Rudin type book for differential geometry and algebra

In summary, for Algebra, Hungerford's Algebra and Roman's Advanced Linear Algebra are recommended. For Differential Geometry, Darling's Differential Forms and Connections, Spivak's Comprehensive Introduction to DG, Serge Lang's Fundamentals of DG, and Barden and Thomas' An Introduction to Differential Manifolds are suggested. Additionally, the book https://www.amazon.com/dp/0821839888/?tag=pfamazon01-20 is recommended for traditional DG topics at an advanced level.
  • #1
I'm currently taking graduate courses on differential geometry and algebra. What books are closest to the style of Rudin for these areas (i.e. rigorous, developing the theory in apropriate generality and being elegant at the same time).

For Algebra, I guess Lang is the bible, but what else is there? I would especially welcome some book that covers some advanced linear algebra, like symplectic and complex structures, matrix groups, etc.

For Differential Geometry, I have already tried a lot of books, but none of them really fit my needs. Kobayashi and Nomizu is almost unreadable for me and it deals mostly with bundles. On the other hand, there's Lee's Introduction to smooth manifolds, which has great list of topics, but I find his way of writing ugly. So topic-wise I'm searching for something like Lee, but done in a more elegant way. Is there anything like that?
 
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  • #2
Suggestions:
Algebra: Hungerford's Algebra, Roman's Advanced Linear Algebra
Differential Geometry: Darling's Differential Forms and Connections, Spivak's Comprehensive Introduction to DG, Serge Lang's Fundamentals of DG, Barden and Thomas' An Introduction to Differential Manifolds.
 
  • #3
Darling's book is very cool, but the emphasis in the later half is on bundles, if the "connections" part didn't clue you in. Nice for physicists who want to get up to speed on this.

I would suggest checking out https://www.amazon.com/dp/0821839888/?tag=pfamazon01-20, who covers more traditional differential geometry topics but at a more advanced level than the typical undergraduate text.
 
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1. What is a "Rudin type book" for differential geometry and algebra?

A "Rudin type book" refers to a textbook that follows the style and approach of the influential mathematician Walter Rudin. Rudin's textbooks are known for their rigorous proofs, clear explanations, and emphasis on fundamental concepts.

2. Is a "Rudin type book" suitable for beginners in differential geometry and algebra?

No, a "Rudin type book" is typically more advanced and rigorous than introductory textbooks. It is best suited for students who already have a strong foundation in mathematics and are looking to deepen their understanding of differential geometry and algebra.

3. Can a "Rudin type book" be used for self-study?

Yes, some students may find a "Rudin type book" to be a helpful resource for self-study. However, it is important to have a strong background in mathematics and to be comfortable with abstract concepts before attempting to use this type of textbook for self-study.

4. How does a "Rudin type book" differ from other textbooks on differential geometry and algebra?

A "Rudin type book" typically places a strong emphasis on developing a solid understanding of fundamental concepts and their interrelationships, rather than just presenting a list of formulas and techniques. It also tends to have a more formal and concise writing style, with a focus on rigorous proofs.

5. Are there any recommended "Rudin type books" for differential geometry and algebra?

Yes, some popular "Rudin type books" for differential geometry and algebra include "Principles of Mathematical Analysis" by Walter Rudin, "Differential Geometry: A First Course" by D. Somasundaram, and "Algebraic Topology: A First Course" by William Fulton.

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