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。     

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．     

CR Submanifolds of Maximal CR Dimension in a Complex Hopf Manifold

We study CR submanifolds M in a Hopf manifold (C H ^N (), J ₀, g ₀) with the Boothby metric g ₀,of maximal CR dimension. Any such M is a CR manifold ofhypersurface type, although embedded in higher codimension, and itsanti-invariant distribution H(M) is spanned by a unit vectorfield U. We classify the CR submanifolds M for which = –J ₀ Uis parallel in the normal bundle under assumptions on thespectrum of the Weingarten operator a . We show that (1) ifa (U) = (1/2)A (where A is the anti-Lee vector) andM fibres in tori over a CR submanifold of the complex projectivespace, then M lies on the (total space of the) pullback of the Hopf fibration via S C P ^{N – 1}, for some geodesic hypersphere S, and (2) if a (U)= 0 and Spec(a ) = {0, c}, for some c R {0}, then M is locally a Riemannian product of totally geodesicsubmanifolds.     

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.     

Sasakian空间型的C-全实伪脐子流形

本文讨论了Ｓａｓａｋｉａｎ空间型的Ｃ-全实伪脐子流形，给出关于第二基本形式长度的一个Ｐｉｎｃｈｉｎｇ定理．     

Contact CR-Warped Product Submanifolds in Sasakian Space Forms

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. Afterwards, I. Hasegawa and the present author obtained 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. In this paper, we improve the above inequality for contact CR-warped products in Sasakian space forms. Some applications are derived. A classification of contact CR-warped products in spheres, which satisfy the equality case, identically, is given.Mathematics Subject Classifications (2000). 53C40, 53C25.     

A Pinching Theorem for Minimal Submanifolds in Sphere

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＭｂｅａｎｎ－ｄｉｍｅｎｓｉｏｎａｌｃｌｏｓｅｄｍｉｎｉｍａｌｙｉｍｍｅｒｓｅｄｓｕｂｍａｎｉｆｏｌｄｉｎｔｈｅｕｎｉｔｓｐｈｅｒｅＳｎ＋ｐ，Ｓｔｈｅｓｅｑｕｒｅｏｆｔｈｅｌｅｎｇｔｈｏｆｔｈｅｓｅｃｏｎｄｆｕｎｄ...     

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.     

Sasakian空间形式中具有平行平均曲率向量的C-全实子流形

本文讨论了Sasakian空间形式中具有平行平均曲率向量的C-全实子流形,得到了一个Simons型公式并且改进了S.Yamaguchi等的一个结果.     

流形上的Riesz变换

设 Ｍ为一完备 Ｒｉｅｍａｎｎ流形, Ｓｔｒｉｃｈａｒｔｚ Ｒ． Ｓ, Ｌｏｈｏｕｅ Ｎ．, Ｂａｋｒｙ Ｄ．及作者等建立了 Ｍ上 Ｒｉｅｓｚ变换Ｒ的 Ｌ~ｐ（１＜ Ｐ＜ ∞）与弱型（１,１）有界性．本文将用分析的方法对曲率非负的流形建立Ｒ的Ｌ*－有界性．     

Radicals of Skew Polynomial and Skew Laurent Polynomial Rings Over Skew Armendariz Rings

In this note we study radicals of skew polynomial ring R[x; α] and skew Laurent polynomial ring R[x, x ^?1; α], for a skew-Armendariz ring R. In particular, among the other results, we show that for an skew-Armendariz ring R, J(R[x; α]) = N ₀(R[x; α]) = Ni?_*(R)[x; α] and J(R[x, x ^?1; α]) = N ₀(R[x, x ^?1; α]) = Ni?_*(R)[x, x ^?1; α].     

Homotopy connectedness theorems for submanifolds of Sasakian manifolds

The homotopy connectedness theorem for invariant immersions in Sasakian manifolds with nonnegative transversal q-bisectional curvature is proved. Some Barth-Lefschetz type theorems for minimal submanifolds and (k, ?)-saddle submanifolds in Sasakian manifolds with positive transversal q-Ricci curvature are proved by using the weak (?-)asymptotic index. As a corollary, the Frankel type theorem is proved.     

On the characterization of spherical curves in 3-dimensional Sasakian spaces

In this paper, we give the spherical characterization of a regular curve in 3-dimensional Sasakian space. Furthermore the differential equation which expresses the mentioned characterization is solved.     

Lower Bound of the First Eigenvalue for the Laplace Operator on Compact Riemannian Manifold

Let M be a compact m-dimensional Riemannian manifold, let d denote, its diameter, -R(R>O) the lower bound of the Ricci curvature, and λ_1 the first eigerivalue for the Laplacian on M. Then there exists a constant C_m=max{2~(1/m-1),2~(1/2)}, Such thatλ_1≥π~2/d~2·1/(2-(11)/(2π~2))+11/2π~2e~cm、(?)     

Internal Decompositions of Skew Lattices

In the present note we study internal decompositions of skew lattices. This provides further insight into the structure of skew lattices. Special attention is devoted to skew lattices in rings.     

Totally Umbilical Hypersurfaces in a Locally Symmetric Manifold

In this paper we obtain some formulas for totally umbilical hypersurfaces in a locally symmetric manifold, and derive some local results on the hypersurfaces from these formulas.