三维空间中的共形曲线论

本文对单位球面S~3中的曲线在Mbius变换下的性质进行了研究.给S~3中的曲线定义了一套由共形弧长参数、共形曲率和共形挠率组成的Mbius不变参数系统,并证明这个系统决定S~3中所有光滑曲线在Mbius变换下的分类.     

De Sitter空间中类空曲线的对偶曲面的不变量

主要从切触几何的视角考察3维de Sitter空间中类空曲线的第一光锥对偶曲面和双曲对偶曲面的不变量的几何性质.     

On Projective Invariants of Curves

It is shown that for the action of the projective group on curves in the plane leaving a point fixed, the differential invariants depending on several points are derivable from the well known 4-point invariants.     

射影空间中代数曲线与超曲面的一些新的性质

发现了代数曲线的新的不变量一特征数,并得到了Pascal定理的不同于3次曲线的Cllasles定理和高次曲线中的Cayley-Bacharach定理等形式的高次推广.进一步研究了平面代数曲线的一些性质.通过定义m次Pascal超曲面,将Pascal定理推广到n维射影空间的m次超曲面中,证明了n-单纯形上的Pascal点位于一个m次Pascal超曲面的充要条件是其每个2维面上的Pascal点分别位于m次平面Pascal空间的一条代数曲线上.进一步,给出了一定条件下m次Pascal超曲面与m-1次Pascal超曲面之间的内在关系.     

The Connectedness of Space Curve Invariants

In this paper we will give necessary conditions for a Borel-fixed monomial ideal to be the generic initial ideal of a reduced, irreducible, non-degenerate curve in P³.     

一些特殊变换群的联合不变量和联合微分不变量

The theory of moving frames developed by Peter J Olver and M Fels has importaut applications to geometry,classical invariant theory.We will use this theory to classify joint invariants and joint differential invariants of some transformation groups.     

长曲线上的手术和基本不变量

本文给出长曲线上进行插入和推进时的基本不变量的变化公式及其应用．     

Applications of Framed Space Curves

In this paper, we discuss some new properties and applications of the linking number of two space curves and of families of planes tangent to a given curve.     

Real Analytic Fibre Bundles: Some Results on Classification and Homotopies

The classification problem for holomorphic fibre bundles over Stein spaces was solved by H. GRAUERT. Along the same lines, the real coherent analytic case was considered by A. TOGNOLI and V. ANCONA. In this paper we propose a different approach, based on classifying spaces, in order to study the previous problem for real analytic fibre bundles over C -analytic subspaces of R ^m. So, let X be a C -analytic subspace of R ^m and G a compact Lie group. The main result is a characterization of the real analytic G-principal fibre bundles over X for which the analytic and topological equivalence coincide. Moreover, we prove that these bundles can be classified also by means of homotopy classes of analytic maps of X into classifying spaces. Among the others results, are worth recording: a relative approximation theorem of continuous cross sections by analytic ones, a theorem about the equivalence between analytical and topological homotopy between cross sections and a covering homotopy theorem.     

Homotopies of Central Quasigroups

The article analyzes homotopies between central quasigroups, and their groups of autotopies. In particular, the cycle types of autotopies of central quasigroups and other group isotopes of prime order are identified.     

A Class Curves with Some Geometric and Physical Properties in a Higher Dimensional Space

In this paper, we have considered a class curves with some geometric properties in a higher dimensional space and obtained the differential equation of such a class curves, which are called the hyperbolas. We have considered also hyperbola-preserving conformal transformation and the relevant physical sense. And therefore obtained other invariant properties under the illustrious concircular transformation.     

Shape-Preserving Approximation by Space Curves

This paper describes a new method for the construction of C ² shape-preserving curves which approximate an ordered set of data in R ³. The curves are obtained using the variable degree polynomial spline spaces recently described in [5].     

某些三维流形不变量的计算

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Quasi-Isomorphism and Some Quasi-Isomorphic Invariants of QTAG-Modules

This article has been retracted.     

A Tale of Three Homotopies

For a Koszul operad \(\mathcal {P}\), there are several existing approaches to the notion of a homotopy between homotopy morphisms of homotopy \(\mathcal {P}\)-algebras. Some of those approaches are known to give rise to the same notions. We exhibit the missing links between those notions, thus putting them all into the same framework. The main nontrivial ingredient in establishing this relationship is the homotopy transfer theorem for homotopy cooperads due to Drummond-Cole and Vallette.     

几何教学中讲授伪欧氏空间中曲线几何性质探究

曲线,曲面理论是古典微分几何教学中的主要研究对象.然而在古典微分几何的教学中,学生往往只是知道如何解题,不知道微分几何学的主要研究工具,以至于不会运用微分几何解决后继课程中的问题.因此在微分几何的教学中有必要增加一些伪欧氏空间中曲线理论.首先讨论在教学中的一个非常重要曲线理论研究工具-费雷内标架,其次运用该标架讨论在四维伪欧氏空间中斜螺线的一些几何性质,最后通过横截性原理与开折理论,结合微分几何基础给出了由偏零斜螺线生成的密切超曲面的局部几何性质.     

Principal Normal Indicatrices of Closed Space Curves

A theorem due to J. Weiner, which is also proven by B. Solomon, implies that a principal normal indicatrix of a closed space curve with nonvanishing curvature has integrated geodesic curvature zero and contains no subarc with integrated geodesic curvature . We prove that the inverse problem always has solutions if one allows zero and negative curvature of space curves and explain why this not is true if nonvanishing curvature is required. This answers affirmatively an open question asked by W. Fenchel in 1950 under the above assumptions but in general this question is found to be answered to the negative.