复射影空间CP~((n+p)/2)中具有2-调和的一般子流形

本文研究了复射影空间中具有2-调和的一般子流形问题.利用活动标架法,获得了这类子流形成为极小子流形的Pinching定理和Simons型积分不等式,此外还得到关于2-调和伪脐一般子流形的一个刚性定理,推广了复射影空间中具有2-调和全实子流形的一些相应结果.     

ON THE CHERN CONNECTION OF FINSLER SUBMANIFOLDS

This paper studies the induced Chern connection of submanifolds in a Finsler manifold and gets the relations between the induced Chern connection and the Chern connection of the induced Finsler metric.Then the authors point out a difference between Finsler submanifolds and Riemann submanifolds.     

Rough singular integrals supported on submanifolds

In this paper, the authors study the boundedness of the singular integral operators associated to submanifolds on Triebel-Lizorkin spaces and Besov spaces. Moreover, the authors also establish an lq-valued inequality for the maximal operators associated to submanifolds in this paper.     

复射影空间中法丛平坦的全实伪脐子流形

该文证明了复射影空间中两种类型的法丛平坦全实伪脐子流形必是极小的,并在紧致的情形确定了它们的具体形状．此外,还说明了复射影空间中的全实全脐子流形一定不是法丛平坦的.     

复空间形式中全实2-调和子流形

本文研究了复空间形式中的全实2-调和子流形,获得了关于此类子流形的一些性质和一个Simons型积分不等式,并研究了其Pinching现象.     

复射影空间CPn+p中的全实2-调和子流形

本文研究了复射影空间中的全实2 -调和子流形问题.利用活动标架法,获得了这类子流形成为极小子流形的关于第二基本形式模长的Pinching定理及一个积分不等式.此外还得到关于全实2-调和伪脐子流形的一些刚性定理,推广了CPn中全实2-调和子流形的一些相应结果.     

常曲率空间中的具有共形第二基本形式的子流形

以把调和态射看作等距浸入的单位法投影的问题为背景,研究了具有共形第二基本形式的子流形,论证了具有共形第二基本形式的高维子流形,一般不是由极小点和全脐点构成,这和曲面的情形形成了鲜明的对照。也给出了常曲率空间中具有平行中曲率的奇数维子流形的一个完全分类。     

共形空间${\mathbb Q}^n_s$中的正则Blaschke拟全脐子流形

[Nie C X,Wu C X,Regular submanifolds in the conformal space Q_p~n,ChinAnn Math,2012,33B(5):695-714]中研究了共形空间Q_s~n中的正则子流形,并引入了共形空间Q_s~n中的子流形理论.本文作者将分类共形空间Q_s~n中的Blaschke拟全脐子流形,证明伪Riemann空间形式中具有常数量曲率和平行的平均曲率向量场的正则子流形是共形空间中的Blaschke拟全脐子流形;反之,共形空间中的Blaschke拟全脐子流形共形等价于伪Riemann空间形式中具有常数量曲率和平行的平均曲率向量场的正则子流形.这一结论可看作是共形空间Q_s~n中共形迷向子流形分类定理的推广.     

On the Structure of Submanifolds with Degenerate Gauss Maps

An n-dimensional submanifold X of a projective space P ^N (C) is called tangentially degenerate if the rank of its Gauss mapping gamma;; X G(n, N) satisfies 0 < rank < n. The authors systematically study the geometry of tangentially degenerate submanifolds of a projective space P ^N (C). By means of the focal images, three basic types of submanifolds are discovered: cones, tangentially degenerate hypersurfaces, and torsal submanifolds. Moreover, for tangentially degenerate submanifolds, a structural theorem is proven. By this theorem, tangentially degenerate submanifolds that do not belong to one of the basic types are foliated into submanifolds of basic types. In the proof the authors introduce irreducible, reducible, and completely reducible tangentially degenerate submanifolds. It is found that cones and tangentially degenerate hypersurfaces are irreducible, and torsal submanifolds are completely reducible while all other tangentially degenerate submanifolds not belonging to basic types are reducible.     

LOOP GROUP ACTIONS AND THE RIBAUCOUR TRANSFORMATIONS FOR FLAT LAGRANGIAN SUBMANIFOLDS

The Ribaucour transformations for flat Lagrangian submanifolds in Cn and CPn via loop group actions are given. As a consequence, the authors obtain a family of new flat Lagrangian submanifolds from a given one via a purely algebraic algorithm. At the same time, it is shown that such Ribaucour transformation always comes with a permutability formula.     

Stability of certain minimal submanifolds in compact symmetric spaces of rank two

In [T. Kimura, M.S. Tanaka, Totally geodesic submanifold in compact symmetric spaces of rank two, Tokyo J. Math., in press], the authors obtained the global classification of the maximal totally geodesic submanifolds in compact connected irreducible symmetric spaces of rank two. In this paper, we determine their stability as minimal submanifolds in compact symmetric spaces of rank two.     

Locally symmetric immersions

We use reflections with respect to submanifolds and related geometric results to develop, inspired by the work of Ferus and other authors, in a unified way a local theory of extrinsic symmetric immersions and submanifolds in a general analytic Riemannian manifold and in locally symmetric spaces. In particular we treat the case of real and complex space forms and study additional relations with holomorphic and symplectic reflections when the ambient space is almost Hermitian. The global case is also taken into consideration and several examples are given.     

关于局部对称共形平坦空间中具有常数量曲率的子流形

该文研究了局部对称共形平坦空间中具有常数量曲率的紧致子流形,证明了这类子流形的某些内蕴刚性定理.     

Normal connection and submanifolds with parallel normal fields in a space of constant curvature

A summary account of the cycle of papers of the authors on submanifolds of a space of constant curvature with a parallel field of p-directions, also containing some new results. In the concluding section there is given a survey of papers on submanifolds of a Riemannian manifold using the concept of normal connection.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 12, pp. 3–30, 1981.     

On Submanifolds with Negative Curvature in Euclidean Space

In this paper we give a survey of results of different authors, among them Tenenblat, on the behaviour of submanifolds with negative curvature.     

关于四元射影空间中的全实2-调和子流形

利用了活动标架法对四元射影空间QPnc中全实2-调和子流形进行了研究,获得了这类子流形成为全实极小子流形的刚性定理,推广了相关文献中的积分不等式．     

复射影空间中具有常数量曲率的完备全实子流形

本文研究了复射影空间中具有常数量曲率的完备全实子流形的问题.利用丘成桐的广义极大值原理和自伴随算子,获得了这类子流形的某些内蕴刚性定理.     

欧氏空间和球面子流形中Yang-Mills场的一类不稳定性结果

本文就欧氏空间和球面中紧致子流形的Yang-Mills场进行了讨论.得到了一类不稳定性结果.     

Curvature Estimates for Minimal Submanifolds of Higher Codimension

The authors derive curvature estimates for minimal submanifolds in Euclidean space for arbitrary dimension and codimension via Gauss map. Thus, Schoen-Simon-Yau＇s results and Ecker-Huisken＇s results are generalized to higher codimension. In this way, Hildebrandt-Jost-Widman＇s result for the Bernstein type theorem is improved.     

Remarks on the Veronese Generating Submanifolds in Minkoski Space

ＲｅｍａｒｋｓｏｎｔｈｅＶｅｒｏｎｅｓｅＧｅｎｅｒａｔｉｎｇＳｕｂｍａｎｉｆｏｌｄｓｉｎＭｉｎｋｏｓｋｉＳｐａｃｅＨｕＣｏｎｇｅ（胡聪娥）（ＨｅｎａｎＵｎｉｖｅｒｓｉｔｙ，Ｋａｉｆｅｎｇ，４７５００１）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒ，ｔｈｅ...