Ahmet Mollaogullari, Çetin Camci
In this study, we show that there is an $α$-Sasakian structure on product manifold $M_{1}\times β(I)$ where $M_{1}$ is a Kaehlerian manifold that has exact 1-form and $β(I)$ is an open curve. After then, we define a new type warped product metric to study the warped product of almost Hermitian manifolds with almost contact metric manifolds.
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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.
Warped product CR-submanifolds in Kaehlerian manifolds were intensively studied only since 2001 after the impulse given by B.Y. Chen. Immediately after, another line of research, similar to that concerning Sasakian geometry as the odd dimensional version of Kaehlerian geometry, was developed, namely warped product contact CR-submanifolds in Sasakian manifolds. In this note we proved that there exists no proper doubly warped product contact CR-submanifolds in trans-Sasakian manifolds.
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.
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…
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 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…
We discuss the geometry of warped foliations. After examining the Levi-Civita connection, we describe the formulae for sectional, Ricci and scalar curvatures. In the final part of this note, we present some examples.
In this paper, we study the geometry of pointwise semi-slant warped products in a locally conformal Kaehler manifold. In particular, we obtain several results which extend Chen's inequality for CR-warped product submanifolds in Kaehler manifolds. Also, we study the corresponding equality cases. Several related results on pointwise semi-slant warped products are also proved in this paper.
Information geometry is an important tool to study statistical models. There are some important examples in statistical models which are regarded as warped products. In this paper, we study information geometry of warped products. We consider the case where the warped product and its fiber space are equipped with dually flat connections and, in the particular case of a cone, characterize the connections on the base space $\mathbb{R}_{>0}$. The resulting connections turn out to be the $α$-connections with $α= \pm{1}$.
The warped product $N_1\times_f N_2$ of two Riemannian manifolds $(N_1,g_1)$ and $(N_2,g_2)$ is the product manifold $N_1\times N_2$ equipped with the warped product metric $g=g_1+f^2 g_2$, where $f$ is a positive function on $N_1$. Warped products play very important roles in differential geometry as well as in physics. A submanifold $M$ of a Kaehler manifold $\tilde M$ is called a $CR$-warped product if it is a warped product $M_T\times_f N_\perp$ of a complex submanifold $M_T$ and a totally real submanifold $M_\perp$ of $\tilde M$. In this article we survey recent results on warped product …
In this paper, we initiate the study of $\p R$-warped products in para-Kähler manifolds and prove some fundamental results on such submanifolds. In particular, we establish a general optimal inequality for $\p R$-warped products in para-Kähler manifolds involving only the warping function and the second fundamental form. Moreover, we completely classify $\p R$-warped products in the flat para-Kähler manifold with least codimension which satisfy the equality case of the inequality.
In the present paper, we study globally framed f-manifolds in the particular setting of indefinite S-manifolds for both spacelike and timelike cases. We prove that if $M = N^{\perp} \times_f N^T$ is a warped CR-submanifold such that $N^{\perp}$ is $ϕ$?-anti-invariant and NT is $ϕ$?-invariant, then M is a CR-product. We show that the second fundamental form of a contact CR warped product of a indefinite S space form satisfies a geometric inequality, $\|h\|^2 \geq p{3\|\nabla lnf\|^2 - \triangle lnf + (c + 2)k + 1}$.
On the geometry of Clairaut warped product Riemannian maps
本文引入Clairaut warped product Riemannian maps，通过推导正则曲线的测地线条件，得出该类映射的成立条件并求出其曲率张量。
This is a survey on the geometry of warped products, without, or essentially with only soft, calculation. Somewhere in the paper, the goal was to give a synthetic account since existing approaches are rather analytic. Somewhere else, we have interpreted statements, especially by means of a physical terminology. This is essentially heuristic, but we think it might be helpful in both directions, that is, in going from a synthetic geometrical language to a relativistic one, and vice-versa.
On the geometry of Riemannian warped product maps
本文引入Clairaut黎曼扭曲乘积映射，基于Ricci曲率建立正则曲线成为测地线的条件，利用曲率张量研究Ricci soliton结构，推导Bochner型公式与标量曲率积分公式，并扩展至conformal映射的几何性质，构造了相关实例。
The warped product $N_1\times_f N_2$ of two Riemannian manifolds $(N_1,g_1)$ and $(N_2,g_2)$ is the product manifold $N_1\times N_2$ equipped with the warped product metric $g=g_1+f^2 g_2$, where $f$ is a positive function on $N_1$. The notion of warped product manifolds is one of the most fruitful generalizations of Riemannian products. Such notion plays very important roles in differential geometry as well as in physics, especially in general relativity. Warped product manifolds have been studied for a long period of time. In contrast, the study of warped product submanifolds was only initia…
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.
In this paper, we study semi-slant submanifolds and their warped products in Kenmotsu manifolds. The existence of such warped products in Kenmotsu manifolds is shown by an example and a characterization. A sharp relation is obtained as a lower bound of the squared norm of second fundamental form in terms of the warping function and the slant angle. The equality case is also considered in this paper. Finally, we provide some applications of our derived results.
The purpose of this paper is to study pointwise pseudo-slant warped product submanifolds of a Kähler manifold $\widetilde{M}$. We derive the conditions of integrability and totally geodesic foliation for the distributions allied to the characterization of a pointwise pseudo-slant submanifold of $\widetilde{M}$. The necessary and sufficient conditions for isometrically immersed pointwise pseudo-slant submanfold of $\widetilde{M}$ to be a pointwise pseudo-slant warped product and a locally Riemannian product are obtained.
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.
In this paper, we generalize the geometry of the product pseudo-Riemannian manifold equipped with the product Poisson structure (\cite{Nas2}) to the geometry of a warped product of pseudo-Riemannian manifolds equipped with a warped Poisson structure. We construct three bivector fields on a product manifold and show that each of them lead under certain conditions to a Poisson structure. One of these bivector fields will be called the warped bivector field. For a warped product of pseudo-Riemannian manifolds equipped with a warped bivector field, we compute the corresponding contravariant Levi-C…
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 …
Geometry of statistical submanifolds of statistical warped product manifolds by optimization techniques
该研究将几何不等式视为优化问题，利用优化技术推导了近Kenmotsu型统计warped乘积流形中统计子流形的最优Casorati不等式和Chen-Ricci不等式，拓展了优化方法在子流形几何领域的应用。
In this note, we introduce a new type of warped products called as sequential warped products to cover a wider variety of exact solutions to Einstein's equation. First, we study the geometry of sequential warped products and obtain covariant derivatives, curvature tensor, Ricci curvature and scalar curvature formulas. Then some important consequences of these formulas are also stated. We provide characterizations of geodesics and two different types of conformal vector fields, namely, Killing vector fields and concircular vector fields on sequential warped product manifolds. Finally, we consid…
In this paper, we prove that there are no proper $CRS$ bi-warped product submanifolds other than contact CR-biwarped products in Sasakian manifolds. On the other hand, we prove that if $M$ is a $CRS$ bi-warped product of the form $M=N_T \times_{f_1}N^{n_{1}}_\perp\times_{f_2} N^{n_{2}}_θ$ in a cosymplectic manifold $\widetilde M$, then its second fundamental form $h$ satisfies the inequality: $$\|h\|^2\geq 2n_1\|\nabla(\ln f_1)\|^2+2n_2(1+2\cot^2θ)\|\nabla(\ln f_2)\|^2,$$ where $N_T,\, N^{n_{1}}_\perp$ and $N^{n_{2}}_θ$ are invariant, anti-invariant and proper pointwise slant submanifolds of $…
Warped product manifolds have been studied for a long period of time. In contrast, the study of warped product submanifolds from extrinsic point of view was initiated by the first author around the beginning of this century in [7, 8]. Since then the study of warped product submanifolds has been investigated by many geometers. The notion of slant submanifolds of almost Hermitian manifolds was introduced in [5]. Bi-slant submanifolds in almost contact metric manifolds were defined in [4] by J. L. Cabrerizo et al. In [26], we studied bi-slant submanifolds and warped product bi-slant submanifolds …
We prove that there exist solutions for a non-parametric capillary problem in a wide class of Riemannian manifolds endowed with a Killing vector field. In other terms, we prove the existence of Killing graphs with prescribed mean curvature and prescribed contact angle along its boundary. These results may be useful for modelling stationary hypersurfaces under the influence of a non-homogeneous gravitational field defined over an arbitrary Riemannian manifold.
In this paper, warped product contact $CR$-submanifolds in Sasakian, Kenmotsu and cosymplectic manifolds are shown to possess a geometric property; namely $\mathcal{D}_T$-minimal. Taking benefit from this property, an optimal general inequality for warped product contact $CR$-submanifolds is established in both Sasakian and Kenmotsu manifolds by means of the Gauss equation, we leave cosyplectic because it is an easy structure. Moreover, a rich geometry appears when the necessity and sufficiency are proved and discussed in the equality case. Applying this general inequality, the inequalities ob…
Higher Fundamental Forms and Warped Product Hypersurfaces
本文利用高阶基本形式刻画乘积流形及共形流形中超曲面嵌入的几何，证明高阶共形基本形式在刻画中起关键作用。
In this paper, we study warped product submanifolds of nearly trans-Sasakian manifolds. The non-existence of the warped product semi-slant submanifolds of the type $N_θ\times{_{f}N_T}$ is shown, whereas some characterization and new geometric obstructions are obtained for the warped products of the type $N_T\times{_{f}N_θ}$. We establish two general inequalities for the squared norm of the second fundamental form. The first inequality generalizes derived inequalities for some contact metric manifolds [16, 18, 19, 24], while by a new technique, the second inequality is constructed to express th…
We study harmonic almost contact structures in the context of contact metric manifolds, and an analysis is carried out when such a manifold fibres over an almost Hermitian manifold, as exemplified by the Boothby-Wang fibration. Two types of almost contact metric warped products are also studied, relating their harmonicity to that of the almost Hermitian structure on the base or fibre.
Geometry of Clairaut Riemannian warped product submersions
本文引入并研究Clairaut黎曼扭曲积浸没，建立其满足Clairaut条件的充要条件，推导源空间曲率张量显式公式，分析相关几何性质并解答文献[2]推广问题。
On warped product on information geometry: statistical manifolds and statistical models
本文研究信息几何中扭曲积构造对散度、对偶结构和对称3-张量的保持性，发现其虽能继承几何结构却不保持规范散度，故在信息论中意义有限。
Recently, M. Atcken studied Contact CR-warped prod- uct submanifolds in cosymplectic space forms and established gen- eral sharp inequalities for CR-warped products in a cosymplectic manifold [1]. In the present paper, we obtain an inequality for the squared norm of the second fundamental form in terms of constant ?$ϕ$-sectional curvature for hemi-slant products in cosymplectic mani- folds. An inequality for hemi-slant warped products in a cosymplectc manifold is also given. The equality case is considered.
The aim of this research is the study of Gray curvature identities, introduced by Alfred Gray in \cite{kn:Gra76} for the class of almost hermitian manifolds. As known till now, there is no equivalent for the class of almost contact manifolds. For this purpose we use the Boohby-Wang fibration and the warped manifolds construction in order to establish which identities could be satisfied by an almost contact manifold.
In this article the degenerate warped products of singular semi-Riemannian manifolds are studied. They were used recently by the author to handle singularities occurring in General Relativity, in black holes and at the big-bang. One main result presented here is that a degenerate warped product of semi-regular semi-Riemannian manifolds with the warping function satisfying a certain condition is a semi-regular semi-Riemannian manifold. The connection and the Riemann curvature of the warped product are expressed in terms of those of the factor manifolds. Examples of singular semi-Riemannian mani…
The twistor space of the sphere S^{2n} is an isotropic Grassmannian that fibers over S^{2n}. An orthogonal complex structure on a subdomain of S^{2n} (a complex structure compatible with the round metric) determines a section of this fibration with holomorphic image. In this paper, we use this correspondence to prove that any finite energy orthogonal complex structure on R^6 must be of a special warped product form, and we also prove that any orthogonal complex structure on R^{2n} that is asymptotically constant must itself be constant. We will also give examples defined on R^{2n} which have i…
The purpose of this article is to study the geometry of gradient almost Yamabe solitons immersed into warped product manifolds $I\times_{f}M^{n}$ whose potential is given by the height function from the immersion. First, we present some geometric rigidity on compact solitons due to a curvature condition on the warped product manifold. In the sequel, we investigate conditions for the existence of totally geodesic, totally umbilical and minimal solitons. Furthermore, in the scope of constant angle immersions, a classification of rotational gradient almost Yamabe soliton immersed into $\mathbb{R}…
Scalar curvature rigidity of warped product metrics
该研究基于旋量几何，证明了严格对数凹扭曲函数下紧区间与维数≥2标准球面的扭曲乘积的标量-平均曲率刚性，推广了已有结果，还解决了Gromov提出的去掉对径点的高维球面标量曲率刚性问题。
Finite Volume Einstein Finsler Warped Product Manifolds of Non-positive or Non-negative Scalar Curvature
本文研究了非对称芬斯勒流形的有限体积爱因斯坦扭曲积结构，并探讨了其在相对论理论中的应用。
Multiply warped products with harmonic Weyl tensor and zero radial Weyl curvature
本文研究具有调和Weyl张量和零径向Weyl曲率的多重扭曲积，证明其至多有三纤维，并利用共形变换和相容性方程刻画局部几何，所得度量源于两类构造，且界是精确的。
In this paper the notion of the intrinsic geometry of an almost contact metric manifold is introduced. Description of some classes of spaces with almost contact metric structures in terms of the intrinsic geometry is given. A new type of almost contact metric spaces, more precisely, Hermitian almost contact metric spaces, is introduced.
Determining the associated metrics we get a local classification of contact metric three manifolds.
We provide some examples of harmonic unit vector fields as normalized gradients of isoparametric functions from a K-contact geometry setting.
In this paper, the notion of an almost contact Kählerian structure is introduced. The interior geometry of almost contact Kählerian spaces is investigated. On the zero-curvature distribution of an almost contact metric structure, as on the total space of a vector bundle, an almost contact Kählerian structure is obtained.
We use classical (Penrose) two-component spinors to set up the differential geometry of two parabolic contact structures in five dimensions, namely $G_2$ contact geometry and Legendrean contact geometry. The key players in these two geometries are invariantly defined directional derivatives defined only in the contact directions. We explain how to define them and their usage in constructing basic invariants such as the harmonic curvature, the obstruction to being locally flat from the parabolic viewpoint. As an application, we calculate the invariant torsion of the $G_2$ contact structure on t…
Warped geometries of Segre-Veronese manifolds
本文研究了Segre-Veronese流形的扭曲几何，提出了参数控制球面切方向与径向切方向权重的模型，并展示了其在计算秩-1张量平均中的应用。
It is well-known that the Einstein condition on warpedgeometries requires the fibres to be necessarily Einstein. However, exact warped solutions have often been obtained using one- and two-dimensional bases. In this paper, keeping the dimensions and signatures of the base and the fibre independently arbitrary, we obtain all Ricci-flat warped metrics with flat base in closed form and show that the number of free parameters is one less than the dimension of the base. Without any assumptions on the base and fibre geometry, we then show that a warped geometry is flat, i.e, has vanishing Riemann cu…
Geometry-Aware Generative Autoencoders for Warped Riemannian Metric Learning and Generative Modeling on Data Manifolds
本文提出几何感知生成自编码器GAGA，通过结合流形学习与生成建模，学习扭曲黎曼度量以刻画数据流形几何，实现均匀采样、测地线生成和群体插值，在单细胞轨迹推断上比现有最优方法提升30%。
Many classical results in relativity theory concerning spherically symmetric space-times have easy generalizations to warped product space-times, with a two-dimensional Lorentzian base and arbitrary dimensional Riemannian fibers. We first give a systematic presentation of the main geometric constructions, with emphasis on the Kodama vector field and the Hawking energy; the construction is signature independent. This leads to proofs of general Birkhoff-type theorems for warped product manifolds; our theorems in particular apply to situations where the warped product manifold is not necessarily …
This paper completes a fundamental construction in Alexandrov geometry. Previously we gave a new construction of metric spaces with curvature bounds either above or below, namely warped products with intrinsic metric space base and fiber, and with possibly vanishing warping functions -- thereby extending the classical cone and suspension constructions from interval base to arbitrary base, and furthermore encompassing gluing constructions. This paper proves the converse, namely, all conditions of the theorems are necessary. Note that in the cone construction, both the construction and its conve…
A Moser-Bernstein problem for Riemannian warped products
本文在黎曼扭曲积中研究Moser-Bernstein问题，求解极小超曲面方程的完整解，推广了经典欧氏空间中的结果。
In the paper we prove, that extrinsic curvature does not impose restrictions on the topology of a contact structure, except the obvious ones.
We show that every K-contact Einstein manifold is Sasakian-Einstein and discuss several corollaries of this result.
New singularity theorems are derived for generic warped-product spacetimes of any dimension. The main purpose is to analyze the stability of (compact or large) extra dimensions against dynamical perturbations. To that end, the base of the warped product is assumed to be our visible 4-dimensional world, while the extra dimensions define the fibers, hence we consider "extra-dimensional evolution". Explicit conditions on the warping function that lead to geodesic incompleteness are given. These conditions can be appropriately rewritten, given a warping function, as restrictions on the intrinsic g…
We introduce a notion of the noncommutative integrability within a framework of contact geometry.
We prove, for a class of contact manifolds, that the universal cover of the group of contact diffeomorphisms carries a natural partial order. It leads to a new viewpoint on geometry and dynamics of contactomorphisms. It gives rise to invariants of contactomorphisms which generalize the classical notion of the rotation number. Our approach is based on tools of Symplectic Topology.
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