Semi-Slant Submanifolds of a Sasakian Manifold

We define and study both bi-slant and semi-slant submanifolds of an almost contact metric manifold and, in particular, of a Sasakian manifold. We prove a characterization theorem for semi-slant submanifolds and we obtain integrability conditions for the distributions which are involved in the definition of such submanifolds. We also study an interesting particular class of semi-slant submanifolds.     

Contact CR-Warped Product Submanifolds in Sasakian Manifolds

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. In the present paper, we obtain 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. The equality case is considered. Also, the minimum codimension of a contact CR-warped product in an odd-dimensional sphere is determined.     

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.     

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.     

S-curvature of Doubly Warped Product of Finsler Manifolds

Doubly warped product of Finsler manifolds is useful in theoretical physics, particularly in general relativity. In this paper, we study doubly warped product of Finsler manifolds with isotropic mean Berwald curvature or weak isotropic S-curvature.     

双扭曲积Hermitian流形

主要研究双扭曲积Hermitian流形的各种曲率,给出了紧致非平凡的双扭曲积Hermitian流形具有常全纯截面曲率的充要条件,得到了一种构造满足第一或第二爱因斯坦条件的Hermitian流形的有效方法.     

Averaging of Legendrian Submanifolds of Contact Manifolds

We give a procedure to ‘average’ canonically C¹-close Legendrian submanifolds of contact manifolds. As a corollary we obtain that, whenever a compact group action leaves a Legendrian submanifold almost invariant, there is an invariant Legendrian submanifold nearby. Mathematics Subject Classification (2000): 53D10.     

Warped Product Bi-slant Immersions in Kaehler Manifolds

A submanifold M of an almost Hermitian manifold \((\widetilde{M},g,J)\) is called slant, if for each point \(p\in M\) and \(0\ne X\in T_p M\), the angle between JX and \(T_p M\) is constant (see Chen in Bull Aust Math Soc 41:135–147, 1990). Later, Carriazo (in: Proceedings of the ICRAMS 2000, Kharagpur, 2000) defined the notion of bi-slant immersions as an extension of slant immersions. In this paper, we study warped product bi-slant submanifolds in Kaehler manifolds and provide some examples of warped product bi-slant submanifolds in some complex Euclidean spaces. Our main theorem states that every warped product bi-slant submanifold in a Kaehler manifold is either a Riemannian product or a warped product hemi-slant submanifold.     

Kaehler流形上的Hamilton力学

利用力学原理、现在微分几何理论和高等微积分把Hamihon力学推广至Kaehler流形上，建立Kaehler流形上Hamihon力学，并得到Hamilton向量场、Hamihon方程等复的数学形式．     

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)     

一类浸入在半黎曼卷积空间的完备超曲面

We deal with complete hypersurfaces immersed in a semi-Riemannian warped product of the type eI×f M~n,where M~n is a connected n-dimensional oriented Riemannian manifold.When the fiber M~n is complete with sectional curvature-k≤K_M for some positive constant k,under appropriate restrictions on the norm of the gradient of the height function h,we proceed with our technique in order to guarantee that complete hypersurface immersed in a semi-Riemannian warped product is a slice.Our approach is based on the well known generalized maximum principle and another suitable maximum principle at the infinity due to Yau.     

Warped products with special Riemannian curvature

We study the geometry of particular classes of Riemannian manifolds obtained as warped products. We focus on the case of constant curvature which is completely classified and on the Einstein case. This study provides nontrivial instances of Einstein manifolds which are warped product of Einstein factors.Supported by a grant from Università di Parma     

SLANT SUBMANIFOLDS OF A RIEMANNIAN PRODUCT MANIFOLD

In this article, the geometry of the slant submanifolds of a Riemannian product manifold is studied. Some necessary and sufficient conditions on slant, bi-slant and semi-slant submanifolds are given. We research fundamental properties of the distributions which are involved in definitions of semi- and bi-slant submanifolds in a Riemannian product manifold.     

Complete Kaehler Submanifolds in CP~n

In this paper we mainly investigate projectively flat complete Kaehler sub-manifolds, in CPn. We give the pinching constants and the local structure.     

Submanifolds associated with graphs

In this paper we present an interesting relationship between graph theory and differential geometry by defining submanifolds of almost Hermitian manifolds associated with certain kinds of graphs. We show some results about the possibility of a graph being associated with a submanifold and we use them to characterize CR-submanifolds by means of trees. Finally, we characterize submanifolds associated with graphs in a four-dimensional almost Hermitian manifold.

     

Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds, II

 A CR-submanifold N of a Kaehler manifold is called a CR-warped product if N is the warped product of a holomorphic submanifold and a totally real submanifold of . This notion of CR-warped products was introduced in part I of this series. It was proved in part I that every CR-warped product in a Kaehler manifold satisfies a basic inequality: . The classification of CR-warped products in complex Euclidean space satisfying the equality case of the inequality is archived in part I. The main purpose of this second part of this series is to classify CR-warped products in complex projective and complex hyperbolic spaces which satisfy the equality.     

Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds, II

 A CR-submanifold N of a Kaehler manifold is called a CR-warped product if N is the warped product of a holomorphic submanifold and a totally real submanifold of . This notion of CR-warped products was introduced in part I of this series. It was proved in part I that every CR-warped product in a Kaehler manifold satisfies a basic inequality: . The classification of CR-warped products in complex Euclidean space satisfying the equality case of the inequality is archived in part I. The main purpose of this second part of this series is to classify CR-warped products in complex projective and complex hyperbolic spaces which satisfy the equality. (Received 13 March 2001; in revised form 10 August 2001)     

Proper Submanifolds in Product Manifolds

The authors obtain various versions of the Omori-Yau's maximum principle on complete properly immersed submanifolds with controlled mean curvature in certain product manifolds,in complete Riemannian manifolds whose k-Ricci curvature has strong quadratic decay,and also obtain a maximum principle for mean curvature flow of complete manifolds with bounded mean curvature.Using the generalized maximum principle,an estimate on the mean curvature of properly immersed submanifolds with bounded projection in N1 in the product manifold N1 ×N2 is given.Other applications of the generalized maximum principle are also given.     

Two Dimensional Submanifolds in Four Manifolds with Boundary

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复射影空间中迷向Kaehler子流形

讨论了复射影空间中迷向Kaehler流形,运用活动标架法获得关于截面曲率,Ricci曲率和第二基本形式模长的Pinching定理,将相关结果作了一定的推广.