Aysel Turgut Vanli, Inan Unal
In this paper, we study on semi-invariant submanifolds of normal complex contact metric manifolds. We give the definition of such submanifolds and we obtain useful relations. Moreover, we give the integrability conditions of distributions.
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We introduce a class of half lightlike submanifolds of almost contact B-metric manifolds and prove that such submanifolds are semi-Riemannian with respect to the associated B-metric. Object of investigations are also minimal of the considered submanifolds and a non-trivial example for them is given.
First, we prove that indefinite Sasakian manifolds do not admit any screen conformal $r$-null submanifolds, tangent to the structure vector field. We, therefore, define a special class of null submanifolds, called; {\it contact screen conformal} $r$-null submanifold of indefinite Sasakian manifolds. Several characterization results, on the above class of null submanifolds, are proved. In particular, we prove that such null submanifolds exist in indefinite Sasakian space forms of constant holomorphic sectional curvatures of $-3$.
This research article introduces the concept of lightlike submanifolds of an indefinite Kenmotsu statistical manifold. Various results on geometry of contact CR and SCR-lightlike submanifolds have been developed. Some characterization theorems on the integrability of distributions and on the geodesicity of these submanifolds have been obtained. Appropriate examples have been presented.
In this paper, we obtain Chen's inequalities for submanifolds in $(κ,μ)$-contact space form with two kinds of generalized semi-symmetric non-metric connections.
Order of contact and ruled submanifolds
本研究借助初等几何测度论，推广了关于密切性与直纹子流形的Monge-Cayley-Salmon定理。
We show that $ϕ$-invariant submanifolds of metric contact pairs with orthogonal characteristic foliations make constant angles with the Reeb vector fields. Our main result is that for the normal case such submanifolds of dimension at least $2$ are all minimal. We prove that an odd-dimensional $ϕ$-invariant submanifold of a metric contact pair with orthogonal characteristic foliations inherits a contact form with an almost contact metric structure, and this induced structure is contact metric if and only if the submanifold is tangent to one Reeb vector field and orthogonal to the other one. Fur…
On complex contact manifolds and their null submanifolds
本文研究不定复切触流形中零子流形的几何，证明四元零子流形恒为全测地，并刻画了屏幕实与屏幕横截反不变子流形上的分布几何。
The present paper is a continuation of our previous work, where a class of half lightlike submanifolds of almost contact B-metric manifolds was introduced. We study curvature properties of totally and screen totally umbilical such submanifolds as well as of the corresponding semi-Riemannian submanifolds with respect to the associated B-metric.
Recently, B.-Y. Chen and O. J. Garay studied pointwise slant submanifolds of almost Hermitian manifolds. By using the notion of pointwise slant submanifolds, we investigate the geometry of pointwise semi-slant submanifolds and their warped products in Sasakian and cosymplectic manifolds. We prove that there exist no proper pointwise semi-slant warped product submanifold other than contact CR-warped products in Sasakian manifolds. We give non-trivial examples of such submanifolds in cosypmlectic manifolds and obtain several fundamental results, including a characterization for warped product po…
Submanifolds with harmonic mean curvature vector field in contact 3-manifolds
研究了三维切触流形中的双调和或多调和曲线曲面。
B.-Y. Chen initiated the study of warped product submanifolds in his fundamental seminal papers \cite{C1,C2,C2.1}. In this paper, we study contact CR-warped product submanifolds of cosymplectic space forms and prove an optimal inequality by using Gauss and Codazzi equations. In addition, we obtain two geometric inequalities for contact CR-warped product submanifolds with a compact invariant factor.
In this paper, we present the invalidities of the results in [3], because of their definition of a semi-invariant submanifold of an almost complex contact metric manifold is not true .
A Note on Doubly Warped Product Contact CR-Submanifolds in trans-Sasakian Manifolds
直到2001年，在B.Y.尘紧接着，另一条研究路线，类似于关于佐佐木几何的奇维版本的凯勒几何，被开发出来，即翘曲的产品接触CR-子流形在佐木流形。本文证明了trans-Sasakian流形中不存在真的双曲积切触CR-子流形。
Some submanifolds of almost contact manifolds with Norden metric
本文研究了具有余维为2的Norden度量的几乎切触流形与全实正规空间的子流形。构造了李群等子流形的例子。
As a generalization of slant submanifolds and semi-slant submanifolds, we introduce the notions of pointwise slant submanifolds and pointwise semi-slant sunmanifolds of an almost contact metric manifold. We obtain a characterization at each notion, investigate the topological properties of pointwise slant submanifolds, and give some examples of them. We also consider some distributions on cosymplectic, Sasakian, Kenmotsu manifolds and deal with some properties of warped product pointwise semi-slant submanifolds. Finally, we give some inequalities for the squared norm of the second fundamental …
The present paper deals with the study of pseudo parallel (in the sense of Chaki and in the sense of Deszcz) contact CR-submanifolds with respect to Levi-Civita connection as well as semisymmetric metric connection of Kenmotsu manifolds and prove that these corresponding two classes are equivalent with a certain condition.
In this paper, we study warped products of contact skew-CR submanifolds, called contact skew CR-warped products. We establish an inequality for the squared norm of the second fundamental form in terms of the warping function and the slant angle. The equality case in the statement of the inequality is investigated and some applications of derived inequality are given. Furthermore, we provide a non-trivial example of such submanifolds.
We introduce higher order mean curvatures of screen almost conformal (SAC) half-lightlike submanifolds of indefinite contact manifolds, admitting a semi-symmetric non-metric connection, and use them to generalize some known results of [6]. Also, we derive a new set of integration formulae via the divergence of some special vector fields tangent to this submanifold. Several examples are also included to illustrate the main concepts.
Contact Calabi-Yau manifolds and Special Legendrian submanifolds
我们考虑了$α$ -Sasakian流形上的Calabi-Yau结构的推广。研究了一类特殊的勒让德子流形的形变，并对5维流形上的不变切触Calabi-Yau结构进行了分类。最后推广到余维$r$ .
This paper has two goals; the first is to generalize results for the existence and nonexistence of warped product submanifolds of almost contact manifolds, accordingly a self-contained reference of such submanifolds is offered to save efforts of potential research. Most of the results of this paper are general and decisive enough to generalize both discovered and not discovered results. Moreover, a discrete example of contact CR-warped product submanifold in Kenmotsu manifold is constructed. For further research direction, we addressed a couple of open problems arose from the results of this p…
We characterize biharmonic anti-invariant surfaces in $3$-dimensional generalized $(κ, μ)$-manifolds with non-zero constant mean curvature by means of the scalar curvature of the ambient space and the mean curvature. In addition, we give a method for constructing infinity many examples of biharmonic submanifolds in a certain $3$-dimensional generalized $(κ, μ)$-manifold. Moreover, we determine $3$-dimensional generalized $(κ, μ)$-manifolds which admit a certain kind of proper biharmonic foliation.
Submanifolds of symplectic manifolds with contact border
构造了具有切触边界的辛流形的辛子流形。这类子流形的边界被证明是接触边界的接触子流形。我们还利用“相对Lefschetz超平面定理”给出了所构造子流形的拓扑特征。我们勾勒出一些应用的结果。
We study invariant submanifolds of manifolds endowed with a normal or complex metric contact pair with decomposable endomorphism field $ϕ$. For the normal case, we prove that a $ϕ$-invariant submanifold tangent to a Reeb vector field and orthogonal to the other one is minimal. For a $ϕ$-invariant submanifold $N$ everywhere transverse to both the Reeb vector fields but not orthogonal to them, we prove that it is minimal if and only if the angle between the tangential component $ξ$ (with respect to $N$) of a Reeb vector field and the Reeb vector field itself is constant along the integral curves…
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]）。
On Contact CR-Product of Sasakian statistical manifold
本文研究Sasakian统计流形中切触CR-子流形的几何性质，刻画其分布可积性，建立关于对偶联络的全测地与脐性结果，并引入切触CR-积的统计版本。
Pre-Lagrangian Submanifolds in Contact Manifolds
本文利用$J-$全纯曲线和Gromov的非线性Fredholm替代，证明了在任何闭切触流形中不存在任何闭Pre-拉格朗日子流形.
Causal Invariance of the Contact Number of Pseudo-Riemannian Submanifolds
该研究证明伪黎曼子流形的普通、类空、类时接触数等价且重合，构造了接触数均为6的洛伦兹曲面，将无穷接触极限纳入半黎曼理论。
In this paper, homology of a contact CR-submanifold of a real hypersurface, which has naturally almost contact metric structure induced from the complex Euclidean space $\mathbb{C}^{m}$, is examined. More precisely, nonexistence of stable integral currents on a compact contact CR-submanifold of real hypersurface of canonical complex space form $\mathbb{C}^m$ is investigated and vanishing theorems concerning the homology groups are obtained.
Averaging of Legendrian submanifolds of contact manifolds
给出了切触流形的Legendrian子流形的正则$C^1$ -闭的“平均”的一个方法.作为推论，我们得到，当一个紧群作用使一个勒让德子流形几乎不变时，在其附近存在一个不变勒让德子流形。
Unlike Legendrian submanifolds, the deformation problem of coisotropic submanifolds can be obstructed. Starting from this observation, we single out in the contact setting the special class of integral coisotropic submanifolds as the direct generalization of Legendrian submanifolds for what concerns deformation and moduli theory. Indeed, being integral coisotropic is proved to be a rigid condition, and moreover the integral coisotropic deformation problem is unobstructed with discrete moduli space.
Equivariant Gluing Constructions of Contact Stationary Legendrian Submanifolds of the (2n+1)-Sphere
$S^{2n+1}$的接触平稳勒让德子流形是在接触变形下体积平稳的勒让德子流形。最简单的接触平稳勒让德子流形（实际上是极小的和勒让德的）是实的赤道$n$ -球面$S_0$。本文给出了一种构造$S^{2n+1}$的接触稳定（但不是极小）Legendrian子流形的方法，该方法是通过将$S_0$的足够多个$U(n+1)$ -旋转副本在适当的横截相交的孤立点处粘在一起.由此产生的子流形是非常对称的;在几何上类似于$S_0$的副本通过窄颈相互连接的“项链”，并在$S^{2n + 1}$周围缠绕了大量的时间，然后在自身上闭合;并且在拓扑上等价于$S^1 \times S^{n-1}$。此外，它们代表了$S^{2n + 1}$的接触定常嵌入子流形的全新例子，从而产生了$C^{n+1}$中嵌入Hamilton定常锥的全新例子。
We establish two main inequalities; one for the norm of the second fundamental form and the other for the matrix of the shape operator. The results obtained are for cosymplectic manifolds and, for these, we show that the contact warped product submanifolds naturally possess a geometric property; namely $\mathcal{D}_1$-minimality which, by means of the Gauss equation, allows us to obtain an optimal general inequality. For sake of generalization, we state our hypotheses for nearly cosymplectic manifolds, then we obtain them as particular cases for cosymplectic manifolds. For the other part of th…
Certain basic inequalities for submanifolds in a (k, u)-contact space form
得到了（k，m）-切触空间形式中子流形的内不变量与外不变量之间的若干基本不等式。作为应用，我们得到了（k，m）-切触空间形式中不变子流形的一些结果.
A rigidity result for coisotropic submanifolds in contact geometry
本文研究接触流形中紧致正则余迷向子流形的余迷向形变，证明了在特征叶层微分同胚的邻近子流形中其具有刚性，并结合经典叶层刚性结果推广至所有邻近子流形。
In this paper I have studied about CR(Cauchy-Riemann)-submanifolds of Lorentzian Concircular Structure manifold ((LCS)n-manifold), Lorentzian Para-Sasakian(LP)-cosymplectic manifold, S-manifold and Generalized Kenmotsu (GKM) manifold. I have discussed some results regarding distribution, structure vector field, totally geodesic submanifold, leaf etc.. I have obtained results on totally umbilical contact CR-submanifold where the anti-invariant distribution has some properties. Next, I have studied some results about D-totally geodesic CR-submanifold (D is the distribution), a contact CR-submani…
We are proposing Tulczyjew's triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a special contact manifold and a Morse family) for a Legendrian submanifold. Contact Hamiltonian and Lagrangian Dynamics are recast as Legendrian submanifolds of the tangent contact manifold. In this picture, the Legendre transformation is determined to be a passage between two d…
We introduce the notions of pointwise almost h-slant submanifolds and pointwise almost h-semi-slant submanifolds as a generalization of slant submanifolds, pointwise slant submanifolds, semi-slant submanifolds, and pointwise semi-slant submanifolds. We have characterizations and investigate the integrability of distributions, the conditions for such distributions to be totally geodesic foliations, the mean curvature vector fields on totally umbilic submanifolds, the properties of h-slant functions and h-semi-slant functions, the properties of non-trivial warped product proper pointwise h-semi-…
J-holomorphic Curves, Legendre Submanifolds and Reeb Chords
本文证明了连通闭切触流形中任意一个Legendre子流形的Reeb向量场至少有一个弦是它的特征.
In this paper, we define and study CR-lightlike submanifolds of a golden semi-Riemannian manifold. We investigate some properties of geodesic CR-submanifolds of a golden semi-Riemannian manifold. Moreover, we obtain many interesting results for totally geodesic and totally umbilical CR-submanifolds on a golden Riemannian manifold. A non-trivial example of CR-lightlike submanifold of a golden semi-Riemannian manifold is also constructed.
Recently Y{ü}ksel et. al. \cite{SY} shows that there doesn't exist any proper semi-slant warped product submanifolds in a Lorentzian paracosymplectic manifold. In the present article, we first define and give preparatory lemmas for a new generalize class of semi-slant submanifolds called pointwise semi-slant submanifolds in a Lorentzian paracosymplectic manifold, and then we ensure by presenting some existence results and characterization theorem that there exist a non-trivial pointwise semi-slant warped product submanifolds in a Lorentzian paracosymplectic manifold counter to warped product s…
Invariant and screen semi-invariant lightlike submanifolds of a metallic semi-Riemannian manifold with a quarter symmetric non-metric connection
本文在金属半黎曼流形上引入带四分之一对称非度量联络的不变与屏幕半不变类光子流形，刻画其分布的可积性与平行性，并给出全测地叶理的结果。
Slant and semi-slant submanifolds in metallic Riemannian manifolds
该文研究金属黎曼流形中斜与半斜子流形的性质，给出相关子流形的刻画、分布可积性条件，并提供对应实例。
Warped product contact CR-submanifolds of globally framed f-manifolds with Lorentz metric
在本文中，我们研究整体框架f-流形在特定的设置不定S-流形的类空和类时的情况。我们证明了：如果$M = N^{\perp} \times_f N^T$是一个弯曲的CR-子流形，使得$N^{\perp}$是$ϕ$？反不变，NT是$ϕ$？-不变，则M是CR-积。本文证明了不定S空间型的接触CR翘曲积的第二基本型满足一个几何不等式$\|h\|^2 \geq p{3\|\nabla lnf\|^2 - \triangle lnf + (c + 2)k + 1}$。
Warped Product Pointwise Semi-slant Submanifolds of Kaehler Manifolds
已知Kaehler流形\cite{Sahin}中不存在翘积半斜子流形。最近，Chen和Garay研究了\cite{CG}中几乎厄米流形的点态斜子流形，得到了许多新的结果.本文首先引入Kaehler流形的点态半斜子流形，然后通过一个例子证明Kaehler流形存在非平凡的点态半斜曲积子流形，与半斜子流形相反.对于Kaehler流形中的这种翘曲积子流形，我们给出了一个刻画定理，并建立了关于第二基本型的平方范数关于翘曲函数的不等式.平等的情况也被考虑。
Pseudo-parallel Lagrangian submanifolds are semi-parallel
证明了Pablo M. Chacon和Guillermo A. [复空间型中伪平行拉格朗日子流形中的Lobos，微分几何应用]说明维数至少为3的复空间型的每个拉格朗日伪平行子流形是半平行的。
The purpose of this paper is to study $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifolds of a paracosymplectic manifold $\widetilde{M}$. We prove that the distributions associated with the definition of $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifold $M$ are always integrable. A necessary and sufficient condition for an isometrically immersed $\mathcal{P}\mathcal{R}$-semi-invariant submanifold of $\widetilde{M}$ to be a $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifold is obtained in terms of the shape operator.
Semi-Invariant Submanifolds in Metric Geometry of Affinors
我们引入了一个推广的结构流形作为最一般的黎曼度量g相关联的仿射（张量场的（1，1）型）F，并开始研究其半不变量子流形。这些子流形是几乎复几何的CR-子流形和几种有趣几何（几乎积、几乎切触等）的半不变量子流形的推广。我们刻画了不变分布和反不变分布的可积性;当F是关于g的协变常数时的特殊情况给出了计算中的主要简化。
Geometry of Invariant Almost Semi Kähler Submanifolds of Flag Manifolds
该文研究旗流形的不变近半Kähler子流形几何，证明了作为其近Hermite子流形的齐性空间必为极小子流形，若这类子流形也是旗流形则是全测地的。
The main purpose of the present paper is to study the geometry of screen transversal lightlike submanifolds and radical screen transversal lightlike submanifolds and screen transversal anti-invariant lightlike submanifolds of Golden Semi-Riemannian manifolds. We investigate the geometry of distributions and obtain necessary and sufficient conditions for the induced connection on these manifolds to be metric connection. We also obtain characterization of screen transversal anti-invariant lightlike submanifolds of Golden Semi- Riemannian manifolds. Finally, we give two examples.
Warped product skew semi-invariant submanifolds of order 1 of a locally product Riemannian manifold
引入局部积黎曼流形的$1$阶翘曲积斜半不变量子流形。给出了一阶斜半不变量子流形是局部翘曲积的一个充要条件。我们还证明了子流形定义中涉及的不变分布在一定条件下是可积的。此外，我们还发现了这类子流形的翘曲函数与第二基本型的平方范数之间的一个不等式。还讨论了平等问题。
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.
Paraquaternionic CR-submanifolds of paraquaternionic Kahler manifolds and semi-Riemannian submersions
本文引入了几乎仿四元厄米流形的仿四元CR-子流形，并给出了关于它们的微分几何的一些基本结果。我们还研究了一类从仿四元数Kaehler流形的仿四元数CR-子流形出发的半黎曼浸没。
We give a complete classification of foliations on open contact manifolds whose leaves are contact submanifolds of the ambient manifold. The results are analogues of Haefliger's classification of foliations on open manifold.
In the present paper, we introduce screen almost semi-invariant (SASI) lightlike submanifolds of indefinite Keahler manifolds. We obtain the neccesary and sufficient condition for the induced connection to be a metric connection on SASI-lightlike submanifolds and construct an example for this manifold. Also we find some conditions for integrability of distributions and investigate certain characterizations.
Recently Hui et al. (\cite{HAP}, \cite{HAN}) studied contact CR-warped product submanifolds and also warped product pseudo-slant submanifolds of a $(LCS)_n$-manifold $\bar{M}$. In this paper we have studied the characterization for both these classes of warped product submanifolds. It is also shown that there do not exists any proper warped product bi-slant submanifold of a $(LCS)_n$-manifold. Although we constructed an example of a bi-slant submanifold of $(LCS)_n$-manifold.
Screen Generic Lightlike Submanifolds of a Locally Bronze Semi-Riemannian Manifold equipped with an (l,m)-type Connection
研究赋予 (l,m) 型联络的局部青铜色半黎曼流形中屏幕一般类光子流形，建立测地性、不存在性、全脐性及极小子流形结构等结论。
The Metallic Ratio is fascinating topic that continually generated news ideas. A Riemannian manifold endowed with a Metallic structure will be called a Metallic Riemannian manifold. The main purpose of the present paper is to study the geometry of transversal lightlike submanifolds and radical transversal lightlike submanifolds and of Metallic Semi-Riemannian manifolds. We investigate the geometry of distributions and obtain necessary and sufficient conditions for the induced connection on these manifolds to be metric connection. We also obtain characterization of transversal lightlike submani…
We derive total mean curvature integration formulae of a three co-dimensional foliation $\mathcal{F}^{n}$ on a screen integrable half-lightlike submanifold, $M^{n+1}$ in a semi-Riemannian manifold $\overline{M}^{n+3}$. We give generalized differential equations relating to mean curvatures of a totally umbilical half-lightlike submanifold admitting a totally umbilical screen distribution, and show that they are generalizations of those given by [4].
An optimal inequality on warped product semi-slant submanifolds of nearly Kaehler manifolds
文[17]证明了Kaehler流形的翘曲积半斜子流形的不存在性，发现它们的存在性是一个有趣的问题。本文利用一个刻划证明了几乎Kaehler流形的翘积半斜子流形的存在性。为此，我们得到了一个不等式的平方范数的第二基本型的翘曲函数和倾斜角。平等的情况下也进行了讨论。
Hineva Inequality for Submanifolds of Real Space Forms with Semi-Symmetric Non-Metric Connection
本文针对具有半对称非度量连接的实空间形式子流形，建立了Hineva不等式，给出了Ricci曲率的下界，并应用于多类子流形。
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