Generalized Cauchy--Riemann lightlike submanifolds of Kaehler manifolds

Summary We introduce a class of submanifolds, namely, Generalized Cauchy--Riemann (GCR) lightlike submanifolds of indefinite Kaehler manifolds. We show that this new class is an umbrella of invariant (complex), screen real [8] and CR lightlike [6] submanifolds. We study the existence (or non-existence) of this new class in an indefinite space form. Then, we prove characterization theorems on the existence of totally umbilical, irrotational screen real, complex and CR minimal lightlike submanifolds. We also give one example each of a non totally geodesic proper minimal GCR and CR lightlike submanifolds.     

Screen conformal lightlike submanifolds of semi-Riemannian manifolds

Since the induced objects on a lightlike submanifold depend on its screen distribution which, in general, is not unique and hence we can not use the classical submanifold theory on a lightlike submanifold in the usual way. Therefore, in present paper, we study screen conformal lightlike submanifolds of a semi-Riemannian manifold, which are essential for the existence of unique screen distribution. We obtain a characterization theorem for the existence of screen conformal lightlike submanifolds of a semi-Riemannian manifold. We prove that if the differential operator D^s is a metric Otsuki connection on transversal lightlike bundle for a screen conformal lightlike submanifold then semi-Riemannian manifold is a semi-Euclidean space. We also obtain some characterization theorems for a screen conformal totally umbilical lightlike submanifold of a semi-Riemannian space form. Further, we obtain a necessary and sufficient condition for a screen conformal lightlike submanifold of constant curvature to be a semi-Euclidean space. Finally, we prove that for an irrotational screen conformal lightlike submanifold of a semi-Riemannian space form, the induced Ricci tensor is symmetric and the null sectional curvature vanishes.     

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.     

Submanifolds in Cauchy Riemann Geometry

In this paper I would like to make a report on the results about hypersurfaces in the Heisenberg group and invariant curves and surfaces in CR geometry. The results are contained in the papers [8, 9, 16] and [14]. Besides, I would also report on the results about the strong maximum principle for a class of mean curvature type operators in [10].     

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.     

On canonical screen for lightlike submanifolds of codimension two

In this paper we study two classes of lightlike submanifolds of codimension two of semi-Riemannian manifolds, according as their radical subspaces are 1-dimensional or 2-dimensional. For a large variety of both these classes, we prove the existence of integrable canonical screen distributions subject to some reasonable geometric conditions and support the results through examples.      

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。     

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。     

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.     

Hermite流形上Coupled Vortex方程的Dirichlet问题

本文研究Coupled Vortex方程的Dirichlet问题,通过热流方法来讨论该方程Dirichlet问题的解的存在唯一性．     

The lightlike flat geometry on spacelike submanifolds of codimension two in Minkowski space

We introduce the notion of the lightcone Gauss–Kronecker curvature for a spacelike submanifold of codimension two in Minkowski space, which is a generalization of the ordinary notion of Gauss curvature of hypersurfaces in Euclidean space. In the local sense, this curvature describes the contact of such submanifolds with lightlike hyperplanes. We study geometric properties of such curvatures and show a Gauss–Bonnet type theorem. As examples we have hypersurfaces in hyperbolic space, spacelike hypersurfaces in the lightcone and spacelike hypersurfaces in de Sitter space.     

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.

     

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 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.     

A vanishing theorem on Kaehler Finsler manifolds

Let M be a connected compact complex manifold endowed with a strongly pseudoconvex complex Finsler metric F. In this paper, we first define the complex horizontal Laplacian □_h and complex vertical Laplacian □_v on the holomorphic tangent bundle T^1,0M of M, and then we obtain a precise relationship among □_h,□_v and the Hodge–Laplace operator on (T^1,0M,,), where , is the induced Hermitian metric on T^1,0M by F. As an application, we prove a vanishing theorem of holomorphic p-forms on M under the condition that F is a Kaehler Finsler metric on M.     

Holomorphic curvature of complex Finsler submanifolds

Let M be a complex n-dimensional manifold endowed with a strongly pseudoconvex complex Finsler metric F. Let M be a complex m-dimensional submanifold of M, which is endowed with the induced complex Finsler metric F. Let D be the complex Rund connection associated with (M, F). We prove that (a) the holomorphic curvature of the induced complex linear connection  on (M, F) and the holomorphic curvature of the intrinsic complex Rund connection ～* on (M, F) coincide; (b) the holomorphic curvature of ～* does not exceed the holomorphic curvature of D; (c) (M, F) is totally geodesic in (M, F) if and only if a suitable contraction of the second fundamental form B(·, ·) of (M, F) vanishes, i.e., B(χ, ι) = 0. Our proofs are mainly based on the Gauss, Codazzi and Ricci equations for (M, F).     

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     

Variation of cone metrics on Riemann surfaces

We discuss the variational properties of the unique conical metric of constant curvature −1 associated to a compact Riemann surface together with a weighted divisor.     

Totally geodesic submanifolds in Lie groups

Summary We study minimal and totally geodesic submanifolds in Lie groups and related problems. We show that: (1) The imbedding of the Grassmann manifold G_F(n,N) in the Lie group G_F(N) defined naturally makes G_F(n,N) a totally geodesic submanifold; (2) The imbedding S⁷→SO(8) defined by octonians makes S⁷a totally geodesic submanifold inSO(8); (3) The natural inclusion of the Lie group G_F(N) in the sphere S^{cN^2-1}(√N) of gl(N,F)is minimal. Therefore the natural imbedding G_F(N)<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>→gl(N,F)is formed by the eigenfunctions of the Laplacian on G_F(N).