LP—Sasaki流形的某些无穷小变换

M.Okumura曾证明了Sasaki流形中保持曲率的无穷小变换必定是无穷小等距变换。K.Matsumoto 在[2]中对于 P-Sasaki 流形讨论了同样的无穷小变换。得到的主要结果为定理 A 在满足φ~2(n-1)~2≠0的 P-Sasaki 流形中,每个保持曲率的无穷小变换必定是无穷小自同构变换。本文将在更为广泛的 LP-Sasaki 流形中讨论这同一问题,主要证明如下定理:     

余辛流形的半不变子流形的Ricci曲率

本文研究了余辛流形的半不变子流形,得到了这类子流形的Ricci曲率与平均曲率平方之间的—个不等式,并讨论了等式成立的充分必要条件．     

关于3维负曲率流形的拓扑型

本文的主要定理是:对每一个正数V,存在一个D>0,如果一个3维黎曼流形M的体积小于V,截面曲率在-1和0之间,而且直径大于D,那么M允许一个双曲结构。     

拟常曲率流形中保持曲率的无穷小变换

1 如所知,在一个 Riemann 流形中,若由′σ~α=σ~α+v~α(σ)dt (1)确定的无穷小变换满足(?)(v)a_(λμ)=2(?)a_(λμ) (2)式中 a_λ是度量张量,(?)是某纯量函数,(?)(v)是关于无穷小变换 v 的李导数,则(1)称为无穷小共形变换,而向量场 v 称为共形 Killing 向量场。如果(?)=const,则称 v 为无穷小位似变换.特别,当(?)=0时,(1)成为无穷小等距变换.在这个情形下,(2)化为 Kil-     

具有平行平均曲率向量场的子流形

本文讨论了具有平行平均曲率向量场的某一类子流形，得到了这类子流形成为全脐点子流形及其余维数减小的充分条件。     

MANIFOLDS WITH POINTWISE PINCHED RICCI CURVATURE

In this article, the author proves a compactness result about Riemannian manifolds with an arbitrary pointwisely pinched Ricci curvature tensor.     

Curvature Computations of 2-Manifolds in $\mathbb{R}^k$

In this paper, we provide simple and explicit formulas for computing Riemannian curvatures, mean curvature vectors, principal curvatures and principal directions for a 2-dimensional Riemannian manifold embedded in IR^k with k ≥ 3.     

正曲率子流形的几何刚性定理

§1Introduction LetMnbeann-dimensionalcompactRiemannianmanifoldisometricallyimmersedinto an(n+p)-dimentionalcompleteandsimplyconnectedRiemannianmanifoldFn+p(c)with constantcurvaturec.DenotebyKMandHthesectionalcurvatureandmeancurvatureofM respectively.In[10],Yauprovedthefollowingstrikingresult.TheoremA.LetMnbeann-dimensionalorientedcompactminimalsubmanifoldin Sn+p(1).IfthesectionalcurvatureofMisnotlessthanp-12p-1,thenMiseitherthetotally geodesicsphere,thestandardimmersionoftheproductoftw…     

具有相对迷向平均Landsberg曲率的流形的一个整体刚性定理

讨论了具有相对迷向平均Landsberg曲率的度量的一些几何性质.证明了任一闭的具有负旗曲率与相对迷向平均Landsberg曲率的流形一定是Riemann流形.     

Hermite流形上的第一或第二k-Ricci曲率

本文研究具有非正或非负的第一和第二k-Ricci曲率的基本性质.首先获得它们与Ricci曲率及全纯截面曲率之间的不等式,然后给出具有正第一或第二k-Ricci曲率的紧致Hermite流形上的消没定理.特别地,本文证明,如果紧致K¨ahler流形上具有正的第二k-Ricci曲率(k2)的Hermite度量,则它一定是射影流形.     

Totally Umbilical Subrnanifolds in a Locally Symmetric Manifold

In this paper we obtain some formulas for totally umbilical submanifolds in a localiy symmetric manifold, and dcrivc some local rcsults on the submanifolds from these formulas.     

Integral Formulas for Submanifolds in Enclidean Space and Their Applications to Uniqueness Theorem

In this paper, the author derives some integral formulas for a pair of submanifolds in Euclidean Space E^n+p, and applies these formulas to generalize the Christoffel theorem and the Hilbert Liebmann-Hsiung theorem.     

A formula for the non-integer powers of the laplacian

We provide an elementary formula for the non-integer powers of the Laplace operator in the Euclidean spaces. Such formulas are helpful in establishing the elliptic nature of some pseudo-differential operators arising from the study of energies of knotted loops in space, and possibly of embedded submanifolds in a Riemannian manifold. Supported in part by an NSF grant.     

INTEGRAL FORMULAS FOR ANTI-INVARIANT SUBMANIFOLDS OF A SASAKIAN SPACE FORM

This paper obtaines the integral formulas of anti-invariant minimal submanifolds of a Sasakian space form, and then applies them to spectral geometry.     

On a Generalization of Zaslavsky’s Theorem for Hyperplane Arrangements

We define arrangements of codimension-1 submanifolds in a smooth manifold which generalize arrangements of hyperplanes. When these submanifolds are removed the manifold breaks up into regions, each of which is homeomorphic to an open disc. The aim of this paper is to derive formulas that count the number of regions formed by such an arrangement. We achieve this aim by generalizing Zaslavsky’s theorem to this setting. We show that this number is determined by the combinatorics of the intersections of these submanifolds.     

Lie symmetries and CR geometry

This paper is divided into three parts. Part I develops a general, new theory (inspired by modern CR geometry) of Lie symmetries of completely integrable pde systems, viewed from their associated submanifolds of solutions. Part II constructs general combinatorial formulas for the prolongations of vector fields to jet spaces. Part III explicitly characterizes the flatness of some systems of the second order. The results presented here are original and were not published elsewhere; most formulas of Parts II and III were verified by means of Maple Release 7. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 47, Complex Analysis, 2007.     

ON SPACELIKE AUSTERE SUBMANIFOLDS IN PSEUDO-EUCLIDEAN SPACE

In this article, we construct some spacelike austere submanifolds in pseduoEuclidean spaces. We also get some indefinite special Lagrangian submanifolds by constructing twisted normal bundle of spacelike austere submanifolds in pseduo-Euclidean spaces.     

On numbers not of the form <Emphasis Type=”Italic”>n</Emphasis>-ω(<Emphasis Type=”Italic”>n</Emphasis>)

Summary We introduce a new class of lightlike submanifolds, namely, Screen Cauchy Riemann (SCR) lightlike submanifolds of indefinite Kaehler manifolds. Contrary to CR-lightlike submanifolds, we show that SCR-lightlike submanifolds include invariant (complex) and screen real subcases of lightlike submanifolds. We study some properties of proper totally umbilical SCR-lightlike submanifolds, their invariant (complex) and screen real subcases.     

Screen Cauchy Riemann lightlike submanifolds

Summary We introduce a new class of lightlike submanifolds, namely, Screen Cauchy Riemann (SCR) lightlike submanifolds of indefinite Kaehler manifolds. Contrary to CR-lightlike submanifolds, we show that SCR-lightlike submanifolds include invariant (complex) and screen real subcases of lightlike submanifolds. We study some properties of proper totally umbilical SCR-lightlike submanifolds, their invariant (complex) and screen real subcases.     

Proper biharmonic submanifolds in a sphere

In this paper, we obtain a constraint of the mean curvature for proper biharmonic submanifolds in a sphere. We give some characterizations of some proper biharmonic submanifolds with parallel mean curvature vector in a sphere. We also construct some new examples of proper biharmonic submanifolds in a sphere.