E^3中曲面的无穷小Bonnet Ⅱ—等距

本文研究欧氏空间E~3中曲面M的无穷小O.BonnetⅡ-等距变形(简称BⅡ-等距)。所谓BⅡ-等距变形是指保持曲面的两主曲率和第Ⅱ基本形式都不变的变形。允许非平凡的这种变形的曲面称为BⅡ曲面。文中按M的Gauss曲率K为零与否(或可展与否)分两种情况讨论。定理1给出非可展曲面为无穷小BⅡ曲面的充要条件:定理2分别对柱面、锥面与切线曲面共三种情况详尽地讨论了可展曲面的无穷小BⅡ-等距变形以及它的自由度。     

R31中混合型曲面保主曲率等距变形

本文研究了Minkowski空间R31曲面的等距变形问题.建立了R31中曲面的共形、等距等概念.推广了O.Bonnet和S.S.Chern关于欧氏空间的结论.对R31出现的新情况--曲面的中曲率梯度类光作了一定探讨,得出的主要结果为:非平坦的、允许保主曲率等距变形的曲面一定不是W-曲面.     

R_1~3中混合型曲面保主曲率等距变形

本文研究了Minkowski空间R_(1)~3曲面的等距变形问题.建立了R_(1)~3中曲面的共形、等距等概念.推广了O.Bonnet和S.S.Chern关于欧氏空间的结论.对R_(1)~3出现的新情况——曲面的中曲率梯度类光作了一定探讨,得出的主要结果为:非平坦的、允许保主曲率等距变形的曲面一定不是W-曲面.     

E~3中曲面无穷小BⅡ──等距的注记

本文继承了文献［１］的讨论，证明了在适当的边界条件下，非可展曲面Ｍ的无穷小ＢⅡ－等距Φ是关于Ｍ的无穷小刚体运动。     

R13中混合型曲面保主曲率等距变形

本文研究了Minkowski空间R₁³曲面的等距变形问题.建立了R₁³中曲面的共形、等距等概念.推广了O.Bonnet和S.S.Chern关于欧氏空间的结论.对R₁³出现的新情况——曲面的中曲率梯度类光作了一定探讨,得出的主要结果为:非平坦的、允许保主曲率等距变形的曲面一定不是W-曲面.     

R~3中正曲率曲面的无穷小变形和几类边值问题

本文讨论了Ｒ３中正曲率曲面的无穷小变形，利用广义解析函数理论，研究了几类有明确几何意义的边值问题的无穷小刚性，其中部分结论是Ｖｅｋｕａ相应结果的推广．     

IR~3中曲面的等距变形和IR~4中曲面的Gauss映射

本文研究了 IR~3中曲面保持平均曲率函数的等距变形,并应用所得结果进一步回答 D.A.Hoffman 和 R.Osserman 提出的关于 IR~4中曲面和到复二次曲面 Q_2的映射问题,改进了 W.Seaman 的结果.     

IR~3中曲面的等距变形和IR~4中曲面的Gauss映射

本文研究了 IR~3中曲面保持平均曲率函数的等距变形,并应用所得结果进一步回答 D.A.Hoffman 和 R.Osserman 提出的关于 IR~4中曲面和到复二次曲面 Q_2的映射问题,改进了 W.Seaman 的结果.     

ON A PAIR OF NON-ISOMETRIC ISOSPECTRAL DOMAINS WITH FRACTAL BOUNDARIES AND THE WEYL-BERRY CONJECTURE

ＯＮＡＰＡＩＲＯＦＮＯＮＩＳＯＭＥＴＲＩＣＩＳＯＳＰＥＣＴＲＡＬＤＯＭＡＩＮＳＷＩＴＨＦＲＡＣＴＡＬＢＯＵＮＤＡＲＩＥＳＡＮＤＴＨＥＷＥＹＬＢＥＲＲＹＣＯＮＪＥＣＴＵＲＥＳＬＥＥＭＡＮ，Ｂ．Ｄ．ＣＨＥＮＨＵＡＭａｎｕｓｃｒｉｐｔｒｅｃ...     

曲面的第Φr—形式及无穷小等距

本考虑光滑曲面片Ｍ上的基本Φｒ－形式及无穷小变形Φ，推广了一些经典的结果。主要有如下两个定理。     

Infinitesimal Isometries Along Curves and Generalized Jacobi Equations

On a Riemannian manifold, a solution of the Killing equation is an infinitesimal isometry. Since the Killing equation is overdetermined, infinitesimal isometries do not exist in general. A completely determined prolongation of the Killing equation is a PDE on the bundle of 1-jets of vector fields. Restricted to a curve, this becomes an ODE that generalizes the Jacobi equation. A solution of this ODE is called an infinitesimal isometry along the curve, which we show to be an infinitesimal rigid variation of the curve. We define Killing transport to be the associated linear isometry between fibers of the bundle along the curve, and show that it is parallel translation for a connection on the bundle related to the Riemannian connection. Restricting to dimension two, we study the holonomy of this connection, prove the Gauss–Bonnet theorem by means of Killing transport, and determine the criteria for local existence of infinitesimal isometries.     

The problem of non-trivial isometries of surfaces preserving principal curvatures

In this paper we deal with the Bonnet problem of determining the surfaces in the Euclidean three dimensional space which can admit at least one nontrival isometry that preserves the principal curvatures(Bonnet surfaces). The problem is considered locally and examined in the general case. The main results are: (a) Necessary and sufficient condition for a surface to be a Bonnet surface is that it admits a special isothermal parameter system. (b) Complete solution of the problem in the class of the isothermic surfaces. Moreover: These results and the methods used provide a new efficient and elegant manner of proving the, already known, fact that all helicoidal surfaces are Bonnet surfaces and determine the already known developable Bonnet surfaces.     

Bonnet Ruled Surfaces

We consider the Bonnet ruled surfaces which admit only one non-trivial isometry that preserves the principal curvatures. We determine the Bonnet ruled surfaces whose generators and orthogonal trajectories form a special net called an A-net.     

Tangential developable surfaces as bonnet surfaces

We study the real Bonnet surfaces which accept one unique nontrivial isometry that preserves the mean curvature, in the three-dimensional Euclidean space. We give a general criterion for these surfaces and use it to determine the tangential developable surfaces of this kind. They are determined implicitly by elliptic integrals of the third kind. Only the tangential developable surfaces of circular helices are explicit examples for which we completely determine the above unique nontrivial isometry. Dedication Dedicated to Siuping Ho for all her invaluable support and encouragement.     

On the problem of isometry of a hypersurface preserving mean curvature

The problem of determining the Bonnet hypersurfaces in R ⁿ⁺¹, for n > 1, is studied here. These hypersurfaces are by definition those that can be isometrically mapped to another hypersurface or to itself (as locus) by at least one nontrivial isometry preserving the mean curvature. The other hypersurface and/or (the locus of) itself is called Bonnet associate of the initial hypersurface. The orthogonal net which is called A-net is special and very important for our study and it is described on a hypersurface. It is proved that, non-minimal hypersurface in R ⁿ⁺¹ with no umbilical points is a Bonnet hypersurface if and only if it has an A-net.     

赋范空间E和l~1(Γ)的单位球面间等距映射的延拓

研究赋范空间E和l~1(Γ)的单位球面之间的等距映射的延拓,得到E和l~1(Γ)的单位球面之间的满等距映射可以延拓为全空间E上的实线性等距算子,从而肯定地回答了相应的Tingley问题.     

S(Hn)上的满数值半径等距

本文研究了矩阵空间到自身的满数值半径等距问题. 利用等距嵌入方法, 获得了自共轭矩阵空间单位球面到自身的满数值半径等距可实线性延拓至全空间上的满数值半径等距, 为Tingley等距延拓问题提供了一种方法.     

Almost isometric flat spaces and perturbation-equivalence

We construct two non-isometric closed subsets of the real line which are almost isometric, and show that any similar example in an Euclidean space is essentially one-dimensional. We then define perturbation-equivalence of almost isometric embeddings, and find a rigid closed subset of the line with an almost isometry onto itself which is not a perturbation of the identity. Finally we show that any almost isometry from an Euclidean space to itself is a perturbation of a sequence of isometries.     

赋β-范空间中单位球面间的等距算子的线性延拓

本文得到了等距映射的线性延拓的一般结果:设E,F是赋范(或β-严格凸赋β-范)线性空间,若V_0:S_1(E)→S_1(F)是等距,且对任意的x,y∈S_1(E),有‖V_0x-|(?)|V_0y‖≤‖x-|(?)|y‖,(?)∈R,则V_0必可延拓到全空间上等距算子(或线性等距算子)。特别,当E,F是赋范线性空间,V_0是满射或F为严格凸空间时,则V_0必可延拓为全空间的线性等距算子,从而推广了文[3～5]中的相应结果。     

非满等距映射的线性延拓

主要研究实赋范空间E和F的单位球面S_1(E)和S_1(F)之间的等距映射的线性延拓问题．得到：若等距映射V_0：S_1(E)→S_1(F)满足一定条件,则V_0可延拓为全空间E上的线性等距映射V：E→F,这是我们首次在非满的情况下考虑Tingley问题．