Book on Differential Geometry/Topology with applications

[Joker93]

Hello!
I want to learn about the mathematics of General Relativity, about Topology and Differential Geometry in general. I am looking for a book that has applications in physics. But, most importantly, i want a book that offers geometrical intuition(graphs and illustrations are a huge plus) but does not shy away from the mathematics. Keep in mind that it should start from zero as i have no knowledge of these stuff.
Somebody told me to look for Nakahara's book but i found that it does not explain anything really properly. Also, i was told to check out Frankel's "Geometry in physics" and at a first glimpse i found it really good but i did not have the time to properly study from it.
If somebody could suggest a good book that has the stuff that i want and if somebody could tell me if Frankel's book is a good(great?) book, i would really appreciate it.
Thanks!

 

[Joker93]

No one?

 

[micromass]

No one?

That's because the books by Nakahara, Frankel and Hassani are exactly the books you're looking for in your OP. If you think those books don't explain the stuff properly, then I'm afraid you'll have to look at pure mathematics books. But it doesn't seem from your post like you want those...

 

[Joker93]

That's because the books by Nakahara, Frankel and Hassani are exactly the books you're looking for in your OP. If you think those books don't explain the stuff properly, then I'm afraid you'll have to look at pure mathematics books. But it doesn't seem from your post like you want those...

I like Frankel's book, but i just asked just in case there is any other hidden gem out there.
In your opinion, which is the best and worst of the three?

 

[micromass]

I like Frankel's book, but i just asked just in case there is any other hidden gem out there.
In your opinion, which is the best and worst of the three?

Frankly, I dislike all three. But I'm a pure mathematician, so you should probably want to ignore my opinion.

 

[Joker93]

Frankly, I dislike all three. But I'm a pure mathematician, so you should probably want to ignore my opinion.

haha! On the contrary, i think that a pure mathematician in a physics forum would be ideal!
What would you suggest? Keep in mind that i would want a book that also offers geometrical intuition so i can understand what everything means.

 

[micromass]

haha! On the contrary, i think that a pure mathematician in a physics forum would be ideal!
What would you suggest? Keep in mind that i would want a book that also offers geometrical intuition so i can understand what everything means.

I understand. Geometrical intuition is crucial of course. It would be very silly to read a math book without trying to understand what everything means geometrically.

The thing is that mathematics is an entire building built on certain foundations. These foundations are usually what provides the intuition for the more abstract concepts. So the notions of manifolds or topological spaces are abstractions and are directly motivated by what they are supposed to abstract.

I am of course completely willing to offer you a pathway through various math books. But the only way to properly do it is to start with the very foundations and go from there. So if you're willing to go this route, you need to tell me what math you are comfortable with and what not so much. It would help even more if you would link me books or texts that you feel understand well.

Finally, I wish to stress that knowing all this math I'm about to suggest is not necessary if your only goal is to learn GR. If you're interested in the mathematics behind it though, or if you wish to understand stuff more rigorously, then math books seem your best bet.

 

[bacte2013]

If you are interested in the geometric approach to the topology, I recommend the following books:
https://www.amazon.com/dp/1439848157/?tag=pfamazon01-20
https://www.amazon.com/dp/0387979263/?tag=pfamazon01-20

The second book is more advanced and fast-paced than the first, but it has outstanding exposition connecting the fields.

I do not know much about the differential geometry, but I think Hubbard/Hubbard has very good introduction to the basic concepts such as manifolds.

 

[Joker93]

I understand. Geometrical intuition is crucial of course. It would be very silly to read a math book without trying to understand what everything means geometrically.

The thing is that mathematics is an entire building built on certain foundations. These foundations are usually what provides the intuition for the more abstract concepts. So the notions of manifolds or topological spaces are abstractions and are directly motivated by what they are supposed to abstract.

I am of course completely willing to offer you a pathway through various math books. But the only way to properly do it is to start with the very foundations and go from there. So if you're willing to go this route, you need to tell me what math you are comfortable with and what not so much. It would help even more if you would link me books or texts that you feel understand well.

Finally, I wish to stress that knowing all this math I'm about to suggest is not necessary if your only goal is to learn GR. If you're interested in the mathematics behind it though, or if you wish to understand stuff more rigorously, then math books seem your best bet.

I wish to fully understand the mathematics behind physics. FULLY. If that is the rigorous way or the "physics" way, so be it.

I am comfortable with single-variable calculus, calculus of variations, complex analysis, linear algebra, Fourier analysis, differential equations(just the way to solve them and i already forgot half of the solutions that do not creep up often), i know a bit of multivariable calculus(although i will be taking a course shortly) and a bit of partial differential equations. Keep in mind that i am comfortable with these as they are being taught to physicists. My courses where not even remotely close to being rigorous(apart from linear algebra and calculus which where a little rigorous).

Please, do suggest any books that you want, either pure mathematics books or applied mathematics books. Just state in which field of mathematics each book goes into(applied or pure).

Thanks!

 

[atyy]

I like Fecko's book. It's covers the material similar to Nakahara and Frankel, but the presentation is funnier, so you may like it.
http://www.cambridge.org/us/academi...fferential-geometry-and-lie-groups-physicists

 

[Joker93]

I like Fecko's book. It's covers the material similar to Nakahara and Frankel, but the presentation is funnier, so you may like it.
http://www.cambridge.org/us/academi...fferential-geometry-and-lie-groups-physicists

Does it also share the same care for geometrical intuition?

 

[mathwonk]

here on the author's website you can read for yourself two sample chapters. i glanced at the one on tangent spaces to manifolds and rather liked it.

http://sophia.dtp.fmph.uniba.sk/~fecko/book.html