First Course in Geometric Topology and Differential Geometry

"[The author] avoids aimless wandering among the topics by explicitly heading towards milestone theorems... [His] directed path through these topics should make an effective course on the mathematics of surfaces. The exercises and hints are well chosen to clarify the central threads rather than diverting into byways." - Computing Reviews "Many examples and illustrations as well as exercises and hints to solutions are providing great support... By well-placed appendices the reader is relieved of the strain to immediately understand some extensive proofs or to learn adjoining mathematical facts... The book is suitable for students of mathematics, physics and of the teaching profession as well as university teachers who might be interested in using certain chapters...to present the topic in a seminar or in not too advanced special lectures about the topic...It is the great clarity of thought in this book, the simplicity and concreteness of the representation with respect to the capacity for teaching of students, and some other aspects that make this work stand out from comparable efforts." - ZAA "The exposition is clear, nicely organized, and generally easy to read." ---Zentralblatt Math

• 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.