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 โ Coordinate Free -- 5.7 Directional Derivatives โ 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 โ Second Attempt -- 6.4 Computations of Curvature Using Coordinates -- 6.5 Theorema Egregium and the Fundamental Theorem of Surfaces -- VII. Geodesics -- 7.1 Introduction โ โStraight Linesโ 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

1. Mathematics
2. Geometry
3. Differential geometry
4. Topology
5. Mathematics
6. Geometry
7. Differential Geometry
8. Topology