On Ricci curvature ofC-totally real submanifolds in Sasakian space forms

LetM ⁿ be a Riemanniann-manifold. Denote byS(p) and Ric(p) the Ricci tensor and the maximum Ricci curvature onM _n, respectively. In this paper we prove that everyC-totally real submanifold of a Sasakian space formM ^2m+1(c) satisfies , whereH ² andg are the square mean curvature function and metric tensor onM ⁿ, respectively. The equality holds identically if and only if eitherM ⁿ is totally geodesic submanifold or n = 2 andM ⁿ is totally umbilical submanifold. Also we show that if aC-totally real submanifoldM ⁿ ofM ²ⁿ⁺¹ (c) satisfies identically, then it is minimal.     

3—Sasakian流形的3—C—全实子流形

引入并研究了３－Ｓａｓａｋｉａｎ流形的３－Ｃ－全实子流形，借助了第二基本形式的长度，给出了单位球面Ｓ＾４ｍ＋３－Ｃ－全实子流形的一个Ｐｉｎｃｈｉｎｇ定理。     

Totally Real Submanifolds in a Quaternion Space Form

In this paper, we prove a theorem for n-dimensional totally real minimal submanifold immersed in quaternion space form.     

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.     

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.     

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。     

Some remarks on pseudo-umbilical totally real submanifolds in a complex projective space

This paper studies the relationship between the pseudo-umbilical totally real submanifolds and the minimal totally real submanifolds in a complex projective space. Two theo- rems which claim that some types of pseudo-umbilical totally real submanifolds must be minimal submanifolds are proved.     

On Jordan ideals and submodules: Algebraic and analytic aspects

Let

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be an algebra, and let X be an arbitrary

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-bimodule. A linear space Y ? X is called a Jordan

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-submodule if Ay + yA ∈ Y for all A ∈

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and y ∈ Y. (For X =

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, this coincides with the notion of a Jordan ideal.) We study conditions under which Jordan submodules are subbimodules. General criteria are given in the purely algebraic situation as well as for the case of Banach bimodules over Banach algebras. We also consider symmetrically normed Jordan submodules over C*-algebras. It turns out that there exist C*-algebras in which not all Jordan ideals are ideals.

     

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.     

四元数射影空间的全实伪脐子流形

给出了四元数射影空间的紧致全实伪脐子流形的关于第二基本形式长度的一个Ｐｉｎｃｈｉｎｇ定理．     

Codimension reduction and second fundamental form of CR submanifolds in complex space forms

Considering n-dimensional real submanifolds M of a complex space form which are CR submanifolds of CR dimension , we study the condition h(FX,Y)+h(X,FY)=0 on the structure tensor F naturally induced from the almost complex structure J of the ambient manifold and on the second fundamental form h of submanifolds M.     

Kaplansky Density and Kadison Transitivity Theorems for Irreducible Representations of Real C＊-Algebras

We prove analogs of the Kaplansky Density Theorem and the Kadison Transitivity Theorem for irreducible representations of a real C＊-algebra on a real Hilbert space. Specifically, if a C＊-algebra is acting irreducibly on a real Hilbert space, then the Hilbert space has either a real, complex, or quaternionic structure with respect to which the density and transitivity theorems hold.     

The Eta Invariant and the Real Connective K-Theory of the Classifying Space for Quaternion Groups

We express the real connective K-theory groups o_4k–1(B Q ) ofthe quaternion group Q of order = 2 ^j 8 in terms of therepresentation theory of Q by showing o_4k–1(B Q ) = Sp(S ^4k+3/Q )where is any fixed point free representation of Q in U(2k + 2).     

p-Banach Spaces and p-Totally Convex Spaces. II

We show that the results about the set S : ={ [0, 1] ^{1 / p} x + (1 – )^{1 / p} z ^{1 / p} y + (1 – )^{1 / p} z}, where x, y, z elements of a p-absolutely convex space D and `' is a congruence relation on D are the best possible. Finally, we give an explicit construction of the left adjoint of the comparison functor Ô _p : B an _p T C _p (resp. Ô _{p, fin} : V ec _p A C _p ).     

Computation of generalized real radicals of polynomial ideals

For an ordered field (K,T) and an idealI of the polynomial ring , the construction of the generalized real radical ofI is investigated. When (K,T) satisfies some computational requirements, a method of computing is presented. Project supported by the National Natural Science Foundation of China (Grant No. 19661002) and the Climbing Project.     

de Sitter空间中具平行平均曲率向量的完备类空子流形

本文证明了ｄｅＳｉｔｔｅｒ空间中具平行平均曲率向量的完备类空子流形在Ｈ２〉Ｃ时其第二基本形式模长平方是上有界的，从而推广了Ｕ－ＨａｎｇＫＩ＾（４）及Ｑ．Ｍ．Ｃｈｅｎｇ＾（３）中的结果。     

Ricci Curvature of Certain Submanifolds in Kenmotsu Space Forms

In this paper, we obtain some sharp inequalities between the Ricci cur- vature and the squared mean curvature for bi-slant and semi-slant submanifolds in Kenmotsu space forms. Estimates of the scalar curvature and the k-Ricci curvature, in terms of the squared mean curvature, are also proved respectively.     

Real <Emphasis Type="Italic">AW</Emphasis><Superscript>*</Superscript>-Algebras of type I

Let R be a real AW ^*-algebra, and suppose that its complexification M = R + iR is also a (complex) AW ^*-algebra. We prove that R is of type I if and only if so is M.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 79–81, 2004Original Russian Text Copyright © by Sh. A. Ayupov