Contact CR-Warped Product Submanifolds in Sasakian Space Forms

Recently, B.-Y. Chen studied warped products which are CR-submanifolds in Kaehler manifolds and established general sharp inequalities for CR-warped products in Kaehler manifolds. Afterwards, I. Hasegawa and the present author obtained a sharp inequality for the squared norm of the second fundamental form (an extrinsic invariant) in terms of the warping function for contact CR-warped products isometrically immersed in Sasakian manifolds. In this paper, we improve the above inequality for contact CR-warped products in Sasakian space forms. Some applications are derived. A classification of contact CR-warped products in spheres, which satisfy the equality case, identically, is given.Mathematics Subject Classifications (2000). 53C40, 53C25.     

A note on warped product submanifolds of Kenmotsu manifolds

Warped product manifolds are known to have applications in Physics. For instance, they provide an excellent setting to model space-time near a black hole or a massive star (cf. [HONG, S. T.: Warped products and black holes, Nuovo Cimento Soc. Ital. Fis. B 120 (2005), 1227–1234]). The studies on warped product manifolds with extrinsic geometric point of view are intensified after B. Y. Chen’s work on CR-warped product submanifolds of Kaehler manifolds. Later on, similar studies are carried out in the setting of Sasakian manifolds by Hasegawa and Mihai. As Kenmotsu manifolds are themselves warped product spaces, it is interesting to investigate warped product submanifolds of Kenmotsu manifolds. In the present note a larger class of warped product submanifolds than the class of contact CR-warped product submanifolds is considered. More precisely the existence of warped product submanifolds of a Kenmotsu manifold with one of the factors an invariant submanifold is ensured, an example of such submanifolds is provided and a characterization for a contact CR-submanifold to be a contact CR-warped product submanifold is established.     

GENERIC WARPED PRODUCT SUBMANIFOLDS OF LOCALLY CONFORMAL KEAHLER MANIFOLDS

Warped product manifolds are known to have applications in physics. For instance, they provide an excellent setting to model space-time near a black hole or a massive star （cf. [9]）. The studies on warped product manifolds with extrinsic geometric point of view were intensified after the B.Y. Chen＇s work on CR-warped product submanifolds of Kaehler manifolds （cf. [6], [7]）. Later on, similar studies were carried out in the setting of 1.c.K. manifolds and nearly Kaehler manifolds （el. [3], [11]）. In the present article, we investigate a larger class of warped product submanifolds of 1.c.K. manifolds, ensure their existence by constructing an example of such manifolds and obtain some important properties of these submanifolds. With regard to the CR-warped product submanifold, a special case of generic warped product submanifolds, we obtain a characterization under which a CR-submanifold is reducesd to a CR-warped product submanifold.     

Generic warped product submanifolds of locally conformal keahler manifolds

Warped product manifolds are known to have applications in physics. For instance, they provide an excellent setting to model space-time near a black hole or a massive star (cf. [9]). The studies on warped product manifolds with extrinsic geometric point of view were intensified after the B.Y. Chen's work on CR-warped product submanifolds of Kaehler manifolds (cf. [6], [7]). Later on, similar studies were carried out in the setting of l.c.K. manifolds and nearly Kaehler manifolds (cf. [3], [11]). In the present article, we investigate a larger class of warped product submanifolds of l.c.K. manifolds, ensure their existence by constructing an example of such manifolds and obtain some important properties of these submanifolds. With regard to the CR-warped product submanifold, a special case of generic warped product submanifolds, we obtain a characterization under which a CR-submanifold is reducesd to a CR-warped product submanifold.     

Non-existence of Warped Product Semi-Slant Submanifolds of Kaehler Manifolds

In this paper, we show that there are no warped product semi-slant submanifolds of Kaehler manifolds. Contrary to this result,we provide an elementary example of a CR-warped product submanifold of a Kaehler manifold     

Harmonic Riemannian maps on locally conformal Kaehler manifolds

We study harmonic Riemannian maps on locally conformal Kaehler manifolds (lcK manifolds). We show that if a Riemannian holomorphic map between lcK manifolds is harmonic, then the Lee vector field of the domain belongs to the kernel of the Riemannian map under a condition. When the domain is Kaehler, we prove that a Riemannian holomorphic map is harmonic if and only if the lcK manifold is Kaehler. Then we find similar results for Riemannian maps between lcK manifolds and Sasakian manifolds. Finally, we check the constancy of some maps between almost complex (or almost contact) manifolds and almost product manifolds.     

Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds

 In this paper we study warped product CR-submanifolds in Kaehler manifolds and introduce the notion of CR-warped products. We prove several fundamental properties of CR-warped products in Kaehler manifolds and establish a general inequality for an arbitrary CR-warped product in an arbitrary Kaehler manifold. We then investigate CR-warped products in a general Kaehler manifold which satisfy the equality case of the inequality. Finally we classify CR-warped products in complex Euclidean space which satisfy the equality. (Received 24 August 2000; in revised form 19 February 2001)     

近Kaehler流形S^3× S^3上的殆切触拉格朗日子流形

对于近Kaehler流形S^3× S^3上的一个拉格朗日子流形M ,给出由M 上的一个单位向量场典范引出的殆切触度量结构是α-Sasakian 的充要条件。当这个殆切触度量结构为切触度量结构时,给出了这个切触度量结构是Sasakian结构的充分必要条件。     

Inequalities for cohomology classes of Kaehler manifolds

A set of inequalities for deRham cohomology classes of Kaehler manifolds is proved. As an application, an inequality for Chern numbers of a Kaehler manifold is obtained.     

Hermitian metric rigidity on compact foliated manifolds

We prove a Hermitian metric rigidity theorem for leafwise symmetric Kaehler metrics on compact manifolds with smooth foliations. This provides applications to the study of the geometry of foliations as well as Kaehler manifolds that contain some symmetric geometry.     

Kaehler流形的全脐实超曲面

给出了Ｋａｅｈｌｅｒ流形的全脐实超曲面的一个分类定理。     

Submanifolds of generalized Hopf manifolds,type numbers and the first Chern class of the normal bundle

Summary Any orientable real hypersurface M of a complex Hopf manifold (carrying the locally conformal Kaehler (l.c.K.) metric discovered by I.Vaisman [33]) has a natural f-structure P as a generic Cauchy-Riemann submanifold; we show (cf. our § 5) that if P anti-commutes with the Weingarten operator, then the type number of the hypersurface is less equal than 1. Moreover, M carries the natural almost contact metrical structure observed by Y.Tashiro [30]; if its almost contact vector is an eigenvector of the Weingarten operator corresponding to a nowhere vanishing eigenfunction and the holomorphic distribution is involutive, then M is foliated with globally conformai Kaehler manifolds (cf. our § 5), provided that some restrictions on the type number of M are imposed. We derive (cf. our § 6) a «Simons type» formula and apply it to compact orientable hypersurfaces with non-negative sectional curvature (in a complex Hopf manifold) and parallel mean curvature vector. Several examples of submanifolds of l.c.K. manifolds are exhibited in § 3. Our § 7 studies complex submanifolds of generalized Hopf manifolds; for instance, we show that the first Chern class of the normal bundle of a complex submanifold having a flat normal connection is vanishing.     

Conformal vector fields on Kaehler manifolds

It is known that a conformal vector field on a compact Kaehler manifold is a Killing vector field. In this paper, we are interested in finding conditions under which a conformal vector field on a non-compact Kaehler manifold is Killing. First we prove that a harmonic analytic conformal vector field on a 2n-dimensional Kaehler manifold (n ≠ 2) of constant nonzero scalar curvature is Killing. It is also shown that on a 2n-dimensional Kaehler Einstein manifold (n > 1) an analytic conformal vector field is either Killing or else the Kaehler manifold is Ricci flat. In particular, it follows that on non-flat Kaehler Einstein manifolds of dimension greater than two, analytic conformal vector fields are Killing.     

Contactly geodesic transformations of almost-contact metric structures

We introduce the notion of contactly geodesic transformation of the metric of an almost-contact metric structure as a contact analog of holomorphically geodesic transformations of the metric of an almost-Hermitian structure. A series of invariants of such transformations is obtained. We prove that such transformations preserve the normality property of an almost-contact metric structure. We prove that cosymplectic and Sasakian manifolds, as well as Kenmotsu manifolds, do not admit nontrivial contactly geodesic transformations of the metric, which is a contact analog of the well-known result for Kählerian manifolds due to Westlake and Yano.     

The Structure of a Class of K-contact Manifolds

In this paper we study a class of K-contact manifolds, namely -conformally flat K-contact manifolds and we show that a compact -conformally flat K-contact manifold with regular contact vector field is a principal S¹-bundle over an almost Kaehler space of constant holomorphic sectional curvature.     

Geometry of formal Kuranishi theory

The principle “ambient cohomology of a Kaehler manifold annihilates obstructions” has been known and exploited since pioneering work of Kodaira. This paper unifies and modestly extends known results in the context of abstract deformations of compact Kaehler manifolds and submanifolds.     

Twistor theory and Blaschke manifolds

Using the twistor theory on quaternionic Kaehler manifolds and some recent results on Blaschke manifolds and compact manifolds whose holonomy group is Spin (7), we prove that a Blaschke manifold of nonnegative scalar curvature whose holonomy group is exceptional is isometric to a projective space.     

The structure of some classes of <Emphasis Type="Italic">K</Emphasis>-contact manifolds

We study projective curvature tensor in K-contact and Sasakian manifolds. We prove that (1) if a K-contact manifold is quasi projectively flat then it is Einstein and (2) a K-contact manifold is ξ-projectively flat if and only if it is Einstein Sasakian. Necessary and sufficient conditions for a K-contact manifold to be quasi projectively flat and φ-projectively flat are obtained. We also prove that for a (2n + 1)-dimensional Sasakian manifold the conditions of being quasi projectively flat, φ-projectively flat and locally isometric to the unit sphere S ²ⁿ⁺¹ (1) are equivalent. Finally, we prove that a compact φ-projectively flat K-contact manifold with regular contact vector field is a principal S ¹-bundle over an almost Kaehler space of constant holomorphic sectional curvature 4.     

Sequential warped product submanifolds having holomorphic,totally real and pointwise slant factors

Periodica Mathematica Hungarica - We introduce sequential warped product submanifolds of Kaehler manifolds, provide examples and establish Chen’s inequality for such submanifolds. The...     

Totally real bisectional curvature,Bochner-Kaehler and Einstein-Kaehler manifolds

By studying various curvature properties of Kaehler manifolds, we establish many new simple geometric characterizations of Bochner-Kaehler and Einstein-Kaehler spaces and of complex space forms.