Differential Geometry

William C. Graustein

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486450117
• eBook ISBN: 9780486153230
• Topic: Mathematics
• Subtopic: Game Theory
• Index: Mathematics

This first course in differential geometry presents the fundamentals of the metric differential geometry of curves and surfaces in a Euclidean space of three dimensions. Written by an outstanding teacher and mathematician, it explains the material in the most effective way, using vector notation and technique. It also provides an introduction to the study of Riemannian geometry.
Suitable for advanced undergraduates and graduate students, the text presupposes a knowledge of calculus. The first nine chapters focus on the theory, treating the basic properties of curves and surfaces, the mapping of surfaces, and the absolute geometry of a surface. The final chapter considers the applications of the theory to certain important classes of surfaces: surfaces of revolution, ruled surfaces, translation surfaces, and minimal surfaces. Nearly 200 problems appear throughout the text, offering ample reinforcement of every subject.

CHAPTER I

INTRODUCTION

1. The nature of differential geometry. Differential geometry may be roughly described as the study of curves and surfaces of general type by means of the calculus. In contrast to it, there is algebraic geometry, which employs algebra as its principal tool and restricts itself to the consideration of a much narrower class of curves and surfaces. Thus, the theory of conic sections or quadric surfaces, with which the reader is familiar from analytic geometry, belongs to algebraic geometry, whereas that of the curvature of a general curve, or that of the tangent plane to a general surface, pertains to differential geometry.

A geometric configuration has two different kinds of properties, those which pertain to the configuration as a whole, and those which are definable for restricted portions of it. Thus, the order of a plane algebraic curve—the number of points in which it is cut by a straight line—, is a property of the curve in its entirety. On the other hand, the curvature of a curve at a point depends only on the shape of the curve in the neighborhood of the point.

Generally speaking, algebraic geometry is concerned with properties of the whole of a configuration, whereas differential geometry deals with properties of a restricted portion of it. Algebraic geometry is essentially a geometry of the whole or a geometry in the large, and differential geometry, a geometry in the small.

Euclidean geometry, either in the synthetic form of the preparatory school or the analytic form of the college, is algebraic geometry. So also is the ordinary projective geometry with which the reader is perhaps conversant. But these two geometries differ essentially in content. Euclidean geometry deals with properties of figures which are unchanged by rigid motions, for example, with distance, angle, and area. Projective geometry deals only with properties of figures which are unchanged by projections, such as the property that a point lie on a line, or that a number of lines be concurrent. The former is a quantitative, or metric, geometry, whereas the latter is concerned with properties of position and has nothing to do with measurement.

The distinction between metric and projective geometry is applicable, also, to differential geometry. Thus, there is a metric, or Euclidean, differential geometry and a projective differential geometry. In this book we shall be concerned only with metric differential geometry. In other words, we shal...

Table of contents

• Title Page
• Copyright Page
• PREFACE
• Table of Contents
• CHAPTER I - INTRODUCTION
• CHAPTER II - SPACE CURVES
• CHAPTER III - CURVES AND SURFACES ASSOCIATED WITH A SPACE CURVE
• CHAPTER IV - FUNDAMENTALS OF THE THEORY OF SURFACES
• CHAPTER V - CURVATURE. IMPORTANT SYSTEMS OF CURVES
• CHAPTER VI - THE FUNDAMENTAL THEOREM
• CHAPTER VII - GEODESIC CURVATURE. GEODESICS
• CHAPTER VIII - MAPPING OF SURFACES
• CHAPTER IX - THE ABSOLUTE GEOMETRY OF A SURFACE
• CHAPTER X - SURFACES OF SPECIAL TYPE
• INDEX