Cambridge University Press
     978-0-521-72149-3 — Elementary Differential Geometry
     Christian Bär 
     Frontmatter
     More Information
                            ElementaryDifferentialGeometry
                            Thelinkbetweenthephysicalworldanditsvisualisationisgeometry.Thiseasy-to-read,
                            generously illustrated textbook presents an elementary introduction to differential
                            geometry with emphasis on geometric results. Avoiding formalism as much as possi-
                            ble, the author harnesses basic mathematical skills in analysis and linear algebra to
                            solve interesting geometric problems, which prepare students for more advanced study
                            in mathematics and other scientific fields such as physics and computer science.
                               The wide range of topics includes curve theory, a detailed study of surfaces, cur-
                            vature, variation of area and minimal surfaces, geodesics, spherical and hyperbolic
                            geometry, the divergence theorem, triangulations, and the Gauss–Bonnet theorem.
                            The section on cartography demonstrates the concrete importance of elementary dif-
                            ferential geometry in applications. Clearly developed arguments and proofs, colour
                            illustrations, and over 100 exercises and solutions make this book ideal for courses and
                            self-study. The only prerequisites are one year of undergraduate calculus and linear
                            algebra.
                            Christian Bär is Professor of Geometry in the Institute for Mathematics at the
                            University of Potsdam, Germany.
     © in this web service Cambridge University Press                                         www.cambridge.org
     Cambridge University Press
     978-0-521-72149-3 — Elementary Differential Geometry
     Christian Bär 
     Frontmatter
     More Information
                            Elementary
                            Differential Geometry
                                             ¨
                            ChristianBar
                            Universität Potsdam, Germany
     © in this web service Cambridge University Press                                         www.cambridge.org
         Cambridge University Press
         978-0-521-72149-3 — Elementary Differential Geometry
         Christian Bär 
         Frontmatter
         More Information
                              University Printing House, Cambridge CB2 8BS, United Kingdom
                              One Liberty Plaza, 20th Floor, New York, NY 10006, USA
                              477 Williamstown Road, Port Melbourne, VIC 3207, Australia
                              314-321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi - 110025, India
                              79 Anson Road, #06-04/06, Singapore 079906
                              Cambridge University Press is part of the University of Cambridge. 
                              It furthers the University’s mission by disseminating knowledge in the pursuit of  
                              education, learning and research at the highest international levels of excellence.
                              www.cambridge.org  
                              Information on this title: www.cambridge.org/9780521721493
                              Originally published in German as Elementare Differentialgeometrie by Walter de Gruyter 2001
                              © Walter de Gruyter GmbH & Co. KG 2000
                              First published in English by Cambridge University Press 2010
                              English translation © C. Bär 2010
                              This publication is in copyright. Subject to statutory exception  
                              and to the provisions of relevant collective licensing agreements,  
                              no reproduction of any part may take place without the written  
                              permission of Cambridge University Press.
                              Reprinted with corrections 2011
                              A catalogue record for this publication is available from the British Library
                              Library of Congress Cataloging in Publication data
                              Bär, Christian.
                                  Elementary differential geometry / Christian Bär.
                                p. cm.
                              ISBN 978-0-521-89671-9 (Hardback) – ISBN 978-0-521-72149-3 (Pbk.)
                              1.  Geometry, Differential–Textbooks.  I. Title.
                              QA641.B325 2010
                              516.3´6–dc22
                                                                   2010001343
                              ISBN  978-0-521-89671-9  Hardback
                              ISBN  978-0-521-72149-3  Paperback
                              Cambridge University Press has no responsibility for the persistence or  
                              accuracy of URLs for external or third-party internet websites referred to in  
                              this publication, and does not guarantee that any content on such websites is,  
                              or will remain, accurate or appropriate.
         © in this web service Cambridge University Press                                                                                                                            www.cambridge.org
     Cambridge University Press
     978-0-521-72149-3 — Elementary Differential Geometry
     Christian Bär 
     Frontmatter
     More Information
                            Contents
                            Preface                                                                  pagevii
                            Notation                                                                       xi
                        1   Euclideangeometry                                                               1
                            1.1  Theaxiomaticapproach                                                       1
                            1.2  TheCartesianmodel                                                        13
                        2   Curvetheory                                                                   22
                                              n
                            2.1  CurvesinR                                                                22
                            2.2  Plane curves                                                             34
                            2.3  Spacecurves                                                              57
                        3   Classicalsurfacetheory                                                        81
                            3.1  Regular surfaces                                                         81
                            3.2  Thetangentplane                                                          93
                            3.3  Thefirstfundamentalform                                                   98
                            3.4  Normalfieldsandorientability                                             103
                            3.5  Thesecondfundamentalform                                                106
                            3.6  Curvature                                                               110
                            3.7  Surface area and integration on surfaces                                126
                            3.8  Someclassesofsurfaces                                                   132
                        4   Theinnergeometryofsurfaces                                                   149
                            4.1  Isometries                                                              149
                            4.2  Vector fields and the covariant derivative                               152
                            4.3  RiemanncurvaturetensorandTheoremaEgregium                               160
                            4.4  Riemannianmetrics                                                       168
                            4.5  Geodesics                                                               171
                            4.6  Theexponentialmap                                                       183
                            4.7  Parallel transport                                                      192
                            4.8  Jacobi fields                                                            196
                                                                                                            v
     © in this web service Cambridge University Press                                        www.cambridge.org