Differential Geometry: Curves - Surfaces - Manifolds (Student Mathematical Library, Volume 16)

Our first knowledge of differential geometry usually comes from the study of the curves and surfaces in $I\!\!R^3$ that arise in calculus. Here we learn about line and surface integrals, divergence and curl, and the various forms of Stokes' Theorem. If we are fortunate, we may encounter curvature and such things as the Serret-Frenet formulas. With just the basic tools from multivariable calculus, plus a little knowledge of linear algebra, it is possible to begin a much richer and rewarding study of differential geometry, which is what is presented in this book. It starts with an introduction to the classical differential geometry of curves and surfaces in Euclidean space, then leads to an introduction to the Riemannian geometry of more general manifolds, including a look at Einstein spaces. An important bridge from the low-dimensional theory to the general case is provided by a chapter on the intrinsic geometry of surfaces. The first half of the book, covering the geometry of curves and surfaces, would be suitable for a one-semester undergraduate course. The local and global theories of curves and surfaces are presented, including detailed discussions of surfaces of rotation, ruled surfaces, and minimal surfaces. The second half of the book, which could be used for a more advanced course, begins with an introduction to differentiable manifolds, Riemannian structures, and the curvature tensor. Two special topics are treated in detail: spaces of constant curvature and Einstein spaces. The main goal of the book is to get started in a fairly elementary way, then to guide the reader toward more sophisticated concepts and more advanced topics. There are many examples and exercises to helpalong the way. Numerous figures help the reader visualize key concepts and examples, especially in lower dimensions. For the second edition, a number of errors were corrected and some text and a number of figures have been added. Our First Knowledge Of Differential Geometry Usually Comes From The Study Of The Curves And Surfaces In I\!\!r^3 That Arise In Calculus. Here We Learn About Line And Surface Integrals, Divergence And Curl, And The Various Forms Of Stokes' Theorem. If We Are Fortunate, We May Encounter Curvature And Such Things As The Serret-frenet Formulas. With Just The Basic Tools From Multivariable Calculus, Plus A Little Knowledge Of Linear Algebra, It Is Possible To Begin A Much Richer And Rewarding Study Of Differential Geometry, Which Is What Is Presented In This Book. It Starts With An Introduction To The Classical Differential Geometry Of Curves And Surfaces In Euclidean Space, Then Leads To An Introduction To The Riemannian Geometry Of More General Manifolds, Including A Look At Einstein Spaces. An Important Bridge From The Low-dimensional Theory To The General Case Is Provided By A Chapter On The Intrinsic Geometry Of Surfaces. The First Half Of The Book, Covering The Geometry Of Curves And Surfaces, Would Be Suitable For A One-semester Undergraduate Course. The Local And Global Theories Of Curves And Surfaces Are Presented, Including Detailed Discussions Of Surfaces Of Rotation, Ruled Surfaces, And Minimal Surfaces. The Second Half Of The Book, Which Could Be Used For A More Advanced Course, Begins With An Introduction To Differentiable Manifolds, Riemannian Structures, And The Curvature Tensor. Two Special Topics Are Treated In Detail: Spaces Of Constant Curvature And Einstein Spaces. The Main Goal Of The Book Is To Get Started In A Fairly Elementary Way, Then To Guide The Reader Toward More Sophisticated Concepts And More Advanced Topics. There Are Many Examples And Exercises To Help Along The Way. Numerous Figures Help The Reader Visualize Key Concepts And Examples, Especially In Lower Dimensions. For The Second Edition, A Number Of Errors Were Corrected And Some Text And A Number Of Figures Have Been Added. From a review of the German edition: The book covers all the topics which could be necessary later for learning higher level differential geometry. The material is very carefully sorted and easy-to-read. —Mathematical Reviews This carefully written book is an introduction to the beautiful ideas and results of differential geometry. The first half covers the geometry of curves and surfaces, which provide much of the motivation and intuition for the general theory. Special topics that are explored include Frenet frames, ruled surfaces, minimal surfaces, and the Gauss-Bonnet theorem. The second part is an introduction to the geometry of general manifolds, with particular emphasis on connections and curvature. The final two chapters are insightful examinations of the special cases of spaces of constant curvature and Einstein manifolds. The text is illustrated with many figures and examples. For the second edition, a number of errors were corrected and some text and a number of figures have been added. The prerequisites are undergraduate analysis and linear algebra. "This carefully written book is an introduction to the beautiful ideas and results of differential geometry. The first half covers the geometry of curves and surfaces, which provide much of the motivation and intuition for the general theory. Special topics that are explored include Frenet frames, ruled surfaces, minimal surfaces, and the Gauss-Bonnet theorem." "The second part is an introduction to the geometry of general manifolds, with particular emphasis on connections and curvature. The final two chapters are insightful examinations of the special cases of spaces of constant curvature and Einstein manifolds." "The text is illustrated with many figures and examples. For the second edition, a number of errors were corrected and some text and a number of figures have been added. The prerequisites are undergraduate analysis and linear algebra."--Book jacket.