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《麯麵幾何學》揭示瞭幾何和拓撲之間的相互關係，為廣大讀者介紹瞭現代幾何的基本概況。書的開始介紹瞭三種簡單的麵，歐幾裏得麵、球麵和雙麯平麵。運用等距同構群的有效機理，並且將這些原理延伸到常麯率的所有可以用閤適的同構方法獲得的麯麵。緊接著主要是從拓撲和群論的觀點齣發，講述一些歐幾裏得麯麵和球麵的分類，較為詳細地討論瞭一些有雙麯麯麵。由於常麯率麯麵理論和現代數學有很大的聯係，該書是一本理想的學習幾何的入門教程，用最簡單易行的方法介紹瞭麯率、群作用和覆蓋麵。這些理論融閤瞭許多經典的概念，如，復分析、微分幾何、拓撲、組閤群論和比較熱門的分形幾何和弦理論。《麯麵幾何學》內容自成體係，在預備知識部分包括一些綫性代數、微積分、基本群論和基本拓撲。

內頁插圖

目錄

Preface

Chapter 1.The Euclidean Plane
1.1 Approaches to Euclidean Geometry
1.2 Isometries
1.3 Rotations and Reflections
1.4 The Three Reflections Theorem
1.5 Orientation-Reversing Isometries
1.6 Distinctive Features of Euclidean Geometry
1.7 Discussion

Chapter 2.Euclidean Surfaces
2.1 Euclid on Manifolds
2.2 The Cylinder
2.3 The Twisted Cylinder
2.4 The Torus and the Klein Bottle
2.5 Quotient Surfaces
2.6 A Nondiscontinuous Group
2.7 Euclidean Surfaces
2.8 Covering a Surface by the Plane
2.9 The Covering Isometry Group
2.10 Discussion

Chapter 3.The Sphere
3.1 The Sphere S2 in R3
3.2 Rotations
3.3 Stereographic Projection
3.4 Inversion and the Complex Coordinate on the Sphere
3.5 Reflections and Rotations as Complex Functions
3.6 The Antipodal Map and the Elliptic Plane
3.7 Remarks on Groups， Spheres and Projective Spaces
3.8 The Area of a Triangle
3.9 The Regular Polyhedra
3.10 Discussion

Chapter 4.The Hyperbolic Plane
4.1 Negative Curvature and the Half-Plane
4.2 The Half-Plane Model and the Conformal Disc Model
4.3 The Three Reflections Theorem
4.4 Isometries as Complex Fnctions
4.5 Geometric Description of Isometries
4.6 Classification of Isometries
4.7 The Area of a Triangle
4.8 The Projective Disc Model
4.9 Hyperbolic Space
4.10 Discussion

Chapter 5.Hyperbolic Surfaces
5.1 Hyperbolic Surfaces and the Killing-Hopf Theorem
5.2 The Pseudosphere
5.3 The Punctured Sphere
5.4 Dense Lines on the Punctured Sphere
5.5 General Construction of Hyperbolic Surfaces from Polygons
5.6 Geometric Realization of Compact Surfaces
5.7 Completeness of Compact Geometric Surfaces
5.8 Compact Hyperbolic Surfaces
5.9 Discussion

Chapter 6.Paths and Geodesics
6.1 Topological Classification of Surfaces
6.2 Geometric Classification of Surfaces
6.3 Paths and Homotopy
6.4 Lifting Paths and Lifting Homotopies
6.5 The Fundamental Group
6.6 Generators and Relations for the Fundamental Group
6.7 Fundamental Group and Genus
6.8 Closed Geodesic Paths
6.9 Classification of Closed Geodesic Paths
6.10 Discussion

Chapter 7.Planar and Spherical TesseUations
7.1 Symmetric Tessellations
7.2 Conditions for a Polygon to Be a Fundamental Region
7.3 The Triangle Tessellations
7.4 Poincarrs Theorem for Compact Polygons
7.5 Discussion

Chapter 8.Tessellations of Compact Surfaces
8.1 Orbifolds and Desingularizations
8.2 From Desingularization to Symmetric Tessellation
8.3 Desingularizations as (Branched) Coverings
8.4 Some Methods of Desingularization
8.5 Reduction to a Permutation Problem
8.6 Solution of the Permutation Problem
8.7 Discussion

References
Index

前言/序言

　　Geometry used to be the basis ofa mathematical education；today it IS not even a standard undergraduate topic.Much as I deplore this situation，1welcome the opportunity to make a fresh start.Classical geometry is nolonger an adequate basis for mathematics or physics-both of which arebe coming increasingly geometric-and geometry Can no longer be divorced from algebra，topology，and analysis.Students need a geometry of greater scope and the factthattherei Sno room for geometryin the curriculumus-til the third or fourth year at least allows 118 to as8ume some mathematical background.
　　What geometry should be taught？I believe that the geometry of surfaces of constant curvature is an ideal choice，for the following reasons：
　　1.It is basically simple and traditional.We are not forgetting euclideangeometry but extending it enough to be interesting and useful.Theextensions offer the simplest possible introduction to fundamentals ofmodem geometry：curvature.group actions，and covering 8paces.
　　2.The prerequisites are modest and standard.A little linear algebra fmostly 2×2 matrices），calculus as far as hyperbolic functions，basic group theory（subgroups and cosets），and basic topology（open，closed，and compact sets）.
　　3.（Most important.）The theory of surfaces of constant curvature has maximal connectivity with the rest of mathematics.Such surfaces model the variants of euclidean geometry obtained by changing the parallel axiom；they are also projective geometries，Riemann surfaces， and complex algebraic curves.They realize all of the topological types of compact two-dimensional manifolds.Historically，they are the 80urce of the main concepts of complex analysis，differential geometry，topology，and combinatorial group theory.（They axe also the sOuroe of some hot research topics of the moment，such as[ractal geometry and string theory.）
　　The only problem with such a deep and broad topic is that it cannot be covered completely by a book of this size.Since.however，this IS the size 0f book I wish to write，I have tried to extend my formal coverage in two wavs：by exercises and by informal discussions.

數學經典教材：麯麵幾何學（影印版） [Geometry of Surfaces] 下載 mobi epub pdf txt 電子書

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經典教材。

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為進一步研究幾何奠定基礎

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經典的微分幾何內容。

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