從高斯-波涅與黎曼-羅赫定理看指標定理

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張育展(Yu-Chan Chang) 查詢紙本館藏

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論文名稱

從高斯-波涅與黎曼-羅赫定理看指標定理
(The Index Theorem from Gauss-Bonnet and Riemann-Roch Theorem)

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摘要(中)

本論文主要證明二個在幾何上重要的定理：Gauss-Bonnet定理與Riemann-Roch定理，且與指標定理做一個連結。第一章主要是用陳省身在1943年發表的”內蘊”手法來證明二維流型上的Gauss-Bonnet定理。第二章主要是介紹古典的Riemann-
Roch定理，以三種不同的形式給出，並在第三章證明第三種上同調化的形式。第四章是藉由計算二個橢圓算子的指標得到流形上的拓樸不變量，此為Atiyah-
Singer指標定理。

摘要(英)

In this thesis, we prove two important theorems in geometry. In chapter one, we state the Gauss-Bonnet theorem on even dimensional manifold and give the detail of the proof of two dimensional case. The proof is based on the paper "A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifold", published by S.S. Chern in 1943. A little history of this theorem is included. Chapter two and three mainly focus on Riemann-Roch theorem on one-dimensional complex manifold, Riemann surface. We establish some basics on Riemann surface in chapter two, such as divisors, holomorphic line bundles, sheaves and cohomology on sheaves, also Hodge theorem in the end of this chapter. The proof of Riemann-Roch is in the chapter three. In chapter four, we show a theorem by calculating two analytic indices of two operators, which give us Gauss-Bonnet and Riemann-Roch theorem. This theorem is the Atiyah-Singer index theorem, proved by Atiyah and Singer in 1963.

關鍵字(中)

★ 高斯-波涅
★ 指標定理
★ 黎曼-羅赫

關鍵字(英)

★ index theorem
★ Riemann-Roch
★ Gauss-Bonnet

論文目次

1 Gauss-Bonnet Theorem 1
1.1 History of Gauss-Bonnet Theorem . . . . . . . . . . . . . . . . . 1
1.2 Fundamental of Riemannian Geometry . . . . . . . . . . . . . . . 2
1.3 Intrinsic Proof for Two-Manifold . . . . . . . . . . . . . . . . . . 4
2 Riemann-Roch Theorem 7
2.1 Riemann-Roch Theorem . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Holomorphic Line Bundles and Sheaves . . . . . . . . . . . . . . 8
2.3 Cohomology On Sheaves . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Hodge Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Proof of Riemann-Roch Theorem 16
3.1 Serre's Duality Theorem . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Proof of Riemann-Roch Theorem . . . . . . . . . . . . . . . . . . 17
4 Atiyah-Singer Index Theorem 20
4.1 Atiyah-Singer Index Theorem . . . . . . . . . . . . . . . . . . . . 20
4.2 Two Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Reference 23

參考文獻

[1] S. S. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for
closed Riemannian manifolds, Annals of Math. 45(1944), 747-752
[2] S. S. Chern, On the curvatura integra in a Riemannian manifold, Annal
of Math., 46(1945), 674-684
[3] K. Kodaira, J. Morrow, Complex manifolds, Holt, Rinehart, and Winston,
New York, 1971
[4] Do Carmo, Di®erential geometry of curves and surfaces, Prentice Hall,
1976, ISBN:0-13-212589-7
[5] P.A. Gri±ths, J. Harris, Principles of algebraic geometry, Wiley, 1978,
ISBN:0-471-05059-8
[6] Lars V. Ahlfors, Complex analysis, third edition, McGraw-Hill, 1979,
ISBN:0-07-085008-9
[7] S. S. Chern, Lectures on di®erential geometry, Chinese version, Lian-Jing
Press, 1990, ISBN:957-08-0296-0
[8] Wu-Hung-Hsi, Introdution to compact Riemann surface, Chinese version,
Lian-Jian Press, 1990, ISBN:957-08-0297-9
[9] Wu Hung-Hsi, Chen Wei-Huan, Topics in Riemannian geometry, Beijing
Universty Press, 1993, ISBN:7-301-02081-3
[10] Caniel Henry Gottlieb, All the way with Gauss-Bonnet and the sociology
of mathematics, The American Mathematical Monthly, Vol. 103, No.6
(1996), 457-469
[11] Yu Yan-Lin, The index theorem and the heat equation method, World
Scienti¯c Publishing, 2001, ISBN:9810246102
[12] Mei Jia-Qiang, Lectures on Rieann surface, Nanjing University,
http://math.nju.edu.cn/ meijq/RiemannSurface.pdf
[13] Wu Hung-Hsi, Historical developent of the Gauss-Bonnet theorem, science
in China press, April, 2008, Vol. 51, No.4, 777-784

指導教授

邱鴻麟(Hung-Lin Chiu)

審核日期

2010-7-18

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