Elementary topics in differential geometry - Thorpe, John A; Thorpe, John A (New York, Springer-Verlag, 1979)
This introductory text develops the geometry of n-dimensional oriented surfaces in Rn+1. By viewing such surfaces as level sets of smooth functions, the author is able to introduce global ideas early without the need for preliminary chapters developing sophisticated machinery. The calculus of vector fields is used as the primary tool in developing the theory. Coordinate patches are introduced only after preliminary dicussions of geodesics, parallel transport, curvature, and convexity. Differential forms are introduced only as needed for use in integration. The text,which draws significantly on students' prior knowledge of linear algebra, multivariate calculus, and differential equations, is designed for a one-semester course at the junior/senior level. Table of Contents Chapter 1 Graphs and Level Sets Chapter 2 Vector Fields Chapter 3 The Tangent Space Chapter 4 Surfaces Chapter 5 Vector Fields on Surfaces; Orientation Chapter 6 The Gauss Map Chapter 7 Geodesies Chapter 8 Parallel Transport Chapter 9 The Weingarten Map Chapter 10 Curvature of Plane Curves Chapter 11 Arc Length and Line Integrals Chapter 12 Curvature of Surfaces Chapter 13 Convex Surfaces Chapter 14 Parameterized Surfaces Chapter 15 Local Equivalance of Surfaces and Parameterized Surfaces Chapter 16 Focal Points Chapter 17 Surface Area and Volume Chapter 18 Minimal Surfaces Chapter 19 The Exponential Map Chapter 20 Surfaces with Boundary Chapter 21 The Gauess-Boness Theorem Chapter 22 Regid Motio
Bibliography 245p; Index 249-253p
Keywords: Differential geometry; Mathematics