Sasakian空间形式中子流形的全实截面曲率(英文)

By using the methods introduced by Chen[Chen Bang-yen,A series of Ka¨hlerian invarianrts and their applications to Khlerian geometry,Beitrge Algebra Geom,2001,42（1）：165-178],we establish some inequalities for invariant submanifolds in a Sasakian space form involving totally real sectional curvature and the scalar curvature.Moreover,we consider the case of equalities.     

Second variation and Legendrian stabilities of minimal Legendrian submanifolds in Sasakian manifolds

First, we derive a new second variation formula which holds for minimal Legendrian submanifolds in Sasakian manifolds. Using this, we prove that any minimal Legendrian submanifold in an η-Einstein Sasakian manifold with “nonpositive” η-Ricci constant is stable. Next we introduce the notion of the Legendrian stability of minimal Legendrian submanifolds in Sasakian manifolds. Using our second variation formula, we find a general criterion for the Legendrian stability of minimal Legendrian submanifolds in η-Einstein Sasakian manifolds with “positive” η-Ricci constant.     

The third fundamental form metric for hypersurfaces in nonflat space forms

For hypersurfaces with regular Weingarten operator in nonflat space forms we study the relations between the intrinsic geometry of the third fundamental form metric and the extrinsic geometry of the hypersurface. We prove a theorema-egregium-type result for this metric and, in particular, give a local classification of hypersurfaces in case of an Einstein structure of this metric.Partially supported by the project 19701003 of NSFC.The geometry groops at TU Berlin and KU Leuven cooperate within the GA DGET program.     

4-dimensional minimal CR submanifolds of the sphere S satisfying Chen's equality

In this paper, we classify 4-dimensional minimal CR submanifolds M of the nearly Kähler 6-sphere S⁶(1) which satisfy Chen's equality, i.e. , where δM(p)=τ(p)−infK(p) for every p∈M.     

On submanifolds whose shape operator is unipotent

The object of this article is to characterize submanifolds of the Euclidean space whose shape operator A^ξ satisfies the equation (A^ξ)² = k||ξ ||²Id, where k > 0 is constant.     

Codimension two Kähler submanifolds of space forms

In this article we study isometric immersions from Kähler manifolds whose (1, 1) part of the second fundamental form is parallel, theppmc isometric immersions. When the domain is a Riemann surface these immersions are precisely those with parallel mean curvature. P. J. Ryan has classified the Kähler manifolds that admit isometric immersions, as real hypersurfaces, in space forms. We classify the codimension twoppmc isometric immersions into space forms.     

Hypersurfaces in statistical manifolds

The condition for the curvature of a statistical manifold to admit a kind of standard hypersurface is given as a first step of the statistical submanifold theory. A complex version of the notion of statistical structures is also introduced.     

Biharmonic hypersurfaces in Sasakian space forms

We consider the Boothby–Wang fibration of a strictly regular Sasakian space form N and find the characterization of biharmonic Hopf cylinders over submanifolds of . Then, we determine all proper-biharmonic Hopf cylinders over homogeneous real hypersurfaces in complex projective spaces.     

Doubly warped product <Emphasis Type="Italic">CR</Emphasis>-submanifolds in locally conformal Kähler manifolds

In this paper we study doubly warped product CR submanifolds in locally conformal K?hler manifolds, and we found a B.Y. Chen’s type inequality for the second fundamental form of these submanifolds. Beneficiary of a CNR-NATO Advanced Research Fellowship pos. 216.2167 Prot. n. 0015506.     

A Characterization of Minimal Legendrian Submanifolds in S2n+1

Let x: L ⁿ S²ⁿ⁺¹ R²ⁿ⁺² be a minimal submanifold in S²ⁿ⁺¹. In this note, we show that L is Legendrian if and only if for any A su(n + 1) the restriction to L of Ax, (–1)x satisfies f = 2(n + 1)f. In this case, 2(n + 1) is an eigenvalue of the Laplacian with multiplicity at least (n(n + 3)). Moreover if the multiplicity equals to ;(n(n + 3)), then L ⁿ is totally geodesic.     

余辛空间形式中的子流形不等式

For submanifolds in a cosymplectic space form tangent to the structure vector field ξ, two important inequalities with Ricci curvature, scalar curvature and the squared mean curvature are obtained. These results are also applied to get corresponding consequences for anti-invariant submanifolds.     

<Emphasis Type="Italic">CR</Emphasis>-Warped Products in Complex Projective Spaces with Compact Holomorphic Factor

A submanifold of a Kaehler manifold is called a CR-warped product if it is the warped product N_T ×_fN of a complex submanifold N_T and a totally real submanifold N. There exist many CR-warped products N_T ×_fN in CP^h+p, h = dim_CN_T and p = dim_RN (see [5, 6]). In contrast, we prove in this article that the situation is quite different if the holomorphic factor N_T is compact. For such CR-wraped products in CP^m (4), we prove the following: (1) The complex dimension m of the ambient space is at least h + p + hp. (2) If m = h + p + hp, then N_T is CP^h(4). We also obtain two geometric inequalities for CR-warped products in CP^m with compact N_T.     

Riemannian submersions with minimal fibers and affine hypersurfaces

Let π : M → B be a Riemannian submersion with minimal fibers. In this article we prove the following results: (1) If M is positively curved, then the horizontal distribution of the submersion is a non-totally geodesic distribution; (2) if M is non-negatively (respectively, negatively) curved, then the fibers of the submersion have non-positive (respectively, negative) scalar curvature; and (3) if M can be realized either as an elliptic proper centroaffine hypersphere or as an improper hypersphere in some affine space, then the horizontal distribution is non-totally geodesic. Several applications are also presented.     

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)     

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)