A First Course in Geometric Topology and Differential Geometry

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I. Topology of Subsets of Euclidean Space.- 1.1 Introduction.- 1.2 Open and Closed Subsets of Sets in ?n.- 1.3 Continuous Maps.- 1.4 Homeomorphisms and Quotient Maps.- 1.5 Connectedness.- 1.6 Compactness.- II. Topological Surfaces.- 2.1 Introduction.- 2.2 Arcs, Disks and 1-spheres.- 2.3 Surfaces in ?n.- 2.4 Surfaces Via Gluing.- 2.5 Properties of Surfaces.- 2.6 Connected Sum and the Classification of Compact Connected Surfaces.- Appendix A2.1 Proof of Theorem 2.4.3 (i).- Appendix A2.2 Proof of Theorem 2.6.1.- III. Simplicial Surfaces.- 3.1 Introduction.- 3.2 Simplices.- 3.3 Simplicial Complexes.- 3.4 Simplicial Surfaces.- 3.5 The Euler Characteristic.- 3.6 Proof of the Classification of Compact Connected Surfaces.- 3.7 Simplicial Curvature and the Simplicial Gauss-Bonnet Theorem.- 3.8 Simplicial Disks and the Brouwer Fixed Point Theorem.- IV. Curves in ?3.- 4.1 Introduction.- 4.2 Smooth Functions.- 4.3 Curves in ?3.- 4.4 Tangent, Normal and Binormal Vectors.- 4.5 Curvature and Torsion.- 4.6 Fundamental Theorem of Curves.- 4.7 Plane Curves.- V. Smooth Surfaces.- 5.1 Introduction.- 5.2 Smooth Surfaces.- 5.3 Examples of Smooth Surfaces.- 5.4 Tangent and Normal Vectors.- 5.5 First Fundamental Form.- 5.6 Directional Derivatives - Coordinate Free.- 5.7 Directional Derivatives - Coordinates.- 5.8 Length and Area.- 5.9 Isometries.- Appendix A5.1 Proof of Proposition 5.3.1.- VI. Curvature of Smooth Surfaces.- 6.1 Introduction and First Attempt.- 6.2 The Weingarten Map and the Second Fundamental Form.- 6.3 Curvature - Second Attempt.- 6.4 Computations of Curvature Using Coordinates.- 6.5 Theorema Egregium and the Fundamental Theorem of Surfaces.- VII. Geodesics.- 7.1 Introduction - "Straight Lines" on Surfaces.- 7.2 Geodesics.- 7.3 Shortest Paths.- VIII. TheGauss-Bonnet Theorem.- 8.1 Introduction.- 8.2 The Exponential Map.- 8.3 Geodesic Polar Coordinates.- 8.4 Proof of the Gauss-Bonnet Theorem.- 8.5 Non-Euclidean Geometry.- Appendix A8.1 Geodesic Convexity.- Appendix A8.2 Geodesic Triangulations.- Further Study.- References.- Hints for Selected Exercises.- Index of Notation.