Modern Differential Geometry Textbook Recommendation

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Could you provide recommendations for a good modern introductory textbook on differential geometry, geared towards physicists. I know physicists and mathematicians do mathematics differently and I would like to see how it is done by a physicists standard. I have heard Chris Ishams “Modern Diff Geometry for Physicists” is good in this respect but I don’t know how modern or at what level this is at. Theodore Frankels Geometry of Physics is mentioned a lot and highly regarded as being the most complete and comprehensive. I would like to get your opinions on what textbooks you think will be leading the way forward in physics classes on Diff Geom for Physicists?
 

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George Jones
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I would like to get your opinions on what textbooks you think will be leading the way forward in physics classes on Diff Geom for Physicists?
This is highly subjective, i.e., it is highly dependent on the course and instructor. For example, even though everything can be treated in the context of bundles, I think that (semi)Riemannian geometry should be separated out from the material on bundles. I think this for two reasons: 1) this is pedagogically better; 2) this is the way differential geometry underlying general relativity (semi-Riemannian) and gauge field theories (bundles) traditionally is treated. Of the books I mention below, Fecko, Nakahara, and Frankel all do this, while Isham doesn't.

I quite like Isham's book, but it might be a bit terse for self-study. Frankel proceeds at (I think) at a slightly more leisurely pace.

"Geometry, Topology, and Physics" by Nakahara is possibly the most standard text.

Folks here know that I am a big fan of "Differential Geometry and Lie Groups for Physicists" by Marian Fecko.

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).
I have found that this format works well for me, but other folks might have different opinions, though I know that some others here at PF also like Fecko.

Fecko is reviewed at the Canadian Association of Physicists website,

http://www.cap.ca/BRMS/Reviews/Rev857_554.pdf
 
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Thanks George, I appreciate your feedback. What do you mean by everything can be treated with bundles? Are bundles a kind of unifying mathematical tool? Do any of the books above take the approach to bundles?
 
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