Differential Geometry: Curves — Surfaces — Manifolds (Student Mathematical Library) 3rd Edition by Wolfgang Kuhnel (PDF)

Description

User’s Reviews

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

⭐The first four chapters of this book provide enough material for a one-semester course in the classical differential geometry of curves and surfaces; the following four chapters then provide a seamless transition into the study of Riemannian manifolds. Altogether, there is enough material for a two-semester course. The book would serve well as the primary text for a first course in differential geometry, but it is so well written that it can also be used for independent study. The essential prerequisites are a solid course in linear algebra, a background in multivariable calculus, and elementary point-set topology. Answers to selected exercises are provided.A course in the classical differential geometry of curves and surfaces in Euclidean 3-space is no longer part of the required undergraduate mathematics curriculum at most universities in the United States (if it ever was). A student who has not taken such a course before studying Riemannian manifolds lacks some necessary historical and background knowledge. To give but one example, it is essential to understand the classical Gaussian curvature of surfaces in order to understand the important sectional curvature function in Riemannian geometry.The aspect of Kuhnel’s book that I appreciate the most is the great versatility it provides for students who have not taken a classical differential geometry course. As already stated, the first four chapters (196 pages) contain a rigorous introduction to classical differential geometry of curves and surfaces. If the reader does not need this review, then he may skip directly to Chapter 5 and begin the study of Riemannian manifolds, referring back to the first four chapters as needed. If, however, he has no background in the classical theory, then he may acquire the background he needs from Chapters 1 through 4. It is a complete, self-contained introduction. Although this is always a difficult judgment and is somewhat subjective, I would say that the curve and surface material in the first four chapters is presented at a level of mathematical sophistication that is slightly higher than the classic book by Manfredo do Carmo. It is definitely more challenging than the popular introduction “Elementary Differential Geometry” by Barrett O’Neill. Chapters 5 through 8 on Riemannian manifolds make far more demands on the reader’s mathematical maturity than the first four chapters. Some of this material is very eclectic and mathematically sophisticated.Kuhnel does not restrict his discussion of classical differential geometry to subsets of Euclidean 3-space. In Chapter 2 he studies curves in R^n for general n, and he provides higher-dimensional generalizations of the Frenet-Serret equations from R^3. The curvature and torsion functions are generalized through what Kuhnel calls the Frenet curvature functions of the curve. Kuhnel also studies curves in Minkowski space, and in general n-dimensional pseudo-Euclidean spaces. This material will be of interest to students interested in special and general relativity.Section 4F devotes 15 pages to a thorough analysis of the Gauss-Bonnet Theorem, which Kuhnel calls “one of the most important theorems in all of differential geometry.” Section 4G devotes 10 pages to selected topics in the global theory of surfaces; this section is primarily inspirational, and I find it one of the most compelling sections in the book. The list of topics includes: (1) An interpretation of Gaussian curvature as an infinitesimal change of the volume element under the Gauss map; (2) A definition of “tight” surfaces as orientable closed surfaces of minimal total absolute Gaussian curvature among all surfaces of the same genus. A theorem is given providing conditions equivalent to tightness; (3) Cohn-Vossen’s result on complete, noncompact surfaces in R^3 showing that the total curvature is no larger than 2(pi) times the Euler characteristic, thus illustrating a beautiful relation between geometry and topology. There are other results, some merely presented without proof. The choice of topics is clearly designed to excite the reader’s imagination and inspire further research beyond this book.Perhaps the single most important feature of this book is its global coherence and unity of purpose. The classical material in the first four chapters is not presented as an end in itself, but with the subsequent study of Riemannian manifolds firmly in mind as the primary goal. There are many notes and references throughout the book designed to help the reader connect the rather abstract machinery of Riemannian manifolds to more intuitively understandable precursors in the classical geometry of curves and surfaces. This unity of purpose, along with consistent notation throughout, is a real advantage for students who are totally new to the subject.The final four chapters move rapidly into a thorough introduction to Riemannian manifolds. There is no substantive discussion of topological manifolds or of smooth manifolds without a metric tensor. Kuhnel’s goal is the study of Riemannian manifolds, and he introduces a Riemannian metric almost immediately after defining manifolds. Kuhnel’s choice of topics in these final chapters is intriguing; it is definitely not the list of subjects to be found in a typical introduction to Riemannian geometry. After the development of the usual geometric machinery in Chapters 5 and 6, Chapter 7 studies spaces of constant curvature, while Chapter 8 is devoted to Einstein manifolds.Numerous remarks make it clear that Kuhnel wants his book to be of assistance to readers who are interested in general relativity. Section 8A will be of special interest to the relativity students. Kuhnel begins Chapter 8 with a slightly amended version of a question that can be found in Besse’s “Einstein Manifolds” when he asks, for a given manifold M,“Is there a ‘best’ metric whose curvature has the property of being most evenly distributed about the manifold?”Kuhnel employs the calculus of variations on the so-called Hilbert-Einstein functional S(g), where S(g) is the total scalar curvature of a metric g, to answer this question. As a corollary, he derives Einstein’s general relativistic field equations in Section 8B, following the approach of the mathematician David Hilbert. The reader may know that historians of science have long debated who actually discovered Einstein’s equations first, Einstein or the mathematician David Hilbert. Leaving the resolution of priority to the historians, it is quite interesting to see Hilbert’s variational argument presented in such meticulous detail. This is a valuable gift to students of general relativity.Despite the scattered material on general relativity, the prospective reader should know that Kuhnel’s book does not claim to provide a systematic development of semi-Riemannian (specifically, Lorentzian) geometry, as used in general relativity. Barrett O’Neill’s “Semi-Riemannian Geometry with Applications to Relativity” still remains one of the few books to provide such a dedicated development.In Chapter 8, Kuhnel also includes a beautiful discussion of the algebraic decomposition of the Riemann curvature tensor. This material will be profoundly helpful to new students who are attempting to understand Ricci curvature, scalar curvature, sectional curvature, and Weyl curvature. This section is followed by a detailed discussion of the Weyl curvature tensor and its use in assigning Petrov types to four-manifolds. Relatively few introductions to Riemannian geometry discuss the Weyl curvature tensor at all, despite its importance in general relativity.I will conclude by congratulating Bruce Hunt on his translation of the original German text into English. His translation is so successful that it has produced one of the clearest, most concise differential geometry texts that I have studied.This is a beautifully conceived book that has gone through many revisions and improvements (in Germany) before producing this third American edition. I recommend it enthusiastically, especially to to readers who need to begin with an introduction to the classical differential geometry of curves and surfaces.

⭐I did not like this for the bibliography page was binded when was still folded so it did not open at all. I had to slice the paper in order to see the contents. The page is awfully crinkled as you can see in the photos, and it is worse in person. Why didn’t you check before you send me this book? This was not hard to be noticed when you just had a second to turn over the book. That particular page is cut larger than the others and doubled. I do not have time to ship it back to return and get it replaced so I am writing this review instead. Very disappointed.

⭐Esta es la segunda edición de un libro de texto muy bien planteado pero que tenia carencias. El añadido de soluciones a los ejercicios supone una gran aportación para los autodidactas como yo.

⭐

⭐Claimed to be new. Clearly used.

Keywords

Free Download Differential Geometry: Curves — Surfaces — Manifolds (Student Mathematical Library) 3rd Edition in PDF format
Differential Geometry: Curves — Surfaces — Manifolds (Student Mathematical Library) 3rd Edition PDF Free Download
Download Differential Geometry: Curves — Surfaces — Manifolds (Student Mathematical Library) 3rd Edition PDF Free
Differential Geometry: Curves — Surfaces — Manifolds (Student Mathematical Library) 3rd Edition PDF Free Download
Download Differential Geometry: Curves — Surfaces — Manifolds (Student Mathematical Library) 3rd Edition PDF
Free Download Ebook Differential Geometry: Curves — Surfaces — Manifolds (Student Mathematical Library) 3rd Edition