DIFFERENTIAL GEOMETRY FOR PHYSICISTS AND MATHEMATICIANS: MOVING FRAMES AND DIFFERENTIAL FORMS: FROM EUCLID PAST RIEMANN

I'm not done reading and studying this book but I have perused it and read the introductory chapter. I like the author's conversational approach and he has some ideas that he presents that I've not seen in print before, my background being a BS Degree in Applied Mathematics and Physics, with some additional study in Civil and Electrical Engineering. I have studied math through the level of advanced calculus and some advanced 400/500 level courses in differential equations, partial differential equations, special functions of mathematical physics, complex analysis, and vector analysis. I share the author's view that much energy is wasted and the real reason for the math is obscured in making formal proofs and not getting down to the business of applications early on. In short, anyone new to the study of any branch of mathematics would prefer being shown what it can be used for first, then gradually introduced to the proofs later in their studies.The book contains no problem sets, only a number of illustrative examples scattered throughout the text. It might be better titled "Lectures on Differential Geometry for Physicists and Mathematicians." The author even cites Feynman's "Lectures on Physics" and references a passage from one of Feynman's lectures in the first chapter. In many ways, it is very similar in its approach to Feynman's Lectures, though the material it covers is certainly of much different character and at a somewhat more advanced level, while it remains accessible to perhaps an advanced undergraduate student with the mathematical maturity that would be expected of perhaps an honors student in intermediate calculus or differential equations. By this I mean to say it isn't the easiest read but with a bit of patience, it can be understood.I have a few criticisms intended for the author. There are some glaring flaws in spelling and English grammar stemming from his mother language obviously being Spanish rather than English. However, the manner of writing, the numerous misspellings, and grammatical errors should in no way be viewed as detrimental to the reading of the book. I quickly made up my mind that the author views getting the material into print far more important than satisfying academia's stuffy approach to making proper sentences. This is mathematics, not a course in English literature after all! Perhaps my biggest criticism is in his overeducated way of throwing complicated words into his sentences where a simpler word would have been a better choice. But having been guilty of a lot of that myself until late in my university education, I know how difficult it can be to trim the fat out of the language as well as sentence structure. Some might consider it arrogance but I'd say that view would be wrong. He is, after all, trying to help in the best way he knows to get some very valuable views into the hands of those who are best equipped to use it.In short, I find the author very real and down to earth, though rightfully proud of his work. And why shouldn't he be? It is, after all, rather unique!My suggestions for improvement in a future edition would be the following:1.) Have someone proof read the manuscript and correct the English, eliminating overly pompous writing in favor of a direct, modern, and very clear style while keeping the tone informal. Simpler is always better when it comes to writing.2.) Add some graded problem sets at the end of each chapter that contain numerous application examples and brief, detailed solutions to every problem in an appendix at the back of the book. Have the solutions thoroughly checked by a couple of colleagues. Don't try to add proofs. Proofs will only make the book less accessible to the intended audience of physicists and applied mathematicians. If there is a desire to add proof problems, collect all of them in an appendix at the back of the book and include detailed steps to each proof in the solutions appendix..

I have a graduate degree in Physics and am familiar with the cartan calculus and bought the book hoping to get a clear explanation of the tetrad and Newman Penrose formalism. This book may be okay for someone who is an expert but is not one to try to learn something from. The notation also is archaic.

I am totally disappointed with the book. It is not a book from which you can really learn anything. It rather looks as a random collection of buzz words with the aim to impress the reader. Certainly you will not learn from it "differential geometry", neither for physicists nor for mathematicians. Instead of providing some mathematical rigor it is full of complaints about other approaches. But it does not provide anything to replace the criticized ideas. At the end of the book there are hints about another book "being written" by the same author, with a promise of similar chaos.The main idea of the book is: Cartan was right, but not right enough. He introduced differential forms, but only number valued, while we need vector valued and tensor valued. Then we learn that Kahler was right, but not right enough. He also has forgotten vector valued differential forms. No mention about books on differential geometry on tensor valued differential forms like those of Kolar, Michor, and Mangiarotti etc.Kahler was right about spinors, but the reader will not find a definition of a spinor or of Kahler co-differential. Just a couple of impressions.Teleparallelism is the way, but no examples of great successes of teleparallel connections are given. Teleparallelism is supposed to help in fixing the problem of conservation of energy-momentum in general relativity, but these are just declaration, without any real work in this direction.Tensors are bad, Clifford algebras are better. But no pointers to the literature that treats both, tensors and Clifford algebras, deeply and from a wide perspective.The books seems to be written with the aim of impressing the reader. "Look how I am able to swim in the ocean of concepts". Nevertheless the book has some value as an entertaining literature. That is why I am giving it two stars instead of just one.

This book covers many interesting topics and could have been extremely interesting. Unfortunately, it is very poorly written and contains many misleading typographical errors and inconsistencies. More seriously, the author freely adorns his text with the jargon of fibre bundle and frame bundle theory without ever explaining in detail what that jargon means. He also keeps referring to pullbacks and pushforwards without defining what functions are doing the pulling back or the pushing forward, or what are the domains of these functions. This is most unfortunate and extremely frustrating for someone interested in getting a detailed understanding of this fascinating and important theory. Some of the sections, which ought to have been the most interesting, degenerate into a garbled string of indecipherable assertions. It is almost as if the author has been "baptized by the spirit" and feels compelled to speak to us in unknown tongues. The nett effect of these episodic breakdowns is that this book teaches us how to sound like experts without providing us with any expertise of substance.That said, if one first learns the theory of frame bundles from a superb book like Lawrence Conlon's Differentiable Manifolds (get the superior 2nd edition), one may be able to derive some benefit from Vargas' insights as to how Cartan generalized the Klein geometries to manifolds and developed his own unique (but generally ignored) approach to geometry.