曲面幾何學(xué)

前言

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.

內(nèi)容概要

　　《曲面幾何學(xué)》揭示了幾何和拓?fù)渲g的相互關(guān)系，為廣大讀者介紹了現(xiàn)代幾何的基本概況。書(shū)的開(kāi)始介紹了三種簡(jiǎn)單的面，歐幾里得面、球面和雙曲平面。運(yùn)用等距同構(gòu)群的有效機(jī)理，并且將這些原理延伸到常曲率的所有可以用合適的同構(gòu)方法獲得的曲面。緊接著主要是從拓?fù)浜腿赫摰挠^點(diǎn)出發(fā)，講述一些歐幾里得曲面和球面的分類(lèi)，較為詳細(xì)地討論了一些有雙曲曲面。由于常曲率曲面理論和現(xiàn)代數(shù)學(xué)有很大的聯(lián)系，該書(shū)是一本理想的學(xué)習(xí)幾何的入門(mén)教程，用最簡(jiǎn)單易行的方法介紹了曲率、群作用和覆蓋面。這些理論融合了許多經(jīng)典的概念，如，復(fù)分析、微分幾何、拓?fù)?、組合群論和比較熱門(mén)的分形幾何和弦理論?！肚鎺缀螌W(xué)》內(nèi)容自成體系，在預(yù)備知識(shí)部分包括一些線性代數(shù)、微積分、基本群論和基本拓?fù)洹?/pre>


作者簡(jiǎn)介
作者：（澳大利亞）史迪威（John Stillwell）


書(shū)籍目錄
PrefaceChapter 1. The Euclidean PlaneChapter 2. Euclidean SurfacesChapter 3. The SphereChapter 4. The Hyperbolic PlaneChapter 5. Hyperbolic SurfacesChapter 6. Paths and GeodesicsChapter 7. Planar and Spherical TessellationsChapter 8. Tessellations of Compact SurfacesReferencesIndex


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