Book Recommandation - Differential Geometry

[WannabeNewton]

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.

 

[micromass]

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.

 

[George Jones]

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.

For theoretical physicists, a good comprise between physics-style and math-style presentations of math might be Fecko's book. Unfortunately, I think that it is at its pedagogically worst in its first chapter. It was from Fecko's book that I leaned how to do the calculations with Killing vectors to which I earlier pointed you.

As micromass said, Lee's book is nice, but, for GR, you would need the sequel, Riemannian Manifolds: an Introduction to Curvature. I have Lee's books on my bookshelf, and I often dip into them. I found the second book particularly useful for its treatment of conjugate points.

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.

 

[WannabeNewton]

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.

Hi. Does the spivak book have any solutions to the exercises as I am self - studying? Thanks.

 

[WannabeNewton]

Thanks George I'll take a look at Fecko's book.

 

[mathwonk]

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/

 

[WannabeNewton]

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/

Yeah it would be but I am currently a junior in high school so I really have no resource to check or aid myself other than this website in the event that I really can't understand a problem.

 

[mathwonk]

schaums outline series used to have a differential geometry "solved problems" book but i don't recommend it, kind of old fashioned. loosen up. try to do without answers. you'll be way ahead of the game.