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Lectures on Surfaces



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Lectures on Surfaces almost everything you wanted to know about them Anatole Katok Vaughn Climenhaga Department of Mathematics Pennsylvania State University University Park Pennsylvania 16802 E mail address katok a math psu edu Department of Mathematics Pennsylvania State University University Park Pennsylvania 16802 E mail address climenha math psu edu 2000 Mathematics Subject Classification Primary 51 01 53 01 57N05 Secondary 53A05 57R05 Key words and phrases surfaces geometry topology Euler characteristic MASS Topographic data for illustration of earth in Figure 1 12 is taken from the National Oceanic and Atmospheric Administration s ETOPO2v2 data set available at www ngdc noaa gov Contents Preface Chapter 1 xi Various Ways of Representing Surfaces and Basic Examples Lecture 1 a First examples b Equations vs other methods c Planar models d Projective plane and flat torus as factor spaces 1 1 1 4 8 9 Lecture 2 a Equations for surfaces and local coordinates b Other ways of introducing local coordinates c Parametric representations d Metrics on surfaces 11 11 14 16 17 Lecture 3 a More about the Mo bius strip and projective plane b A first glance at geodesics c Parametric representations of curves d Difficulties with representation by embedding e Regularity conditions for parametrically defined surfaces 18 18 20 22 24 Lecture 4 28 27 v vi Contents a Remarks on metric spaces and topology b Homeomorphisms and isometries c Other notions of dimension d Geodesics 28 31 32 33 Lecture 5 a Isometries of the Euclidean plane b Isometries of the sphere and the elliptic plane 34 34 38 Lecture 6 a Classification of isometries of the sphere and the elliptic plane b Area of a spherical triangle 39 Lecture 7 a Spaces with lots of isometries b Symmetric spaces c Remarks concerning direct products 43 43 45 47 Chapter 2 Combinatorial Structure and Topological Classification of Surfaces 39 41 51 Lecture 8 a Topology and combinatorial structure on surfaces b Triangulation c Euler



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