Differential Geometry Of Curves And Surfaces

by Masaaki Umehara, Kotaro Yamada, Wayne Rossman

Overview

'In a class populated by students who already have some exposure to the concept of a manifold, the presence of chapter 3 in this text may make for an unusual and interesting course. The primary function of this book will be as a text for a more conventional course in the classical theory of curves and surfaces.'
MAA ReviewsThis engrossing volume on curve and surface theories is the result of many years of experience the authors have had with teaching the most essential aspects of this subject. The first half of the text is suitable for a university-level course, without the need for referencing other texts, as it is completely self-contained. More advanced material in the second half of the book, including appendices, also serves more experienced students well.Furthermore, this text is also suitable for a seminar for graduate students, and for self-study. It is written in a robust style that gives the student the opportunity to continue his study at a higher level beyond what a course would usually offer. Further material is included, for example, closed curves, enveloping curves, curves of constant width, the fundamental theorem of surface theory, constant mean curvature surfaces, and existence of curvature line coordinates.Surface theory from the viewpoint of manifolds theory is explained, and encompasses higher level material that is useful for the more advanced student. This includes, but is not limited to, indices of umbilics, properties of cycloids, existence of conformal coordinates, and characterizing conditions for singularities.In summary, this textbook succeeds in elucidating detailed explanations of fundamental material, where the most essential basic notions stand out clearly, but does not shy away from the more advanced topics needed for research in this field. It provides a large collection of mathematically rich supporting topics. Thus, it is an ideal first textbook in this field.

Product Details

ISBN-13:	9789814740241
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	06/29/2017
Edition description:	New Edition
Pages:	328
Product dimensions:	6.00(w) x 9.00(h) x 0.60(d)

Table of Contents

Preface v

I Curves 1

1 What exactly is a "curve"? 1

2 Curvature and the Frenet formula 10

3 Closed curves* 28

4 Geometry of spirals* 40

5 Space curves 50

II Surfaces 57

6 What exactly is a "surface"? 57

7 The first fundamental form 68

8 The second fundamental form 77

9 Principal and asymptotic directions 89

10 Geodesies and the Gauss-Bonnet theorem 99

11 Proof of the Gauss-Bonnet theorem* 117

III Surfaces from the Viewpoint of Manifolds* 131

12 Differential forms 131

13 Levi-Civita connections 138

14 The Gauss-Bonnet formula for 2-manifolds 148

15 Poincaré-Hopf index theorem 152

16 The Laplacian and isothermal coordinates 165

17 The Gauss and Codazzi equations 172

18 Cycloids as brachistochrones 181

19 Geodesic triangulations of compact Riemannian 2-manifoids 184

Appendix A Supplements 193

A.1 A review of calculus 193

A.2 The fundamental theorems for ordinary differential equations 198

A.3 Euclidean spares 200

Appendix B Advanced Topics on Curves and Surfaces 213

B.1 Evolutes and the cycloid pendulum 213

B.2 Convex curves and curves of constant width 219

B.3 Line integrals and the isoperimetric inequality 224

B.4 First fundamental forms and maps 231

B.5 Curvature line coordinates and asymptotic line coordinates 236

B.6 Surfaces with K = 0 244

B.7 A relationship between surfaces with constant Gaussian curvature and with constant mean curvature 251

B.8 Surfaces of revolution of negative constant Gaussian curvature 259

B.9 Criteria of typical singularities 261

B.10 Proof of the fundamental theorem for surfaces 268

Answers to Exercises 279

Bibliography 301

List of Symbols 305

Index 307