On Two Conjectures Concerning the Veronese Generating submanifolds

The notion of finite type submanifolds was introduced by B.Y.Chen.In this paper the con-jectures on scalar curvature of Veronses generating submanifolds in E^6 and the minimal conjecture on Veronese space-like submanifold ∑ and Veronese pseudo-Riemannian submanifold-↑∑ in E1^6 are proved.We have ∑ is minimal in H^5.-↑∑ is minimal in S1^5,∑ and -↑∑ are of 1-type in E1^6.     

SUBMANIFOLDS IN S^n＋p WITH PARALLEL MOEBIUS FORM

In this paper,the rigidity theorems of the submanifolds in S^n p with parallel Moebius form and constant MObius scalar curvature are given.     

On Submanifolds of the Unit Sphere with Vanishing M?bius Form and Parallel Para-Blaschke Tensor

The para-Blaschke tensor are extended in this paper from hypersurfaces to general higher codimensional submanifolds in the unit sphere Sn, which is invariant under the M?bius transformations on Sn. Then some typical new examples of umbilic-free submanifolds in Snwith vanishing M?bius form and a parallel para-Blaschke tensor of two distinct eigenvalues, D₁ and D₂, are constructed. The main theorem of this paper is a simple characterization of these newly found examples in te...     

Regular Submanifolds in Conformal Space ${\mathbb Q}^n_p$

The authors study the regular submanifolds in the conformal space Q_p~n and introduce the submanifold theory in the conformal space Q_p~n.The first variation formula of the Willmore volume functional of pseudo-Riemannian submanifolds in the conformal spaceQ_p~n is given.Finally,the conformal isotropic submanifolds in the conformal space Q_p~n are classified.     

GEOMETRIC PROPERTIES FOR GAUSSIAN IMAGE OF SUBMANIFOLDS IN S^n＋p（1）

The geometric properties for Gaussian image of submanifolds in a sphere are investigated.The computation formula,geometric equalities and inequalities for the volume of Gaussian image of certain submanifolds in a sphere are obtained.     

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.     

关于球面上紧致极小子流形内蕴刚性的注记

In this paper,some intrinsic rigidity theorems for compact minimal submanifolds in a sphere are gien,so that the classical corresponding results due to S.T.Yau and Ejiri.N are improved.     

Remarks on the rank properties of formal CR maps Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

We prove several new transversality results for formal CR maps between formal real hypersurfaces in complex space. Both cases of finite and infinite type hypersurfaces are tackled in this note.     

CLASSIFICATIONS OF DUPIN HYPERSURFACES IN LIE SPHERE GEOMETRY

This is a survey of local and global classification results concerning Dupin hypersurfaces in Sⁿ(or Rⁿ) that have been obtained in the context of Lie sphere geometry.The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres.Along with these classification results,many important concepts from Lie sphere geometry,such as curvature spheres,Lie curvatures,and Legendre lifts of submanifolds of Sⁿ(or Rⁿ),are describ...     

Low-type submanifolds of real space forms via the immersions by projectors

We undertake a comprehensive study of submanifolds of low Chen-type (1, 2, or 3) in non-flat real space forms, immersed into a suitable (pseudo) Euclidean space of symmetric matrices by projection operators. Some previous results for submanifolds of the unit sphere (obtained in [A. Ros, Eigenvalue inequalities for minimal submanifolds and P-manifolds, Math. Z. 187 (1984) 393–404; M. Barros, B.Y. Chen, Spherical submanifolds which are of 2-type via the second standard immersion of the sphere, Nagoya Math. J. 108 (1987) 77–91; I. Dimitrić, Spherical hypersurfaces with low type quadric representation, Tokyo J. Math. 13 (1990) 469–492; J.T. Lu, Hypersurfaces of a sphere with 3-type quadric representation, Kodai Math. J. 17 (1994) 290–298]) are generalized and extended to real projective and hyperbolic spaces as well as to the sphere. In particular, we give a characterization of 2-type submanifolds of these space forms with parallel mean curvature vector. We classify 2-type hypersurfaces in these spaces and give two sets of necessary conditions for a minimal hypersurface to be of 3-type and for a hypersurface with constant mean curvature to be mass-symmetric and of 3-type. These conditions are then used to classify such hypersurfaces of dimension n5. For example, the complete minimal hypersurfaces of the unit sphere Sⁿ⁺¹ which are of 3-type via the immersion by projectors are exactly the 3-dimensional Cartan minimal hypersurface and the Clifford minimal hypersurfaces M_k,n−k for n≠2k. An interesting characterization of horospheres in is also obtained.     

Classification results for biharmonic submanifolds in spheres

We study biharmonic submanifolds of the Euclidean sphere that satisfy certain geometric properties. We classify: (i) the biharmonic hypersurfaces with at most two distinct principal curvatures; (ii) the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudoumbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curvature vector field. We also study the type, in the sense of B-Y. Chen, of compact proper biharmonic submanifolds with constant mean curvature in spheres. Dedicated to Professor Vasile Oproiu on his 65th birthday The first author was supported by a INdAM doctoral fellowship, Italy. The second author was supported by PRIN 2005, Italy. The third author was supported by Grant CEEX ET 5871/2006, Romania     

双曲空间Hn+p(-1)中具常数量曲率的完备子流形

设Mn是Hn p(-1)中具有常标准数量曲率的n维完备子流形,本文证明了这种完备子流形的某些内蕴刚性定理和分类定理,并对超曲面的情形进行了研究．     

Vanishing theorems on Riemannian manifolds, and geometric applications

A general Liouville-type result and a corresponding vanishing theorem are proved under minimal regularity assumptions. The latter is then applied to conformal deformations of stable minimal hypersurfaces, to the L² cohomology of complete manifolds, to harmonic maps under various geometric assumptions, and to the topology of submanifolds of Cartan-Hadamard spaces with controlled extrinsic geometry.     

Generalized chen submanifolds

As a generalization of Chen submanifolds,k-th Chen submanifolds are defined. A characterization for them is proved. Spherical 2nd Chen submanifolds are discussed. For a compact submanifoldM with parallel second fundamental form it is proved thatM is ak-th Chen submanifold if and only ifM is ofk-type.Dedicated to Prof. A. Barlotti for his 70- th birthdayThe first author was partially supported by the Provincial Scientific Research Fund from Guangdong Province, China.     

Generalized Cauchy-Riemann lightlike submanifolds of indefinite Sasakian manifolds

We study a class of submanifolds, called Generalized Cauchy-Riemann (GCR) lightlike submanifolds of indefinite Sasakian manifolds as an umbrella of invariant, screen real, contact CR lightlike subcases [8] and real hypersurfaces [9]. We prove existence and non-existence theorems and a characterization theorem on minimal GCR-lightlike submanifolds.     

Non-existence of stable currents in submanifolds of the Euclidean spaces

We prove the non-existence theorems of stable integral currents for certain classes of hypersurfaces or higher codimensional submanifolds in the Euclidean spaces.     

Scalar curvature of a class of minimal submanifolds in rank 1 symmetric spaces

In this paper, we give some rigidity theorems which concern with compact minimal coisotropic submanifolds in ℂPⁿ, compact minimal quaternionic coisotropic submanifolds in ℚPⁿ and compact minimal hypersurfaces in P² (Cay).      

Convergence and finite determination of formal CR mappings

It is shown that a formal mapping between two real-analytic hypersurfaces in complex space is convergent provided that neither hypersurface contains a nontrivial holomorphic variety. For higher codimensional generic submanifolds, convergence is proved e.g. under the assumption that the source is of finite type, the target does not contain a nontrivial holomorphic variety, and the mapping is finite. Finite determination (by jets of a predetermined order) of formal mappings between smooth generic submanifolds is also established.

     

单位球面中极小子流形的曲率拼挤

本文证明了单位球面中极小子流形的一些拼挤定理,特别注意到单位球面中的极小超曲面、给出了截曲率的拼挤常数,我们也改进了由Ｎ．Ｅｊｉｒｉ得到的Ｒｉｃｃｉ曲率拼挤常数。