A First Course in Differential Geometry

This book is designed to introduce differential geometry to beginning graduate students as well as to advanced undergraduate students. In the last couple of decades, differential geometry, along with other branches of mathematics, has been greatly developed. In this book we will study only the traditional topics: namely, curves and surfaces in a three-dimensional Euclidean space E³. Unlike most classical books on the subject, however, more attention is paid here to the relationships between local and global properties, as opposed to local properties only. Although we restrict our attention to curves and surfaces in E³, most global theorems for curves and surfaces in this book can be extended to higher dimensional spaces, to more general curves and surfaces, or to both. Moreover, geometric interpretations are given along with analytic expressions. This enables students to make use of geometric intuition, which is a precious tool for studying geometry and related problems.