First course in geometric topology and differentia

'[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 (March 1998) '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.' -- ZAA (1997) The uniqueness of this text in combining geometric topology and differential geometry lies in its unifying thread: the notion of a surface. With numerous illustrations, exercises and examples, the student comes to understand the relationship between modern axiomatic approach and geometric intuition. The text is kept at a concrete level, 'motivational' in nature, avoiding abstractions. A number of intuitively appealing definitions and theorems concerning surfaces in the topological, polyhedral, and smooth cases are presented from the geometric view, and point set topology is restricted to subsets of Euclidean spaces. The treatment of differential geometry is classical, dealing with surfaces in R 3 . The material here is accessible to math majors at the junior/senior level. Contents: Introduction To the Student Chapter I. Topology of Subsets of Euclidean Space Introduction Open and Closed Subsets of Sets in R n Continuous Maps Homeomorphisms and Quotient Maps Connectedness Compactness Chapter II. Topological Surfaces Introduction Arcs, Disks and 1-spheres Surfaces in R n Surfaces Via Gluing Properties of Surfaces Connected Sum and the Classification of Compact Connected Surfaces Appendix Proof of Theorem 2.4.3 (i) Appendix Proof of Theorem 2.6.1 Chapter III. Simplicial Surfaces Introduction Simplices Simplicial Complexes Simplicial Surfaces The Euler Characteristic Proof of the Classification of Compact Connected Surfaces Simplicial Curvature and the Simplicial Gauss-Bonnet Theorem Simplicial Disks and the Brouwer Fixed Point Theorem Chapter IV. Curves in R 3 Introduction Smooth Functions Curves in R 3 Tangent, Normal and Binormal Vectors Curvature and Torsion Fundamental Theorem of Curves Plane Curves Chapter V. Smooth Surfaces Introduction Smooth Surfaces Examples of Smooth Surfaces Tangent and Normal Vectors First Fundamental Form Directional Derivatives - Coordinate Free Directional Derivatives - Coordinates Length and Area Isometries Appendix Proof of Proposition 5.3.1 Chapter VI. Curvature of Smooth Surfaces Introduction and First Attempt The Weingarten Map and the Second Fundamental Form Curvature - Second Attempt Computations of Curvature Using Coordinates Theorema Egregium and the Fundamental Theorem of Surfaces Chapter VII. Geodesics Introduction 'Straight Lines' on Surfaces Geodesics Shortest Paths Chapter VIII. The Gauss-Bonnet Theorem Introduction The Exponential Map Geodesic Polar Coordinates Proof of the Gauss-Bonnet Theorem Appendix Geodesic Convexity Appendix Geodesic Triangulations Appendices A. Affine Linear Algebra Further Study References