Modern Differential Geometry Textbook Recommendation

In summary, the books "Differential Geometry and Lie Groups for Physicists" by Marian Fecko, "Geometry, Topology, and Physics" by Nakahara, and "Modern Diff Geometry for Physicists" by Chris Isham are all recommended for introductory texts on differential geometry geared towards physicists.
  • #1
kay bei
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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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  • #2
kay bei said:
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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Related to Modern Differential Geometry Textbook Recommendation

1. What is differential geometry?

Differential geometry is a branch of mathematics that studies the properties of curves and surfaces in higher-dimensional spaces. It uses techniques from calculus and linear algebra to analyze the geometric structure of these objects.

2. Why is a modern textbook on differential geometry recommended?

A modern textbook on differential geometry is recommended because it provides a comprehensive and up-to-date understanding of the subject. It may include recent developments and applications, making it more relevant for current research and practical use.

3. What are the key topics that should be covered in a modern differential geometry textbook?

A modern differential geometry textbook should cover topics such as manifolds, tangent spaces, curvature, connections, and Riemannian geometry. It should also include applications in physics, engineering, and other fields.

4. Are there any specific authors or books that are highly recommended for studying modern differential geometry?

There are many excellent authors and books on modern differential geometry, and the choice may depend on individual preferences. Some popular options include "Differential Geometry of Curves and Surfaces" by Manfredo do Carmo, "Riemannian Geometry" by Peter Petersen, and "Modern Differential Geometry for Physicists" by Chris J. Isham.

5. How can studying modern differential geometry benefit a scientist?

Studying modern differential geometry can benefit a scientist in many ways. It can provide a deeper understanding of geometric structures and their applications in various fields such as physics, engineering, and computer science. It can also enhance problem-solving skills and critical thinking abilities, which are essential for scientific research and analysis.

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