A First Course in Geometric Topology and Differential Geometry 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. The Gauss 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.