本文選題：勒讓德浸入 + 漸屈線��； 參考：《東北師范大學(xué)》2016年博士論文

【摘要】：本文主要研究了半歐氏空間中的光滑曲線和光滑曲面在奇點(diǎn)鄰近的微分幾何.2009年,幾何學(xué)家Saji, Umehara, Yamada在美國數(shù)學(xué)年刊發(fā)表的文章中系統(tǒng)的闡述了曲面在尖楞處的曲率函數(shù)的定義并且解釋了高斯曲率在尖楞和燕尾處的特征.這篇文章是研究子流形在奇點(diǎn)鄰近微分幾何性質(zhì)的一個(gè)里程碑.在這個(gè)時(shí)期,許多數(shù)學(xué)工作者投入到了子流形在奇點(diǎn)鄰近幾何性質(zhì)的研究當(dāng)中.本文首先關(guān)注了雙曲平面上帶有奇點(diǎn)的光滑曲線,給出了奇點(diǎn)鄰近曲率和漸屈線的概念,描述了三種偽球上的漸屈線的不同特征,進(jìn)而研究了曲線的奇點(diǎn)和測地頂點(diǎn)之間的關(guān)系.其次,本文考慮了指標(biāo)為二的四維半歐氏空間中的偽類光曲線和偏類光曲線,研究了它們的光錐高斯曲面的性質(zhì),揭示了它們的類光超曲面的奇點(diǎn)與一些幾何不變量之間的關(guān)系.最后,本文從類光幾何的角度考慮了四維Anti de Sitter空間中類空曲面的拐點(diǎn)和H奇點(diǎn),解決了拐點(diǎn)的分類和識(shí)別問題,探索了拐點(diǎn)和H奇點(diǎn)的關(guān)系。本文共分為四章.第一章引言,主要介紹奇點(diǎn)理論應(yīng)用研究的內(nèi)容,發(fā)展概況和本文的背景,并簡要闡述了全文的研究內(nèi)容和結(jié)構(gòu)安排。第二章主要介紹了光滑曲線和光滑曲面相關(guān)的子流形的微分幾何和奇點(diǎn)理論的一些基本概念和結(jié)論。第三章主要研究了光滑曲線及其生成的子流形在奇點(diǎn)鄰近的微分幾何.簡要介紹了歐氏平面上奇異曲線的幾何性質(zhì),進(jìn)而研究了雙曲平面上的奇異曲線.在奇點(diǎn)處定義了曲率的概念,并進(jìn)一步研究了多重漸屈線和四頂點(diǎn)定理.對于指標(biāo)為二的四維半歐氏空間中偽類光曲線和偏類光曲線,我們應(yīng)用Legendrian奇點(diǎn)理論解決了它們的類光超曲面的奇點(diǎn)分類問題。第四章主要研究了四維Anti de Sitter空間中類空曲面的拐點(diǎn)的識(shí)別問題.我們知道曲面上每一點(diǎn)都對應(yīng)著一個(gè)曲率橢圓,當(dāng)曲率橢圓退化成徑向線段時(shí),對應(yīng)的點(diǎn)被稱為拐點(diǎn).依賴于退化的曲率橢圓與拐點(diǎn)的位置關(guān)系,我們可以將拐點(diǎn)分為三類,即實(shí)型拐點(diǎn)、虛型拐點(diǎn)、平坦型拐點(diǎn).首先,我們用傳統(tǒng)的辦法給出了判斷拐點(diǎn)類別的方法.其次,我們從類光幾何的角度分別揭示了實(shí)型拐點(diǎn)、虛型拐點(diǎn)、平坦型拐點(diǎn)的等價(jià)條件.最后,我們給出平均方向曲線的微分方程,并且指出H奇點(diǎn)是由拐點(diǎn)和穩(wěn)定點(diǎn)構(gòu)成的。
[Abstract]:In this paper, we study the differential geometry of smooth curves and smooth surfaces near singularities in semi-Euclidean spaces. In an article published in the American Journal of Mathematics, the geometric scientist Saji, Umehara, Yamada has systematically expounded the definition of curvature function at the tip of the surface and explained the characteristics of the curvature of the Gao Si at the tip of the corrugated and the swallow-tail. This paper is a milestone in the study of differential geometry of submanifolds near singularities. During this period, many mathematical workers devoted themselves to the study of the geometric properties of submanifolds near singularities. In this paper, we first focus on the smooth curves with singularities on the hyperbolic plane, give the concepts of the adjacent curvature and the involute line of the singularities, and describe the different characteristics of the evolutional lines on the three pseudo spheres. Furthermore, the relationship between the singularity of the curve and the geodesic vertex is studied. Secondly, in this paper, we consider the pseudo-photoluminescence curve and the biased photophore curve in the four-dimensional semi-Euclidean space with index two, and study the properties of their optical cone Gao Si surfaces. The relationship between the singularity of their photonic hypersurfaces and some geometric invariants is revealed. Finally, in this paper, the inflection points and H singularities of space-like surfaces in four-dimensional Anti de Sitter spaces are considered from the point of view of light-like geometry. The problem of classification and recognition of inflection points is solved, and the relationship between inflection points and H singularities is explored. This paper is divided into four chapters. The first chapter introduces the content, development and background of the application of singularity theory, and briefly describes the research content and structure of this paper. In chapter 2, some basic concepts and conclusions of differential geometry and singularity theory of submanifolds related to smooth curves and smooth surfaces are introduced. In chapter 3, we study the differential geometry of smooth curves and their generated submanifolds near singularities. The geometric properties of singular curves on Euclidean plane are briefly introduced, and the singular curves on hyperbolic plane are studied. The concept of curvature at singularities is defined, and the multiple involute and four-vertex theorems are further studied. For pseudo-photophore curves and partial optical-like curves in four-dimensional semi-Euclidean space with index two, we apply Legendrian singularity theory to solve the singularity classification problem of their photohypersurfaces. In chapter 4, the problem of recognizing inflection points of space-like surfaces in four dimensional Anti de Sitter spaces is studied. We know that every point on the surface corresponds to an ellipse of curvature, and when the ellipse of curvature degenerates into a radial segment, the corresponding point is called the inflection point. Depending on the position relationship between the degenerate curvature ellipse and the inflection point, we can divide the inflection point into three categories, that is, the real inflection point, the virtual inflection point and the flat inflection point. First of all, we use the traditional method to determine the inflection point category. Secondly, we reveal the equivalent conditions of real type inflection point, virtual type inflection point and flat type inflection point from the point of view of similar optical geometry. Finally, we give the differential equation of the mean direction curve, and point out that the singularity of H is composed of the inflection point and the stable point.
【學(xué)位授予單位】：東北師范大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O186.1

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