Beginner's foray into Diff Geometry

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Discussion Overview

The discussion revolves around recommendations for beginner-friendly resources in differential geometry (DG), particularly in the context of learning for applications in general relativity (GR) and other fields. Participants share their experiences with various textbooks and express their preferences based on different backgrounds and goals.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant finds "Differential Geometry" by Erwin Kreyszig difficult and outdated, seeking alternative recommendations.
  • Another participant suggests "Elementary Differential Geometry" by O'Neill and "Differential Geometry of Curves and Surfaces" by Do Carmo as better options.
  • A participant mentions a personal recommendation for a book linked to Amazon, indicating a positive experience.
  • Discussion about the importance of aligning book choices with specific goals in DG, such as applications in physics, chemistry, or mathematics.
  • Suggestions include John Lee's trio of books for those with some topology knowledge and "Curvature in Mathematics and Physics" by Shlomo Sternberg for a more physics-oriented approach.
  • One participant shares a free online text that uses classical notation, which may be beneficial for beginners.
  • There is a query about whether the learning is specifically for GR, with a participant expressing a desire to learn DG in general as well.
  • Another participant emphasizes the need for a solid foundation in linear algebra and multivariable calculus before tackling DG or GR.
  • A recommendation for "Gravity: An Introduction to Einstein's General Relativity" by James B. Hartle is made, with a note on its suitability for beginners if they have the necessary mathematical background.

Areas of Agreement / Disagreement

Participants express differing opinions on the suitability of various textbooks, with no consensus on a single recommended resource. There is also a shared understanding of the importance of prior mathematical knowledge, but specific pathways to learning DG remain contested.

Contextual Notes

Participants highlight the challenges posed by outdated notation and approaches in older texts, as well as the varying levels of mathematical background required for different books.

Who May Find This Useful

Beginners in differential geometry, students preparing for general relativity, and those interested in the mathematical foundations of physics may find this discussion helpful.

tade
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I would like to learn DG so I picked up "Differential Geometry" by Erwin Kreyszig.

I'm finding it too difficult to understand, especially the notation. What books on DG would you guys recommend to beginners?
 
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That book is horrible and extremely outdated. Get O'Neill "Elementary Differential Geometry" or Do Carmo "Differential Geometry of Curves and Surfaces".
 
What are your goals and background? Differential geometry books with different goals are quite different. Are you most interested in applications to physics, chemistry, biology, engineering, mathematics, or something else?
I would say if you know some topology (or want to learn some) you could go with the John Lee trio
Introduction to Topological Manifolds
Introduction to Smooth Manifolds
Riemannian Manifolds: An Introduction to Curvature

If you like some physical stuffs and know a little calculus and linear algebra a good start would be
Curvature in Mathematics and Physics by Shlomo Sternberg. I have only read a bit of it but the price, quality, and minimal required background make it a good choice for a first exposure.

A supplement of some interest is Riemannian Geometry: A Beginners Guide Frank Morgan

Some times people suggest something like Elementary Differential Geometry by Pressley. I do not care much for that book. It is easier because it is so limited in scope. I do not think time with it is well spent.

Yes old timey books can be challenging due to the notation and approach.
 
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Here's a free online text:

http://www.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf

The notation used is the classical notation.
 
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By the way tade, is this for GR?
 
WannabeNewton said:
By the way tade, is this for GR?

Initially, yes. But I would like to learn DG in general too.
 
And preferably something noob-friendly. :-p
 
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tade said:
Initially, yes. But I would like to learn DG in general too.
Ah ok. That's cool. I was going to say that if this was for GR specifically then most GR texts by themselves offer good introductions to tensor calculus and differential geometry as used in GR. What's your background in math by the way?
 
  • #10
WannabeNewton said:
Ah ok. That's cool. I was going to say that if this was for GR specifically then most GR texts by themselves offer good introductions to tensor calculus and differential geometry as used in GR. What's your background in math by the way?
12th grade AP calc.
 
  • #11
Ok well you will need to know linear algebra and multivariable calculus first before jumping into either classical differential geometry of surfaces or GR.
 
  • #12
If your single-variable calculus is good, then I really recommend this book: https://www.amazon.com/dp/0130414085/?tag=pfamazon01-20

It is a solid intro to multivariable calculus, linear algebra and diffential forms. I think it perfectly fills up the gaps that still Lie between you and differential geometry.
 
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  • #13
Do you think Gravity: An Introduction to Einstein's General Relativity by James B. Hartle is a good book for noobs?

Though I probably need to finish Hubbard's book first.
 
  • #14
If you know LA and Calc 3 then yes, I would highly highly recommend that book. It's brilliant.
 

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