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陳文豪博士 Gauss-Bonnet Theorem and Its Applications Gauss-Bonne

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陳文豪博士 Gauss-Bonnet Theorem and Its Applications Gauss-Bonne 東 海 大 學 數 學 系 研 究 所 碩 士 論 文 指 導 教 授： 陳 文 豪 博 士 Gauss-Bonnet Theorem and Its Applications Gauss-Bonnet 定理及其應用 研究生：曾柏耕 中華民國一○○年六月 - 摘 要 - Gauss-Bonnet 定理是一個美麗的定理， 它把曲面上的曲率和曲面的尤拉特徵數 做一個連結。 換句話說， Gauss-Bonnet 定理是幾何和拓樸之間的橋樑。 在本論文中， 我們提出 Gauss-Bonnet 定理的發展及証明， 並討論它的一些應用。 例如，龐加萊-霍 普夫指標定理，毛球定理，和代數基本定理。 除此之外，我們還討論 R3 空間中多面體 的離散型 Gauss-Bonnet 定理。 i Abstract Gauss Bonnet theorem is beautiful because it relates the curvature of a surface with its Euler characteristic. It links differential geometry with topology. In this paper, we present some developments on the proof and some applications of Gauss-Bonnet theorem. For example, the Poincaré-Hopf index theorem, the hairy ball theorem, and the fundamental theorem of algebra. Moreover, we discuss the discrete Gauss-Bonnet theorem about a convex polyhedron in R3. ii - 致 謝 辭 - 時間過得很快，在這幾年的求學生涯中，體會很多，也學到了很多。 感謝我的指導教授 陳文豪老師這些日子的教導與照顧，而在擔任老師的小助教這段時間，不管是微積分、高等微 積分、或幾何學，都讓我有了更深的體悟。 另外也要感謝系上其他老師的教誨，謝謝黃皇男老 師 TeX 的教導以及上課分享的人生閱歷，讓我學到很多， 曹景懿老師、陳淑珍老師、胡馨云老 師、和楊智烜老師的鼓勵讓人充滿鬥志。 感謝口試委員曹景懿老師、陳昇國老師、林盈甄老師， 除了細心地審查論文內容之外，並提出了不少寶貴的意見，使本論文更加完善。 還要我的實習學 校的老師們，林孟儒老師、李浚標主任、林詩舫老師、徐大欽老師、江幸紋老師，和宇秀姐等等 的幫忙及鼓勵。 在研究所求學的路途上，除了老師們外，很感謝一路互相彼此打氣的同學及朋友。 謝謝 我的同門：雅淳、宗憲、啟敏、和世華，除了論文上的幫忙，彼此那種團結的感覺更令人覺得窩 心。 謝謝凡雅、岳永、元宏、謹榕、宏銘、范姜、國峰、家銘、大鈞、培瑛、金龍、聖鴻、書齊 的鼓勵，而平時的互相消遣不失為一種生活的樂趣。 也謝謝大學部學弟妹的幫忙，像依蒨、柏 安、淑澐、育如、浚亦、宜慧、瀚嘉、宸慶等等； 還有信元、耀程、應麟、一醇、仕勛、麗貞、 惠華、又琳、雋心、kero蔡、蔣洋、吱吱吱、瑞昌、宗達、雅萍、璧慈的支持， 還有太多太多的 人需要感謝，他們可能用不同的表達方式，但讓我更往前走了一步，使我在東海有了很多回憶。 在這裡特別感謝勞教處，勞作小組長和校內工讀讓我學到了領導和工作的態度，也認識了很多新 朋友。 最後要感謝我的家人們在我從小到大的求學路上，總是無怨尤的一直支持著，讓我能夠全 心全力的朝我的目標前進，謝謝你們，沒有你們，就沒有現在的我。 iii Contents 1 Introduction 1 2 Preliminaries 4 3 Gauss-Bonnet Theorem 9 3.1 The Local Gauss-Bonnet Theorem . 9 3.2 The Global Gauss-Bonnet Theorem . 19 3.3 The Discrete Gauss-Bonnet Theorem . 25 4 The Applications of Gauss-Bonnet Theorem 31 4.1 The Poincaré-Hopf Index Theorem . 31 4.2 The Hairy Ball Theorem . 37 4.3 The Fundamental Theorem of Algebra . 41 5 Conclusion 46 Bibliography 47 iv Chapter 1 Introduction In this paper, we investigate the Gauss-Bonnet theorem in dierential geometry, the discrete Gauss-Bonnet theorem in discrete dierential geometry and its applications. The Gauss-Bonnet theorem is probably the most beautiful and deepest theorem in the dierential geometry of surfaces. The simplest version of the Gauss-Bonnet theorem states that the sum of the interior angles of a triangle in the Euclidean plane equals π. The local Gauss-Bonnet theorem states as follows: Theorem 1.0.1. [18] Suppose R is a simply connected and regular region with simple, closed, piecewise regular, and positively oriented boundary in an oriented regular surface M. If γ = @R with length `(γ) has exterior angles i at the vertices γ(si), i = 1; 2; : : : ; n, then n X k ds + KdA + = 2π: (1.0.1) ˆ g ¨ i @R R i=1 Gauss published his formula in 1827 and deal with geodesic triangles on surfaces, and then Bonnet generalized it in 1848 to any simple connected region enclosed by arbitrary curves. A few years later, the equation (1.0.1) became: KdA = 2πχ(M); (1.0.2) ¨M where M is an oriented compact surface and χ(M) is the Euler characteristic of M. 1 .H÷1 OLONOMYANDTHE GAUSS-BONNET THEOREM 87 Corollary 1.10. The quantity ` g ds KdA k @M C C Z “M k 1 XD is a topological invariant, i.e., does not change as we deform the surface M . In particular, in the event that @M , so the surface M is closed, we have D; The equationCorollary (1.0.2) 1.11. When providesM is a closed, a remarkable oriented surface relation without boundary, between we the have topology of a compact surface and the integral of its curvature.KdA It is2.M striking /: that the total curvature does not D “M change asIt is we very deform interestingthe that the surface,total curvature for example,does not change as as shown we deform in the Figure surface, for 1.0.1. example, as shown in Figure 1.9. In a topology course, one proves that any closed, oriented surface without boundary must F IGURE 1.9 have the topologicalFigure type 1.0.1: of a sphere The ortotal of a g Gaussian-holed torus for curvature some positive is integer unchanged.g. Thus (cf. Exercise 4), the possible Euler characteristics of such a surface are 2, 0, 2, 4, ...; moreover, the integral KdA M determines the topological type of the surface. ’ H. HopfWe found conclude an this application section with a few of applications Gauss-Bonnet of the Gauss-Bonnet theorem Theorem. in 1885. Example 5. Suppose M is a surface of nonpositive Gaussian curvature. Then there cannot be a geodesic Theorem2-gon 1.0.2.R on M[12]that bounds The suma simply of connected the indices region. of For a if dierentiable there were, by Theorem vector 1.6 weeld wouldV havewith isolated 0 KdA 2 .1 2/ > 0; singular points on a compact surface M isD equal toC the Euler characteristic of M. “R which is a contradiction. (Note that the exterior angles must be strictly less than because there is a unique We apply(smooth) the geodesic theorem with a given to nd tangent a direction.) interestingO statement in meteorology. The statement is the hairyExample ball theorem. 6. Suppose M is topologically equivalent to a cylinder and its Gaussian curvature is negative. Then there is at most one simple closed geodesic in M . Note, first, as indicated in Figure 1.10, that if there is a simple closed geodesic ˛, either it must separate M into two unbounded pieces or else it bounds Theorema disk 1.0.3.R, in which[9] There case we would does have not0 exist > aKdA non-vanishing2.R/ 2 continuous, which is a contradiction. tangent vector On led R D D the other hand, suppose there were two. If they don’t intersect, then they bound a cylinder R and we get ’ V on a0 sphere > KdA2. 2.R/ 0, which is a contradiction. If they do intersect, then we we have a geodesic R S D D 2-gon bounding a simply connected region, which cannot happen by Example 5. O ’ Moreover, we introduce the fundamental theorem of algebra. Theorem 1.0.4. [5] Every nonconstant polynomial P (z) 2 C[z] has at least one complex root. In chapter 2, we review some basic denitions and theorems in dierential geometry. The content of this chapter contains the rst fundamental form, tangent spaces, the Gauss map, the second fundamental form, normal curvatures, geodesic curvatures, principal curvatures and the Gaussian curvature. In chapter 3, we present a topical subject that is Gauss-Bonnet theorem. In section 3.1 and 3.2, we need some topological denitions and prove carefully the local and global Gauss-Bonnet theorem. Besides, in section 3.3, we rst discuss a regular complex with a 2 triangulation on a sphere S2. Then we dene the discrete Gaussian curvature of it, and nd the discrete Gauss-Bonnet theorem related to the total curvature. In chapter 4, we investigate the applications of Gauss-Bonnet theorem: the Poincaré- Hopf index theorem, the hairy ball theorem, and the fundamental theorem of algebra. In section 4.1, we list some singular points and their indices with a tangent vector eld, and then we prove the Poincaré-Hopf index theorem. In section 4.2, we construct a tangent vector eld on a sphere S2 to recheck the hairy ball theorem is true. In addition, we nd two tangent vector elds on a torus T . One is a nowhere zero tangent vector elds on a torus T , and the other is not. In section 4.3, we use Gauss-Bonnet theorem to prove the fundamental theorem of algebra. In chapter 5, we nd the generalizations of Gauss-Bonnet theorem and its application. Besides, we have more topics about discrete dierentiable geometry in other papers. 3 Chapter 2 Preliminaries In this chapter, we introduce the Gaussian curvature in dierential geometry. Denition 2.0.1. [2] A subset S of R3 is a surface if for every point p 2 S, there exists an open set U in R2 and an open set W in R3 containing p such that S \W is homeomorphic to U. That is, 9 σ : U ! W \ S s.t. σ is a homeomorphism: We call σ surface patches or parametrizations. A surface S should mean a smooth surface, and a surface patch σ should mean a regular surface patch, i.e., σu × σv 6= 0. We look at the rst fundamental form. It allows us to make measurements on the surface, for instance, lengths of curves, angles of tangent vectors, areas of regions, without referring back to the ambient space R3 where the surface lies. Denition 2.0.2. [2] Let γ(t) = σ(u(t); v(t)) be a curve in a surface patch σ. Then the rst fundamental form of σ is 2 2 2 ds = g11du + 2g12dudv + g22dv ; 4 where g11 :=< σu; σu >; g12 :=< σu; σv >; g22 :=< σv; σv >; and < ; > is an inner product. It is obvious that g12 = g21 :=< σv; σu >. Denition 2.0.3. [2] The tangent space TpS at a point p of a surface S is the set of tangent vectors at p of all curves in S passing through p. Proposition 2.0.4. [2] Let σ : U ! R3 be a patch of a surface S containing a point p 2 S, and let (u; v) be coordinates in U. The tangent space to S at p is the vector subspace of 3 spanned by the vector and .
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