Differential Geometry (Dover Books on Mathematics) by Heinrich W. Guggenheimer (PDF)

Description

This is a text of local differential geometry considered as an application of advanced calculus and linear algebra. The discussion is designed for advanced undergraduate or beginning graduate study, and presumes of readers only a fair knowledge of matrix algebra and of advanced calculus of functions of several real variables. The author, who is a Professor of Mathematics at the Polytechnic Institute of New York, begins with a discussion of plane geometry and then treats the local theory of Lie groups and transformation groups, solid differential geometry, and Riemannian geometry, leading to a general theory of connections. The author presents a full development of the Erlangen Program in the foundations of geometry as used by Elie Cartan as a basis of modern differential geometry; the book can serve as an introduction to the methods of E. Cartan. The theory is applied to give a complete development of affine differential geometry in two and three dimensions. Although the text deals only with local problems (except for global problems that can be treated by methods of advanced calculus), the definitions have been formulated so as to be applicable to modern global differential geometry. The algebraic development of tensors is equally accessible to physicists and to pure mathematicians. The wealth of specific resutls and the replacement of most tensor calculations by linear algebra makes the book attractive to users of mathematics in other disciplines.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The original edition was published by McGraw-Hill in 1963. A number of introductory Differential Geometry textbooks were published in that time period. I’ve learned from a number of them: Thomas Willmore, Elementary Differential Geometry (1959), Barrett O’Neill, Elementary Differential Geometry (1966) and Erwin Kreyszig, Differential Geometry and Riemannian Geometry (1968). My first instinct is to offer a comparison and contrast between Guggenheimer and those other three textbooks (Willmore, O’Neill, Kreyszig). Such a review would exceed my capabilities. I do, however, recommend all of the above textbooks for study (my favorite being O’Neill ! ). Guggenheimer’s text is quite different from the average textbook. It is at once elementary and advanced. His goal is to present the subject as an “application of advanced calculus and linear algebra” and to present “classical problems with modern methods.” He succeeds in the goal. Fourteen chapters: from curves to connections. Here is a problem from the final, if brief, chapter (connections): ” The holonomy groups of a Cartan-Finsler space, and more generally of any space with an infinitesimal nonlinear connection matrix, need not be a Lie group. Why ? ” (page 363). What else do we get ?(1) Moving Frames: “the notion which will permit us to compute in the vector plane, phenomena of the euclidean plane, is that of Frame.” (page 18).(2) Interesting solved Examples: for instance, “Find the natural equation of a conic.” (pages 64 and 65).(3) Introduction to variational calculus: an interesting example of isoperimetric inequality (page 84).(4) Introduction to Transformation Groups, concluding with an application of Euler Angles: “rotation of coordinates” (see page 100).(5) Two chapters (six and seven): “elaborations of the work of Sophus Lie.” Theorem: “The property of being a straight line is an invariant of euclidean geometry,” and “while one cannot say that this theorem is a startling new discovery, it is nice to see how group theory automatically yields the basic features.” (page 143). Assiduous attention to chapter six and chapter seven will be rewarded. Lie group and germs defined and utilized.(6) We get tensors in chapter nine. Guggenheimer follows “the kernel-index notation of Schouten” (page 177). Be forewarned ! Your route will be “a generalization of the notions of the vector and the matrix of linear algebra.” An interesting approach (ten pages). Then, exterior calculus: “one sees, in reality, integration is not on functions but on differential forms.” (page 191). An excellent discussion.(7) Fifty-five pages elaborate upon surfaces: Gauss-Codazzi-Mainardi equations (page 214). Gauss’ “theorema egregium,” (page 234). Section #10-3, Integration Theory, is an interesting highlight. Stoke’s Theorem( page 249).An exercise: “prove that a convex surface is always star-shaped.” (page 255). One learns additional material in the exercises, that is another positive attribute of this textbook.(8) In succession: Inner (intrinsic) geometry–Hyperbolic geometry highlighted (page 275); Affine geometry, read: “it is seen from these formulas that the words elliptic, hyperbolic, parabolic points, have the same meaning in euclidean and affine geometry.” (page 298); Riemannian geometry–with introduction to curvature, Christoffel symbol, covariant derivative. An interesting proof that “a metric can be brought into the euclidean form if and only if its Riemann curvature vanishes.” (page 324). An interesting approach to Integral Theorems, section #13-5 (attributed to Yano).(9) Answers (short) to “selected exercises” are presented at end-of-book. Hints are supplied for many others.Also, read: “as far as possible, both tensor calculus and exterior differentiation have been replaced by matrix-algebraic operations.” Also, read: “all topological questions have been strictly excluded.” (Preface).There you have a synopsis of this individualistic approach to Differential Geometry. I like this text quite a bit.The notation needs getting used to, but, the problems are a treasure trove of additional material.

⭐I think this must be the least expensive differential geometry book that uses Cartan’s orthonormal frame method. Though more than 40 years old, the notation is essentially modern (there are a few typographical oddities which aren’t really bothersome).This is a very rich book, with fascinating material on nearly every page. In fact, I think it’s a bit too rich for beginners, who should probably start with a more focused text like Millman & Parker or Pressley.Table of Contents for Differential GeometryPrefaceChapter 1. Elementary Differential Geometry 1-1 Curves 1-2 Vector and Matrix Functions 1-3 Some FormulasChapter 2. Curvature 2-1 Arc Length 2-2 The Moving Frame 2-3 The Circle of CurvatureChapter 3. Evolutes and Involutes 3-1 The Riemann-Stieltjès Integral 3-2 Involutes and Evolutes 3-3 Spiral Arcs 3-4 Congruence and Homothety 3-5 The Moving PlaneChapter 4. Calculus of Variations 4-1 Euler Equations 4-2 The Isoperimetric ProblemChapter 5. Introduction to Transformation Groups 5-1 Translations and Rotations 5-2 Affine TransformationsChapter 6. Lie Group Germs 6-1 Lie Group Germs and Lie Algebras 6-2 The Adjoint Representation 6-3 One-parameter SubgroupsChapter 7. Transformation Groups 7-1 Transformation Groups 7-2 Invariants 7-3 Affine Differential GeometryChapter 8. Space Curves 8-1 Space Curves in Euclidean Geometry 8-2 Ruled Surfaces 8-3 Space Curves in Affine GeometryChapter 9. Tensors 9-1 Dual Spaces 9-2 The Tensor Product 9-3 Exterior Calculus 9-4 Manifolds and Tensor FieldsChapter 10. Surfaces 10-1 Curvatures 10-2 Examples 10-3 Integration Theory 10-4 Mappings and Deformations 10-5 Closed Surfaces 10-6 Line CongruencesChapter 11. Inner Geometry of Surfaces 11-1 Geodesics 11-2 Clifford-Klein Surfaces 11-3 The Bonnet FormulaChapter 12. Affine Geometry of Surfaces 12-1 Frenet Formulas 12-2 Special Surfaces 12-3 Curves on a SurfaceChapter 13. Riemannian Geometry 13-1 Parallelism and Curvature 13-2 Geodesics 13-3 Subspaces 13-4 Groups of Motions 13-5 Integral TheoremsChapter 14. ConnectionsAnswers to Selected ExercisesIndex

⭐The book is fine, but it looks slightly used. The cover on it looks very worn / has quite the crease. A bit of a bummer when you purchase a new item.

⭐Honestly, a lot of these Dover books are good and so cheap, you can’t afford not to buy it.

⭐A good introduction book. The reader would need a different book for recent developments in the field.

⭐Although this book first appeared in 1963, it is the most “understandable and modern” DG book from Dover. I have browsed more than ten DG books to learn math required for GR. Some “modern-style” books usually start with concepts of topology. Do you really need to go through a topology book to read a DG book? Other “old” books do not tell you what a tensor is. They just say a tensor is a certain object transformed in such and such way. So what? Actually the modern treatment is more visual in the sense that it provides a geometrical meaning, but only after you have learned a complex language of modern mathematics, which detests an intuitive, visual way of presentation. The book by Guggenheimer is not too abstract, yet it is sufficiently modern to bestow readers with capability of geometric thinking. If you have finished a sophomore level advanced calculus or mathematical physics, you will not have much difficulty in cruising through the Guggenheimer.

⭐Livro bastante completo quanto ao assunto dado. Também tenho o do Erwin Kreyzig mas acredito que esse seja até mais completo. A explicação do conteúdo é bastante clara e simples, sempre partindo de um exemplo fácil e evoluindo gradativamente para um mais complexo. A organização do conteúdo também facilita a compreensão, já que é dividido em capítulos que definem bem o conteúdo, como o 1 que trata da base da geometria diferencial que sera necessaria no decorrer do livro, o 2 que trata de curvatura, e assim por diante. Recomendo.What I received is an old printing with sewn binding. the pages will not fall apart like the newer lower quality glue bound books. Just what I wanted 🙂

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