The discussion revolves around recommendations for introductory differential geometry books suitable for someone preparing to study general relativity. Participants share their experiences with various texts and express their preferences based on different levels of prior knowledge in topology and geometry.
Participants generally agree on the value of different texts for varying levels of prior knowledge, but there is no consensus on a single best book. Some express preferences for specific authors or styles, while others raise concerns about the availability of solutions for self-study.
Participants note that the rigor and pedagogical approach of differential geometry texts can vary significantly, and the effectiveness of learning may depend on individual preferences and backgrounds in topology.
This discussion may be useful for students and self-learners interested in differential geometry, particularly those preparing for advanced studies in general relativity or related fields.
WannabeNewton said:Hi guys. I finished working through D'Inverno's "Introducing Einstein's Relativity" and Schutz's "A First Course in General Relativity" and some of Carroll's "Spacetime and Geometry" but I don't really feel like I learned most of what is out there. I also feel that before I can tackle Wald I need to read up on a proper introductory differential geometry book. If anyone could recommended me one that would be great. Thanks in advance.
George Jones said:As n!kofeyn has stated, contents of differential geometry references vary widely. Another book worth looking at is Differential Geometry and Lie Groups for Physicists by Marian Fecko,
https://www.amazon.com/dp/0521845076/?tag=pfamazon01-20.
This book is not as rigorous as the books by Lee and Tu, but it more rigorous and comprehensive than the book by Schutz. Fecko treats linear connections and associated curvature, and connections and curvature for bundles. Consequently, Fecko can be used for a more in-depth treatment of the math underlying both GR and gauge firld theories than traditionally is presented in physics courses.
Fecko has an unusual format. From its Preface,
A specific feature of this book is its strong emphasis on developing the general theory through a large number of simple exercises (more than a thousand of them), in which the reader analyzes "in a hands-on fashion" various details of a "theory" as well as plenty of concrete examples (the proof of the pudding is in the eating).
The book is reviewed at the Canadian Association of Physicists website,
http://www.cap.ca/BRMS/Reviews/Rev857_554.pdf.
From the review
There are no problems at the end of each chapter, but that's because by the time you reached the end of the chapter, you feel like you've done your homework already, proving or solving every little numbered exercise, of which there can be between one and half a dozen per page. Fortunately, each chapter ends with a summary and a list of relevant equations, with references back to the text. ...
A somewhat idiosyncratic flavour of this text is reflected in the numbering: there are no numbered equations, it's the exercises that are numbered, and referred to later.
Personal observations based on my limited experience with my copy of the book:
1) often very clear, but sometimes a bit unclear;
2) some examples of mathematical imprecision/looseness, but these examples are not more densely distributed than in, say, Nakahara;
3) the simple examples are often effective.
micromass said:If you know some topology, then I recommend "Introduction to smooth manifolds" by John Lee.
If you don't know topology, then I would go for "A comprehensive introduction to differential geometry" by Spivak.
mathwonk said:here (at the bottom of the web page), is a link to a free first course in differential geometry by a student of the great S.S.Chern. Shifrin is an excellent teacher and author and a professional differential geometer as well. I do not know about answers, but most good books do not give answers to exercises. Learning to know whether your answer is right without being told is considered a valuable skill.
http://www.math.uga.edu/~shifrin/