Totally Geodesic Invariant Subm anifolds of Contact Metric Manifold

§１．　ＰｒｅｌｉｍｉｎａｒｉｅｓＬｅｔＭｂｅａ（２ｎ＋１）－ｄｉｍｅｎｓｉｏｎａｌｃｏｎｔａｃｔｍｅｔｒｉｃｍａｎｉｆｏｌｄｗｉｔｈｓｔｒｕｃｔｕｒｅｔｅｎｓｏｒｓ（Φ－，ξ－，η－，ｇ）．ＴｈｅｎｔｈｅｙｓａｔｉｓｆｙΦ－ξ－＝０，η－（ξ－）＝１，Φ－２＝－Ｉ＋η－ξ－，η－（Ｘ）＝ｇ（Ｘ，ξ－），　　　ｇ（Φ－Ｘ，Φ－Ｙ）＝ｇ（Ｘ，Ｙ）－η－（Ｘ）η－（Ｙ），ｇ（Ｘ，Φ－Ｙ）＝ｄη－（Ｘ，Ｙ）（１．１）ＦｏｒａｎｙｖｅｃｔｏｒｆｉｅｌｄｓＸａｎｄＹ…     

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.     

近乎Sasakian流形的CR子流形

本文定义并讨论了近乎Ｓａｓａｋｉａｎ流形的ＣＲ子流形,得到了关开这类子流形的微分几何方面的一些有意义的结果。     

Skew CR Submanifolds of a Sasakian Manifold

ＳｋｅｗＣＲＳｕｂｍａｎｉｆｏｌｄｓｏｆａＳａｓａｋｉａｎＭａｎｉｆｏｌｄＬｉｕＸｉｍｉｎ（刘西民）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＮａｎｋａｉＵｎｉｖｅｒｓｉｔｙ，Ｔｉａｎｊｉｎ，３０００７１）ＬｉａｎｇＸｉｑｕａｎ（梁希泉）（Ｉ...     

Levi Form of CR Submanifolds of Maximal CR Dimension of Complex Space forms

Considering the Levi form on CR submanifolds of maximal CR dimension of complex space forms, we prove that on some remarkable real submanifolds of complex projective space the Levi form can never vanish and we determine all such submanifolds in the case when the ambient manifold is a complex Euclidean space. This revised version was published online in June 2006 with corrections to the Cover Date.     

伪黎曼流形中的紧致类空子流形

We first establish a integral inequality for compact maximal space-like subman ifolds in pseudo-Riemannian manifolds Np(n+p). Then, we investigate compact space-like sub manifolds and hupersurfaces with parallel second fundamental form in Np(n+p) and some other ambient spaces. We obtain some distribution theorems for the square norm of the second fundamental form.     

Sasakian流形中反不变极小子流形的稳定性

本文给出了Ｓａｓａｋｉａｎ流形中反不变极小子流形是稳定或不稳定的一个充分条件．     

The Dimension Function of Holomorphic Spaces of a Real Submanifold of an Almost Complex Manifold

Let M be a real submanifold of an almost complex manifold and let be the maximal holomorphic subspace, for each x M. We prove that c:M,c(x)=dim H _x is upper-semicontinuous.     

殆Hermitian流形中的CR子流形

本文研究了殆Kaehler流形中CR子流形的上同调、CR子波形的分布D及其正交补D⊥的维数大于1的时候,近Kaehler流形中每个全脐非平凡的CR子流形一定是全测地的。最后得到：如果M^～是具有H^～B＞0的近Kaehler流形,那么M^～不允许有混合叶层非凡的CR子流形。     

Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex Euclidean space

We treat m-dimensional real submanifolds M of complex space forms ̿M when the maximal holomorphic tangent subspace is (m−1)-dimensional. On these manifolds there exists an almost contact structure F which is naturally induced from the ambient space and in this paper we study the condition h(FX,Y)−h(X,FY) = g(FX,Y)η, η∊ T⊥(M), on the structure F and on the second fundamental form h of these submanifolds. Especially when the ambient space ̿M is a complex Euclidean space, we obtain a complete classification of submanifolds M which satisfy these conditions.Mathematics Subject Classifications (2000): 53C15, 53C40, 53B20.     

关于局部对称伪黎曼流形中的2-调和类空子流形

研究局部对称伪黎曼流形中的2-调和类空子流形,得到了这类子流形成为极大的Pinching现象及推广的J．Simons型积分不等式．     

拟常曲率Riemann流形中的伪脐点子流形

研究了拟常曲率Riemann流形中具有平行平均曲率向量的伪脐点子流形,得到了一个Simons型公式．     

On the Manifold of Almost Complex Structures

Let (M, g ₀) be a smooth closed Riemannian manifold of even dimension 2_n admitting an almost complex structure. It is shown that the space of all almost complex structures on M determining the same orientation as the one determined by a fixed almost complex structure J ₀ is a smooth locally trivial fiber bundle over the space of almost complex structures orthogonal with respect to g ₀ and determining the same orientation as J ₀.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 66–71.Original Russian Text Copyright © 2005 by N. A. Daurtseva.     

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.     

复射影空间的等参子流形

本文给出了复射影空间Ｐ＿ｎ（Ｃ）上的等参映射定义，并证明了等参映射ｆ在Ｈｏｐｆ主丛π：Ｓ￣（２ｎ＋１）→Ｐ＿ｎ（Ｃ）下的水平提升为Ｓ￣（２ｎ＋１）的等参映射。同时，利用对称空间的表示给出了Ｐ＿ｎ（Ｃ）上等参子流形的例子．     

关于四元射影空间的正曲率全复子流形

本文讨论四元射影空间的全复子流形，证明了四元射影空间的正截面曲率紧致全复子流形一定是全测地的。     

Certain CR submanifolds of maximal CR dimension of complex space forms

We study m-dimensional real submanifolds with (m−1)-dimensional maximal holomorphic tangent subspace in complex space forms. On such a manifold there exists an almost contact structure which is naturally induced from the ambient space and in this paper we study the anti-commutative condition of the almost contact structure and the second fundamental form of these submanifolds and we characterize certain model spaces in complex space forms.     

复射影空间的2-调和全实子流形

研究了复射影空间中2-调和全实子流形,得到了这类子流形的一个积分公式,讨论了伪脐条件下的情形,通过计算第二基本形式模长平方的Laplacian得到一个刚性定理.     

The Minimal Submanifolds in a Locally Symmetric and Conformally Flat Riemannian Manifold

In this paper,we extend two important theorem in[1],[2]to the minimal submanifolds in aLocally symmetric and conformally flat Riemannian mainfold N~(+p).When N~(+p)is a space S_(1)~(+p) of constantcurvature,our theorems reduce to the theorems of[1],[2].