曲面几何学

图书标准信息

• 作者 [澳]史迪威 著
• 出版社 世界图书出版公司
• 出版时间 2010-01
• 版次 1
• ISBN 9787510005312
• 定价 28.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 216页
• 正文语种 简体中文，英语

【内容简介】

《曲面几何学》揭示了几何和拓扑之间的相互关系，为广大读者介绍了现代几何的基本概况。书的开始介绍了三种简单的面，欧几里得面、球面和双曲平面。运用等距同构群的有效机理，并且将这些原理延伸到常曲率的所有可以用合适的同构方法获得的曲面。紧接着主要是从拓扑和群论的观点出发，讲述一些欧几里得曲面和球面的分类，较为详细地讨论了一些有双曲曲面。由于常曲率曲面理论和现代数学有很大的联系，该书是一本理想的学习几何的入门教程，用最简单易行的方法介绍了曲率、群作用和覆盖面。这些理论融合了许多经典的概念，如，复分析、微分几何、拓扑、组合群论和比较热门的分形几何和弦理论。《曲面几何学》内容自成体系，在预备知识部分包括一些线性代数、微积分、基本群论和基本拓扑。

【目录】

Preface

Chapter1.TheEuclideanPlane
1.1ApproachestoEuclideanGeometry
1.2Isometries
1.3RotationsandReflections
1.4TheThreeReflectionsTheorem
1.5Orientation-ReversingIsometries
1.6DistinctiveFeaturesofEuclideanGeometry
1.7Discussion

Chapter2.EuclideanSurfaces
2.1EuclidonManifolds
2.2TheCylinder
2.3TheTwistedCylinder
2.4TheTorusandtheKleinBottle
2.5QuotientSurfaces
2.6ANondiscontinuousGroup
2.7EuclideanSurfaces
2.8CoveringaSurfacebythePlane
2.9TheCoveringIsometryGroup
2.10Discussion

Chapter3.TheSphere
3.1TheSphereS2inR3
3.2Rotations
3.3StereographicProjection
3.4InversionandtheComplexCoordinateontheSphere
3.5ReflectionsandRotationsasComplexFunctions
3.6TheAntipodalMapandtheEllipticPlane
3.7RemarksonGroups，SpheresandProjectiveSpaces
3.8TheAreaofaTriangle
3.9TheRegularPolyhedra
3.10Discussion

Chapter4.TheHyperbolicPlane
4.1NegativeCurvatureandtheHalf-Plane
4.2TheHalf-PlaneModelandtheConformalDiscModel
4.3TheThreeReflectionsTheorem
4.4IsometriesasComplexFnctions
4.5GeometricDescriptionofIsometries
4.6ClassificationofIsometries
4.7TheAreaofaTriangle
4.8TheProjectiveDiscModel
4.9HyperbolicSpace
4.10Discussion

Chapter5.HyperbolicSurfaces
5.1HyperbolicSurfacesandtheKilling-HopfTheorem
5.2ThePseudosphere
5.3ThePuncturedSphere
5.4DenseLinesonthePuncturedSphere
5.5GeneralConstructionofHyperbolicSurfacesfromPolygons
5.6GeometricRealizationofCompactSurfaces
5.7CompletenessofCompactGeometricSurfaces
5.8CompactHyperbolicSurfaces
5.9Discussion

Chapter6.PathsandGeodesics
6.1TopologicalClassificationofSurfaces
6.2GeometricClassificationofSurfaces
6.3PathsandHomotopy
6.4LiftingPathsandLiftingHomotopies
6.5TheFundamentalGroup
6.6GeneratorsandRelationsfortheFundamentalGroup
6.7FundamentalGroupandGenus
6.8ClosedGeodesicPaths
6.9ClassificationofClosedGeodesicPaths
6.10Discussion

Chapter7.PlanarandSphericalTesseUations
7.1SymmetricTessellations
7.2ConditionsforaPolygontoBeaFundamentalRegion
7.3TheTriangleTessellations
7.4PoincarrsTheoremforCompactPolygons
7.5Discussion

Chapter8.TessellationsofCompactSurfaces
8.1OrbifoldsandDesingularizations
8.2FromDesingularizationtoSymmetricTessellation
8.3Desingularizationsas(Branched)Coverings
8.4SomeMethodsofDesingularization
8.5ReductiontoaPermutationProblem
8.6SolutionofthePermutationProblem
8.7Discussion

References
Index