Book Recommendations in Differential Geometry

[Wrichik Basu]

I wanted to study General Relativity, but when I started with it, I found that I must know tensor analysis and Differential geometry as prequisites, along with multivariable calculus.

I already have books on tensors and multivariable calculus, but can anyone recommend me books on differential geometry?

My knowledge so far: Plane Trigonometry complete, single variable calculus - almost finished.

 

[Demystifier]

Most books on general relativity have some chapters on differential geometry, so it is not necessary to learn differential geometry from a specialized book.

 

[MathematicalPhysicist]

Spivak's and Munkres's.
Comprehensive intro to DG
Analysis on Manifolds

respectively.

Edit: I must add that I nearly finished with Munkres's text, and Spivak's I finished only chapter 1. (but in Munkres's I haven't done the exercise and in Spivak's I did just a few of the exercises in chapter 1.).

I don't have the time to do the exercise as I am worked out on other matters.

 

[Demystifier]

Spivak's and Munkres's.
Comprehensive intro to DG
Analysis on Manifolds

I don't think those are appropriate to someone who is just finishing single-variable calculus.

 

[MathematicalPhysicist]

I don't think those are appropriate to someone who is just finishing single-variable calculus.

Obviously he first needs to go through multivariable calculus and linear algebra of finite dimensional vector spaces.

After these two topics he should go to Analysis on manifolds by Munkres.

 

[PeroK]

I wanted to study General Relativity, but when I started with it, I found that I must know tensor analysis and Differential geometry as prequisites, along with multivariable calculus.

I already have books on tensors and multivariable calculus, but can anyone recommend me books on differential geometry?

My knowledge so far: Plane Trigonometry complete, single variable calculus - almost finished.

If you learn GR from Hartle's book, then he minimises the mathematical requirements. That said, GR is still an advanced subject.

https://www.amazon.com/dp/0805386629/?tag=pfamazon01-20

You will definitely need the vector calculus, but you won't need differential geometry as a subject in its own right. Depending on your maths skills, you should get by with the book itself, or a bit of reading up on the side.

And, you won't need tensor analysis until the later chapters. And, likewise, a bit of focused reading up is better than studying the subject in a mathematical form that might be very different from the GR book you are using.

You will, however, need to have Special Relativity absolutely nailed. Any weakness in your understanding of SR will be a serious problem.

My advice is to do vector calculus, then get the GR book and see how you go. Then, choose the maths resources to fit the GR text.

 

[Demystifier]

After these two topics he should go to Analysis on manifolds by Munkres.

Munkres my be a good advice for mathematical physicists, but not for the majority of physicists who just want to pick up a minimal knowledge of mathematics needed to understand the GR physics. And even for mathematical physicists, there are a plenty of differential-geometry books written for physicists, Munkres not being one of them.

 

[George Jones]

After these two topics he should go to Analysis on manifolds by Munkres.

I disagree with this recommendation for two reasons: 1) it is not mathematical enough; 2) it is too mathematical.

It is not mathematical enough because it only treats manifolds as subsets of R^n, and it does not cover anything to do with the (semi)Riemannian manifolds necessary for general relativity.

It is too mathematical because I agree with Demystifier and PeroK.

 

[Wrichik Basu]

If you learn GR from Hartle's book, then he minimises the mathematical requirements. That said, GR is still an advanced subject.

https://www.amazon.com/dp/0805386629/?tag=pfamazon01-20

You will definitely need the vector calculus, but you won't need differential geometry as a subject in its own right. Depending on your maths skills, you should get by with the book itself, or a bit of reading up on the side.

And, you won't need tensor analysis until the later chapters. And, likewise, a bit of focused reading up is better than studying the subject in a mathematical form that might be very different from the GR book you are using.

You will, however, need to have Special Relativity absolutely nailed. Any weakness in your understanding of SR will be a serious problem.

My advice is to do vector calculus, then get the GR book and see how you go. Then, choose the maths resources to fit the GR text.

I started reading GR from this book:

So, do I need to concentrate on differential geometry first the time being, or should I just study vector calculus and start?

 

[Wrichik Basu]

If you learn GR from Hartle's book, then he minimises the mathematical requirements. That said, GR is still an advanced subject.

https://www.amazon.com/dp/0805386629/?tag=pfamazon01-20

You will definitely need the vector calculus, but you won't need differential geometry as a subject in its own right. Depending on your maths skills, you should get by with the book itself, or a bit of reading up on the side.

And, you won't need tensor analysis until the later chapters. And, likewise, a bit of focused reading up is better than studying the subject in a mathematical form that might be very different from the GR book you are using.

You will, however, need to have Special Relativity absolutely nailed. Any weakness in your understanding of SR will be a serious problem.

My advice is to do vector calculus, then get the GR book and see how you go. Then, choose the maths resources to fit the GR text.

Which book is better: Grøn or Hartle?

 

[PeroK]

Which book is better: Grøn or Hartle?

I only know Hartle. And, my advice is based on my being a recent student of GR, rather than an expert.

If you've got Gron, stick with that and see how you get on. It looks quite interesting.

 

[smodak]

Have you finished classical mechanics? If not, you may want to do that first including solving problems. From all your posts in this forum, it seems to me that you are trying to jump ahead and avoid prerequisites. This is not a good idea. Instead of learning something here and learning something there, I would advice you to follow some sort of standard math and physics undergrad curriculum. Once you finish that, start GR. Just my 2 cents.

 

[Wrichik Basu]

Have you finished classical mechanics? If not, you may want to do that first including solving problems. From all your posts in this forum, it seems to me that you are trying to jump ahead and avoid prerequisites. This is not a good idea. Instead of learning something here and learning something there, I would advice you to follow some sort of standard math and physics undergrad curriculum. Once you finish that, start GR. Just my 2 cents.

I have completed reading most chapters in classical, in some, problems are left. I am fond of GR more than classical, and hence...

 

[smodak]

I have completed reading most chapters in classical, in some, problems are left. I am fond of GR more than classical, and hence...

I am not trying to pick a fight with you not am I trying to undermine or discourage you, I am merely trying to point you in the right direction. I do not understand fully what you are saying or exactly what your background is, but do you know and can solve problems using action principle, for example? How can you be fond of GR without even knowing it? Understanding GR requires a bit of mathematical maturity let alone a good grasp of basic physics. You can surely read some popular treatment on the subject including those written by Einstein, but if you jump right into the rigorous GR treatments without, for the lack of a better word, enough practice of the fundamentals, I have a hunch that you may get frustrated. Having said that, start with Gron or Collier and then go to Hartle. You surely need to master SR first and a thus a bit of Electromagnetism.

 

[Wrichik Basu]

I am not trying to pick a fight with you not am I trying to undermine or discourage you, I am merely trying to point you in the right direction. I do not understand fully what you are saying or exactly what your background is, but do you know and can solve problems using action principle, for example? How can you be fond of GR without even knowing it? Understanding GR requires a bit of mathematical maturity let alone a good grasp of basic physics. You can surely read some popular treatment on the subject including those written by Einstein, but if you jump right into the rigorous GR treatments without, for the lack of a better word, enough practice of the fundamentals, I have a hunch that you may get frustrated. Having said that, start with Gron or Collier and then go to Hartle. You surely need to master SR first and a thus a bit of Electromagnetism.

Fight?? No offence. There is no fight. I just want to learn GR, b ut if I cannot do it now without maths, I'll preserve it for reading after 2 years perhaps. No problem.

 

[vanhees71]

The minimal math you need for GR is what you need for classical electrodynamics in the four-dimensional Minkowski-space formulation. Everything else is usually contained in good introductory GR books. My favorite is Landau&Lifshitz vol. II, which introduces the physics without too much of the finer mathematical details which the experts need and like but which distract from learning the physics. There you also get a brillant introduction to classical electrodynamics in a modern way, using the relativistic point of view from the very beginning, which imho is the only approach to the subject for the 21st-century student anyway! So I'd recommend to check out this book to see, whether you have the prerequisites for it, i.e., mostly linear algebra and classical vector analysis.

 

[smodak]

Fight?? No offence. There is no fight. I just want to learn GR, b ut if I cannot do it now without maths, I'll preserve it for reading after 2 years perhaps. No problem.

You misunderstood me. You did not do anything wrong. I felt like I was being too harsh on you. You kept asking for books and I kept telling you to get the prerequisites done. I wanted to tell you that I am merely trying to help. Btw, I am originally from Kolkata as well. Bengali. But I immigrated to the US in 1998.

 

[Daverz]

For a first pass, don't worry about specialized differential geometry books. You may be able to get a lot out of bcrowell's GR book, even if you may have to skip some of the math.

http://www.lightandmatter.com/genrel/

 

[noir1993]

If you want to study differential geometry with the aim of studying general relativity, and have some idea about linear vector spaces and linear transformations, you can look into Part II of Spacetime, Geometry and Gravitation by Pankaj Sharan, Hindustan Book Agency. The international publisher is probably Springer.
I'm recommending this book because it requires only the knowledge of linear algebra. But I would recommend you invest some time in studying vector calculus, advanced classical mechanics (Lagrangian and Hamiltonian formalism), variational principle and special relativity before jumping into GR.

 

[noir1993]

I started reading GR from this book:

View attachment 205160

So, do I need to concentrate on differential geometry first the time being, or should I just study vector calculus and start?

I skimmed through this book. Almost half of the book is devoted to developing concepts like vectors and differential calculus etc. I would suggest that if your are not comfortable with the first four chapters of the book, you should not proceed further. Those four highly condensed chapters contain enough stuff to keep a good high school student busy for at least a year. And Chapters 5-10 is a one semester advanced undergraduate course in its own right! GR starts from Chapter 11. Although I do not like this approach too much but I'm merely pointing this out since you have started reading this book. I do not want to discourage you but I would advise you to proceed with caution so that you don't become frustrated. These things tend to take a bit of time. :)