Curvature in Mathematics and Physics (Dover Books on Mathematics) by Shlomo Sternberg (PDF)

Description

This original text for courses in differential geometry is geared toward advanced undergraduate and graduate majors in math and physics. Based on an advanced class taught by a world-renowned mathematician for more than fifty years, the treatment introduces semi-Riemannian geometry and its principal physical application, Einstein’s theory of general relativity, using the Cartan exterior calculus as a principal tool. Starting with an introduction to the various curvatures associated to a hypersurface embedded in Euclidean space, the text advances to a brief review of the differential and integral calculus on manifolds. A discussion of the fundamental notions of linear connections and their curvatures follows, along with considerations of Levi-Civita’s theorem, bi-invariant metrics on a Lie group, Cartan calculations, Gauss’s lemma, and variational formulas. Additional topics include the Hopf-Rinow, Myer’s, and Frobenius theorems; special and general relativity; connections on principal and associated bundles; the star operator; superconnections; semi-Riemannian submersions; and Petrov types. Prerequisites include linear algebra and advanced calculus, preferably in the language of differential forms.

User’s Reviews

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

⭐A few reviews here and around the web characterize this as “undergrad.” At MIT? If you don’t have STRONG linear algebra and advanced (minimum 3 years) calculus, you’ll be lost by the third chapter.I’m the CTO of Classpros dot com and design math visualizations for Engineering students and this is NOT an undergrad text, unless you’re an EXCEPTIONAL undergrad with a LOT of physics and calc already behind you, as in a senior at a tech college. This is not at all to knock the book, it’s to save purchasers from disappointment when they see the author jumping into Relativity from the viewpoint of Cartan exterior calculus and tensors.That said, this is an EXTRAORDINARY text because differential geometry has become so specialized that few grad students except in limited areas of physics/applied math get to go there. The “expansion” of the field with game programming and sims is a new revolution that is “bringing back” fields as dusty as quaternions and spherical trig (see our article on Wiki on the Lenart Sphere, for example)– and creating exciting new interest in differential forms and geometry.After reviewing and using half a dozen (rare and old) books on differential geometry, this is the ONE book you must have if you are serious about the field, both for breadth and depth and especially currency. Both the most recent applications and the older physics are covered flawlessly. Also, if this were a Springer text, it would be over $100 at this quality and rarity– what a bargain from Dover!Several reviewers have noted this text as worth the price even though it is a “classic.” Ahem. Dover DOES reprise many worthy volumes, but this $13 gem is NEW IN 2012– yes, by a professor that’s taught it for half a century, but this is NOT one of Dover’s well known reprints from 1956, it’s new and original! In fact, the author (humbly) compares it to the “gold standard” of O’Neill (from 1982), which IS $113 even on Amazon (

⭐). If you’re a prof considering a pre-relativity course, please do your students a favor and consider this fine text! Dover is trying to set a trend by offering very high quality texts for prices us un-rich folks can afford! With all due respect to O’Neill, at 10% the cost, nearly the same pages, and more intuitive notation– why not? Give your class a ramen free weekend!Library Picks reviews titles exclusively for the benefit of Amazon shoppers and has nothing to do with Amazon, the authors or publishers, and we always buy the books we review. Since another reviewer published the contents, you’ve probably got what you need to decide. If you’ve had to figure out what is integratable or not based on whether you can find linear differentiable forms using linear algebra and other tools, you’re ready for this fine text. Otherwise more advanced calc and linear algebra are musts first. Unless other reviewers are channeling Albert, I’m not sure why they’re trying to tell you this is “elementary?” Guess the concept is relative, pun intended.

⭐Although billed as a text for undergraduates, only the best-prepared and exceptional undergraduates are likely to get much out of this book. More realistic preparation would be a good course in modern differential geometry.The book uses the modern definition of “differential manifold” throughout, but I can’t find it defined anywhere in the book. The grossly inadequate index contains only 17 items starting with “m” , and these do not include “manifold” ! The closest to a definition seems to be a definition of “parametrized surface” in Chapter 1.I doubt that anyone without a previous acquaintance with differential forms will get much out of this book. Technically, the definitions are given in Chapter 2, but in only a few pages in a very abstract way. Some of that treatment seemed to me downright perverse. For example, the concept of “Lie derivative” of a vector field has a simple geometrical interpretation given in almost all texts. From this follows the related concept of Lie derivative of a differential form field, whose geometrical motivation depends on the previous concept of Lie derivative of a vector field. But Sternberg introduces these concepts “backward” starting with a geometrically unmotivated algebraic definition of Lie derivative of a form field (which relies on the nontrivial Weil formula) and from that produces an algebraic definition of Lie derivative of a vector field. I would be surprised if anyone unfamiliar with the general concept of “Lie derivative” will come away from this discussion with an adequate understanding.That said, there is a lot of value in this book for those already familiar with differential geometry. I enjoyed browsing through it. It includes many pictures of prominent differential geometers, with some biographical sketches. Most of the mathematics is presented at an appropriate level of rigor, perhaps not quite as high as mathematical research, but much higher than typical physics literature. Though the exposition is more informal than that of most mathematics texts, it is usually no less rigorous.Apart from applications to general relativity, most of the physical applications were presented so sketchily that they were hard for me to follow in detail. The “Curvature in … Physics” part of the title should be taken with a grain of salt by readers without extensive experience in the relevant fields of physics.I cannot recommend this book as a textbook, but I have enjoyed browsing through it and am happy to have it in my library. I think that for its price, it is a bargain.

⭐Muy libro.Classic book that will provide you with a rigorous overview of the links between the conceptual properties and interesting empirical applications that make the study of the physics and mathematics of nonlinear surfaces really exciting. I think it’s one of the best — if not the best — book of its kind. It was a brilliant move to include this outstanding book in the Dover series makes because that increases accessibility since the price of each book is affordable unlike most of the comparable excellent titles at say Springer. I’ve gained the ability to start applying this work to biological problems in microbiology (bacterial cell membrane properties such as protein permeability). I highly recommend this book.

⭐This one is a good book but it is intended for Physics audience. It is best for calculations point of view. It’s major drawback is that theoretical foundation and intitutions has not been made very clear before every definition and introduction of the chapters. Yet this text is good in many context.

⭐Es uno de los libros que más me ha sorprendido en los últimos tiempos. Muy bien redactado y trata numerosos temas. Puede servir tanto para físicos teóricos como para matemáticos con ciertos conocimientos de Relatividad. Es un buen complemento para otros libros como Semi-Riemannian Geometry With Applications to Relativity de O’Neill, es del mismo estilo.You need to have an introduction of thenotion of curvature before entering that book because the slope of understanding is quite steep. But with a good preparation, you will enjoy the book.

⭐This book is very clear and precise mathematically. I also really enjoyed that most of the examples in the book where all based on physics problems such as the Schwarzschild solution, and the higgs mechanism.

⭐ok

Keywords

Free Download Curvature in Mathematics and Physics (Dover Books on Mathematics) in PDF format
Curvature in Mathematics and Physics (Dover Books on Mathematics) PDF Free Download
Download Curvature in Mathematics and Physics (Dover Books on Mathematics) 2013 PDF Free
Curvature in Mathematics and Physics (Dover Books on Mathematics) 2013 PDF Free Download
Download Curvature in Mathematics and Physics (Dover Books on Mathematics) PDF
Free Download Ebook Curvature in Mathematics and Physics (Dover Books on Mathematics)