欧氏空间中紧致子流形的积分公式(英文)

本文研究了(n+p)维欧氏空间R~(n+p)中n维定向紧致无边子流形Mn的积分公式的问题.首先定义了M~n沿其单位平均曲率向量场ξ方向的高阶平均曲率H~r(0≤r≤n);然后,利用活动标架与外微分法,获得了关于Mn的一个新的积分公式.新公式推广了余维数p=1即超曲面情况下的经典积分公式.     

欧氏空间中隐没子流形的中曲率向量场

在[4]中导出了隐没在欧氏空间R~(m+p)中的紧致、有向的m维子流形M~m的Minkowski公式其中K_(2r)是黎曼流形M~m的Killing不变量,x是子流形M~m(?)R~(m+p)的定位向量,H_(2r)是第r个中曲率向量场。特别是,H_0正是通常的中曲率向量场。     

关于平均曲率为常数的迷向子流形

设S~(n+p)((?))是具常数截面曲率的n+p维完备单连通的Riemann流形,f:M→S~(n+p)((?))是n维连通Riemann流形M到S~(n+p)((?))的等距浸入。若在f(M)的每点,沿任何切方向的法曲率向量都有相等长度,则f(M)称为迷向子流形,本文证明如下的结果: 设M是n维紧致连通的Riemann流形,f:M→S~(n+p)((?))是迷向浸入,使得f(M)具常数平均曲率H。若M的截面曲率处处不小于1/2(H~2+(?)),则f(M)是全脐点的。     

局部对称黎曼流形中具有平行平均曲率向量的子流形

设Nn+p是截面曲率KN满足的n+p维局部对称完备黎曼流形,p≥2.M是Nn+p的具有平行平均曲率向量的n维紧致子流形.本文讨论了这类子流形关于第二基本形式模长平方的积分不等式及其Pinching问题.     

常曲率空间中具正Ricci曲率的子流形

设S~(n+p)(1)是一单位球面,M~n是浸入S~(n+p)(1)的具有非零平行平均曲率向量的n维紧致子流形.证明了当n≥4,p≥2时,如果M~n的Ricci曲率不小于(n-2)(1+H~2),则M~n是全脐的或者M~n的Ricci曲率等于(n-2)(1+H~2),进而M~n的几何分类被完全给出.     

关于伪脐子流形的一个整体定理

设 M~n 是截面曲率为 c 的(n+p)维黎曼空间 M~(n+p)(c)中 n 维子流形。如在 M~n 上存在函数λ使得:〈h(x,y),H〉=λ〈x,y〉成立,其中λ=H~2,则称 M~n是 M~(n+p)(c)的伪脐子流形。本文得到常曲率空间中紧致伪脐子流形的一个整体定理(定理2.1)。     

关于CP~n的全实n维极小子流形的注记

设CP~n是具有Fubini-Study度量的复n维射影空间,它的常全纯截面曲率c=4。M是CP~n的全实子流形,即p∈M,总有JT_p(W)⊥T_p(M),这里J表示CP~n的殆复结构。又若M的平均曲率向量为零,则称M为全实极小子流形。 沈一兵最近证明了:     

单位球面中具平行单位平均曲率向量的子流形

设M是n-维闭黎曼流形,等距浸入(n+p)-维单位球空间S^n+p,具有平行的单位平均曲率向量。若S≤min{2n/3,2(n-1)^1/2},其中S是M的第二基本形式长度的平方,则M是S^n+p的一个(n+1)-维全测地子流形Sⁿ⁺¹中的超曲面。     

Bochner-Kaehler流形的Kaehler子流形

Bochner-Kaehler流形指Bochner曲率张量消失的Kaehler流形。常全纯截面曲率流形是它的特例。本文得到下面结果: 定理 设M是复n+p维Bochner-Kaehler流形M的复n(≥2)维紧Kaehler子流形。若M每点的所有截面曲率都大于M在该点的全纯截面曲率的上确界的1/8。则M是全测地的。 当M是复射空间CP~(n+p)时,这就是Ros A.和Verstraelen L.证明的K.Ogine猜测。郭孝英、沈一兵最近推广到局部对称的Bochner-Kaehler流行M,(科学通报1987年第2期)。     

由曲率函数和外力场之差支配的凸超曲面的发展

考虑由曲率函数和外力场之差支配的凸超曲面的发展.证明了外力场为常向量场时,初始超曲面的凸性是保持的,且曲率流在有限时间内爆破.对于线性外力场,初始超曲面的凸性保持.而且,若线性常数为负数,则曲率流在有限时间内收敛到一点;若线性常数为正数且初始曲率小于某一与外力场有关的常数,则曲率流光滑地存在于任意有限时间区间,并发散到无穷;若线性常数为正数且初始曲率大于某一与外力场有关的常数,则曲率流在有限时间内爆破.     

Estimates of stability of characterization of additive types of distributions

An estimate of stability of characterization of distribution types is obtained for the case of additive types. Under some conditions, the estimate has the order ε^1/3L(ε), where L(ε) is a slowly varying function. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.     

Study of the blow-up of solutions of systems of equations of hydrodynamic type

Mathematical Notes - We study the initial boundary-value problem for three-dimensional systems of equations of pseudoparabolic type. The system is similar to the Oskolkov system, but differs from...     

一类完全正则半群簇子簇格的性质

本文研究了完全正则半群簇的子簇格［Ｖ＋∩ＰＶ，Ｖ＋∩ＰＶ］的某些格运算性质，我们证明了簇Ｖ＋∩ＰＶ可分解为Ｖ与Ｖ＋∩ＰＶ的并；对任意完全正则半群簇Ｗ，有Ｗ∩（Ｖ∨Ｖ＋∩ＰＶ）＝（Ｗ∩Ｖ）∨（Ｗ∩Ｖ＋∩ＰＶ）．特别地，我们得到了等式Ｖ＋∩ＰＶ＝Ｖ成立的若干条件．     

Characterization of types of asymptotic discernibility of families of hypotheses

We give a characterization of the types of asymptotic discernibility of families of hypotheses in the case of hypothetical measures that are not, in general, mutually absolutely continuous. The case when the logarithm of the likelihood ratio admits an asymptotic expansion of the type of an expansion with local asymptotic normality is examined in detail. Examples are studied.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 64–71, 1987.     

Distribution of types of symmetry of tensors of high degree

The asymptotic distribution of tensors of degree N in symmetry types is studied in this paper.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 155, pp. 181–186, 1986.     

Preservation of equivalence of derivations under reduction of depth of formulas

In this paper we consider derivations in the (&, )-fragment of the intuitionistic propositional calculus. As is known, replacement of any occurrence of a formula [F] in a sequent S by an occurrence of the formula [p], where p is a new propositional variable, with the simultaneous addition to the antecedent of the formula F p or p F depending on the sign of the occurrence of F in S, leaves the derivability unchanged. We give a proof of the fact that the natural extension of this transformation to derivations preserves the relation of equivalence of derivations, i.e., transformed derivations are equivalent if and only if the originals are equivalent. (Derivations are considered equivalent if certain of their normal forms coincide, or, what is the same, if their deductive terms coincide.) It is proved that by the iteration of this transformation, each derivation of an arbitrary sequent S can be transformed into a derivation of a sequent S, depending only on S, whose succedent is a variable, and in the antecedent there occur only formulas of the form a,a & b, a b,,(a b) c, a & b c, a (b & c), wherea, b, c are variables. Here if S is balanced, then S is also balanced. (A sequent is called balanced if each variable occurs in it no more than twice.) The familiar correspondence between certain concepts of the theory of categories and concepts of the theory of proofs allows one to assert that there has been constructed a univalent functor, mapping a free Cartesian closed category into itself.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 197–207, 1979.     

Estimate of growth of Lebesgue functions of linear methods of summation

Sufficient conditions are obtained in terms of matrix elements for the boundedness of Lebesgue functions of linear methods of summation of expansions in orthogonal systems of polynomial type.Translated from Matematicheskie Zametki, Vol. 6, No. 3, pp. 277–286, September, 1969.