einstein流形的条件是完全相同的。
Hypersurfaces of Lorentzian para-Sasakian manifolds
本文分别研究了（1，1，1）几乎切触流形、Lorentz几乎仿切触流形和Lorentz仿Sasakian流形的不变超曲面和非不变超曲面。证明了（1，1，1）殆切触流形的非不变超曲面具有殆积结构.研究仿射余辛流形和正规（1，1，1）几乎切触流形的超曲面.证明了洛伦兹几乎仿接触流形的非不变超曲面是几乎乘积度量流形。给出了Lorentzian para-Sasakian流形的非不变超曲面是局部积流形的若干充要条件。我们建立了Lorentzian para-Sasakian流形的不变超曲面的Lorentzian para-Sasakian结构。最后给出了Lorentzian para-Sasakian流形的不变和非不变超曲面的一些例子。
The Schouten-van Kampen affine connection adapted to an almost (para) contact metric structure
本文研究了几乎接触或仿接触度量结构的Schouten-van Kampen联络。利用这种联系，刻画了几类几乎（仿）切触度量流形。这种连接的某些曲率特性被发现。
Object of study are para-Ricci-like solitons on para-Sasaki-like almost paracontact almost paracomplex Riemannian manifolds, briefly, Riemannian $Π$-manifolds. Different cases when the potential of the soliton is the Reeb vector field or pointwise collinear to it are considered. Some additional geometric properties of the constructed objects are proven. Results for a parallel symmetric second-order covariant tensor on the considered manifolds are obtained. Explicit example of dimension 5 in support of the given assertions is provided.
Slant submanifolds of lorentzian almost contact manifolds
本文研究了Lorentz几乎切触流形的斜子流形。我们把子流形看成是一个类空，然后在子流形上定义了倾斜角，从而推广了A。Lotta（切触几何中的斜子流形[8]）和M. A. Khan et.（Lorentzian Paracontact流形的斜子流形[7]）。
Tensor sphere bundle of Cheeger-Gromoll type
在具有Cheeger-Gromoll型度量的黎曼流形的张量丛上构造了一个度量框架结构，并将其限制在张量球面丛上，得到了张量球面丛上的几乎度量仿接触结构.此外，我们还证明了赋予诱导度量的张量球面丛决不是空间型。
Geometric Characterizations of δ-Almost Yam- abe Solitons with QSNM Connections
本文研究了带四分之一对称非度量联络∇的仿切触度量流形上δ-几乎Yamabe孤子的几何结构，通过特定假设建立分类结果，推导其膨胀/稳态/收缩条件并给出示例验证。
Connections with (skew-symmetric) torsion on non-symmetric Riemannian manifold satisfying the Einstein metricity condition (NGT with torsion) are considered. It is shown that an almost Hermitian manifold is an NGT with torsion if and only if it is a Nearly Kähler manifold. In the case of an almost contact metric manifold the NGT with torsion spaces are characterized and a possibly new class of almost contact metric manifolds is extracted. Similar considerations lead to a definition of a particular classes of almost para-Hermitian and almost paracontact metric manifolds. The conditions are give…
Homogeneous pseudo-Riemannian structures of metrics of Kaluza-Klein type on the three-dimensional anti-de Sitter spacetime
本文研究了三维反德西特时空上的卡鲁扎-克莱因型度量的同构伪黎曼结构及其等距群和约当分解。
Geometry of a weak para-$f$-structure
该研究聚焦弱殆仿f-结构及其子类的几何，通过新张量推导f的协变导数，证明弱仿𝒦-流形的相关叶状结构性质，验证仿𝒮-结构刚性与特定结构的约化关系，并得到弱殆仿切触结构的推论。
Inspired by the concepts of slant distribution and slant submanifold, with their variants of hemi-slant, semi-slant, bi-slant, or almost bi-slant, we introduce the more general concepts of $k$-slant distribution and $k$-slant submanifold in the settings of an almost Hermitian, an almost product Riemannian, an almost contact metric, and an almost paracontact metric manifold and study some of their properties. We prove that, for any proper $k$-slant distribution in the tangent bundle of a Riemannian manifold, there exists another one in its orthogonal complement, and we establish basic relations…
Geometric Realizability of Covariant Derivative Kähler Tensors for almost Pseudo-Hermitian and almost Para-Hermitian Manifolds
几乎伪埃尔米特流形或几乎仿埃尔米特流形的Kähler形式的共变导数满足一定的代数关系。相反，我们证明了满足这些代数关系的任何3-张量都可以几何地实现。
Relations between conjugate connections with respect to the pair of Norden metrics and to the almost complex structure on almost Norden manifolds are studied. Conjugate connections of the Levi-Civita connections induced by the Norden metrics are obtained. Statistical structures on almost Norden manifolds are considered.
Almost Lie structures on an anchored Banach bundle
在适当的假设下，推广了Banach锚丛框架下的线性几乎Poisson结构、几乎李代数胚、几乎微分的概念以及它们之间的关系.然后，我们得到了一个适应的形式主义的机械系统，这是说明了进化问题的“希尔伯特蛇”
We study the local commutation relation between the Lefschetz operator and the exterior differential on an almost complex manifold with a compatible metric. The identity that we obtain generalizes the backbone of the local Kähler identities to the setting of almost Hermitian manifolds, allowing for new global results for such manifolds.
Almost Hermitian Structures and Quaternionic Geometries
Gray和Hervella把几乎厄米特结构（g，I）分为16类。我们系统地研究了这些类之间的相互作用时，一个几乎hyper-Hermitian结构（g，I，J，K）。在一般维度中，我们发现至多167种不同的几乎超厄米结构。特别是，我们得到了一些关系，使hyperKäher或局部共形hyperKähler结构，从而推广了希钦的结果。我们还研究了几乎四元数厄米几何的类型，产生和列表的结果。
The object of investigation are the almost contact manifolds with B-metric in the lowest dimension three, constructed on Lie algebras. It is considered a relation between the classes in the Bianchi classification of three-dimensional real Lie algebras and the classes of a classification of the considered manifolds. There are studied some geometrical characteristics in some special classes.
Some lightlike submanifolds of almost complex manifolds with Norden metric
本文研究了具有Norden度量的殆复流形中关于一个Norden度量非退化而关于另一个Norden度量类光的子流形.给出了某些子流形的诱导几何对象之间的关系。所考虑的子流形的例子构造。
A 4-dimensional Riemannian manifold equipped with a circulant structure, which is an isometry with respect to the metric and its fourth power is the identity, is considered. The almost product manifold associated with the considered manifold is studied. The relation between the covariant derivatives of the almost product structure and the circulant structure is obtained. The conditions for the covariant derivative of the circulant structure, which imply that an almost product manifold belongs to each of the basic classes of the Staikova-Gribachev classification, are given.
On some cohomological properties of almost complex manifolds
本文研究一类特殊的殆复结构，称为纯复结构和全复结构，这是由李铁江和W. Zhang，在辛结构和Hard Lefschetz条件的基础上.利用离散子群给出了李群的紧代数上存在上述类型的殆复结构的充分条件。我们得到了紧诣零流形和解流形上的纯的和完全的殆复结构。这些家庭中的一些参数化的实2-形式是反不变的殆复结构。
Object of investigation are almost hypercomplex manifolds with Hermitian-Norden metrics of the lowest dimension. The considered manifolds are constructed on 4-dimensional Lie groups. It is established a relation between the classes of a classification of 4-dimensional indecomposable real Lie algebras and the classification of the manifolds under study. The basic geometrical characteristics of the constructed manifolds are studied in the frame of the mentioned classification of the Lie algebras.
Radical transversal lightlike hypersurfaces of almost complex manifolds with Norden metric
本文引入了具有Norden度量的殆复流形的根横截类光超曲面。这些超曲面的研究是出于这样一个事实，即不确定的几乎埃尔米特流形这类类光超曲面不存在。我们还证明了具有Norden度量的殆复流形的根横截类光超曲面具有唯一的屏蔽分布和具有Norden度量的Kaehler流形的对称Ricci张量的性质.本文得到了关于类光超曲面与具有Norden度量的殆复流形的非退化超曲面的关系的新结果。所考虑的超曲面的例子。
In this paper the notion of quasi-isometry between two Riemannian manifolds has been introduced. This idea is also imposed to study quasi-isometry between two almost contact metric manifolds. Moving further, some curvature properties of two quasi-isometrically embedded almost contact metric manifolds, $N(k)-$contact metric manifolds and Sasakian manifolds are investigated. Next, an illustrative example of a quasi-isometry between two Sasakian structures is constructed. Finally, a relation between the scalar curvature and the quasi-isometric constants for two quasi-isometric Riemannian manifold…
We study left-invariant generalized Kähler structures on almost abelian Lie groups, i.e., on solvable Lie groups with a codimension-one abelian normal subgroup. In particular, we classify six-dimensional almost abelian Lie groups which admit a left-invariant complex structure and establish which of those have a left-invariant Hermitian structure whose fundamental 2-form is $\partial \bar \partial$-closed. We obtain a classification of six-dimensional generalized Kähler almost abelian Lie groups and determine the 6-dimensional compact almost abelian solvmanifolds admitting an invariant generali…
An almost Golden Riemannian structure $(φ,g)$ on a manifold is given by a tensor field $φ$ of type (1,1) satisfying the Golden section relation $φ^{2}=φ+1$, and a pure Riemannian metric $g$, i.e., a metric satisfying $g(φX,Y)=g(X,φY)$. We study connections adapted to such a structure, finding two of them, the first canonical and the well adapted, which measure the integrability of $φ$ and the integrability of the $G$-structure corresponding to $(φ,g)$.
Our aim is to define and study a structure for some $(4n+3)$-dimensional manifolds which is named almost coquaternion structure. This structure is composed of three almost cocomplex structures $(ϕ_a, ξ_a, η_a)$, $a = 1,2,3$, which satisfy some relations and may be considered as analogous to the almost quaternion structure for $(4n+4)$-dimensional manifolds. The sphere $S^{4n+3}$ is a typical example of differentiable manifold which admits an almost coquaternion structure $(ϕ_a, ξ_a, η_a)$, $a = 1,2,3$. Using the 1-forms $η_a$ of the almost coquaternion structure of the sphere $S^{4n+3}$, C. Te…
The Relation Between the Associate Almost Complex Structure to $HM'$ and $(HM',S,T)$-Cartan Connections
本文研究了伪Finsler流形上的$(HM',S,T)$ -Cartan联络。Bejancu和H.R. Farran，都是由自然的殆复结构所产生的非线性连接$HM'$。证明了与$(HM',S,T)$ -Cartan联络相关联的自然几乎复线性联络是关于Sasaki度量$G$的度量线性联络.最后给出了$(M', J, G)$是Kähler流形的一些条件。
This paper explores the relation between the structure of fibre bundles akin to those associated to a closed almost nonnegatively sectionally curved manifold and rational homotopy theory.
We study the relations between the projective and the almost conformally symplectic structures on a smooth even dimensional manifold. We describe these relations by a single almost conformally symplectic connection with totally trace--free torsion sharing the geodesics (up to parametrization) with the projective class. This connection generalizes a (conformally) Fedosov structure depending on the remaining torsion of this distinguished connection. In fact, we see these structures as the almost symplectic analogy of the conformal Riemannian structures, because there is an analogy of the class o…
Two-Dimensional Almost-Riemannian Structures with Tangency Points
二维准黎曼结构是曲面上的广义黎曼结构，其局部正交标架由可以共线的向量场的李括号生成对给出。研究了紧致定向曲面上几乎黎曼结构的拓扑不变量与定义该结构的曲面上的秩二向量丛之间的关系。我们分析的一般情况下，包括存在的切点，即点的分布和他们的李括号的两个发电机是线性相关的。本文的主要结果给出了紧定向曲面上定向准黎曼结构的一个分类，并给出了该结构所对应的向量丛的欧拉数.此外，我们提出了一个高斯？具有切点的几乎黎曼结构的Bonnet公式。
We analyze the differential relation corresponding to integrability of almost complex structures, reformulated as a directed immersion relation by Demailly and Gaussier. Combining results of Clemente [3], we show that applying h-principle techniques yields the following statement: for an almost complex manifold with arbitrary metric $(X, J, g)$, and for $ε> 0$, there exists a smooth function $f : X \rightarrow \mathbb{R}$ and almost complex structure $J'$ on $X$ such that $J$ and $J'$ are $C^0$-close on the graph of $f$ with respect to the extended metric on $X \times \mathbb{R}$, and such tha…
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