Finally, in Chapter 5, we use the first and second variations of arc length to derive some global properties of surfaces. Chapter 4 unifies the intrinsic geometry of surfaces around the concept of covariant derivative again, our purpose was to prepare the reader for the basic notion of connection in Riemannian geometry. Chapter 3 is built on the Gauss normal map and contains a large amount of the local geometry of surfaces in R³.

Thus, Chapter 2 develops around the concept of a regular surface in R³ when this concept is properly developed, it is probably the best model for differentiable manifolds. We have tried to build each chapter of the book around some simple and fundamental idea. The presentation differs from the traditional ones by a more extensive use of elementary linear algebra and by a certain emphasis placed on basic geometrical facts, rather than on machinery or random details. This book is an introduction to the differential geometry of curves and surfaces, both in its local and global aspects. Cavalcante, participated in the project as if it was a work of her own and I might say that without her this volume would not exist.įinally, I would like to thank my son, Manfredo Jr., for helping me with several figures in this edition. Thanks are also due to John Grafton, Senior Acquisitions Editor at Dover Publications, who believed that the book was still valuable and included in the text all of the changes I had in mind, and to the editor, James Miller, for his patience with my frequent requests.Īs usual, my wife, Leny A. Here I would like to express my deep appreciation and thank them all. For several reasons it is impossible to mention the names of all the people who generously donated their time doing that. In this edition, I have included many of the corrections and suggestions kindly sent to me by those who have used the book.

Theorem of Hopf-Rinowĥ-4 First and Second Variations of Arc Length Bonnet’s Theoremĥ-6 Covering Spaces The Theorems of Hadamardĥ-7 Global Theorems for Curves: The Fary-Milnor Theoremĥ-10 Abstract Surfaces Further GeneralizationsĪppendix: Point-Set Topology of Euclidean Spaces Bibliography and Comments Hints and Answers Index Preface to the Second Edition Global Differential Geometryĥ-3 Complete Surfaces. Geodesic Polar CoordinatesĤ-7 Further Properties of Geodesics Convex NeighborhoodsĪppendix: Proofs of the Fundamental Theorems of the Local Theory of Curves and Surfaces 5. The Intrinsic Geometry of SurfacesĤ-3 The Gauss Theorem and the Equations of CompatibilityĤ-5 The Gauss-Bonnet Theorem and Its ApplicationsĤ-6 The Exponential Map. The Geometry of the Gauss Mapģ-2 The Definition of the Gauss Map and Its Fundamental PropertiesĪppendix: Self-Adjoint Linear Maps and Quadratic Forms 4. Regular SurfacesĢ-2 Regular Surfaces Inverse Images of Regular ValuesĢ-3 Change of Parameters Differentiable Functions on SurfaceĢ-4 The Tangent Plane The Differential of a MapĢ-7 A Characterization of Compact Orientable SurfacesĪppendix: A Brief Review of Continuity and Differentiability 3.

Curvesġ-5 The Local Theory of Curves Parametrized by Arc Lengthġ-7 Global Properties of Plane Curves 2. To Leny, for her indispensable assistance in all the stages of this book Contents Preface to the Second Edition Preface Some Remarks on Using this Book 1. Manufactured in the United States by LSC Communications The author has also provided a new Preface for this edition. do Carmoĭifferential Geometry of Curves and Surfaces: Revised & Updated Second Edition is a revised, corrected, and updated second edition of the work originally published in 1976 by Prentice-Hall, Inc., Englewood Cliffs, New Jersey.

Differential geometry surfaces