# Differential Geometry vs. General Relativity

1. Dec 6, 2014

### pierce15

Hello,

This spring, I will have the opportunity to do a one-on-one independent study in math or physics. I've narrowed down my choices to differential geometry and general relativity. I'm thinking about the future here- will studying general relativity this spring better prepare me for differential geometry in the future, or vice versa? Or does it not make a difference?

Also, while we're here, if anyone has recommendations for differential geometry books, feel free to share them.

2. Dec 6, 2014

### ShayanJ

For understanding General Relativity, you need to know differential geometry but not vice versa! I think its better to take differential geometry, better in a way that its strange for me to that such a question is being asked!

3. Dec 6, 2014

### pierce15

If I studied general relativity, I would learn from a textbook that teaches tensors and necessary differential geometry as necessary (Schutz, A First Course in General Relativity). Sorry that I didn't make that clear in the first post.

4. Dec 6, 2014

### ShayanJ

That'll work. But it brings up the question why do you want to learn differential geometry? Is that course an introductory course to geometry of curves and surfaces in Euclidean space or a course in Riemannian geometry or a course in manifold geometry?
Because if you want to learn differential geometry only because of learning GR and that course is only a course on curves and surfaces or on Riemannian geometry, then there is no point in taking it after learning GR.
But if you want it not only for GR and that course is more general than only Riemannian geometry, then its good to take if after GR.

5. Dec 8, 2014

### pierce15

Either course could really go in whatever direction I decided to take it. My motive in taking either is to prepare partly for future math, and partly for future physics. (With a stronger bias one way depending on whether I learn GR or DG.)

6. Dec 8, 2014

### sara.w

To the OP:

I just finished a course in differential geometry and we studied the text Geometrical Methods of Mathematical Physics (Bernard Schutz). One thing to note from that book is that it is written from more of a physics background than a math background as he is not always very careful in his notation. If your instructor is very careful in studying the text he will pick up on the small deficiencies.

The good thing about that kind of course is that it sets you up for classical mechanics, quantum mechanics, and general relativity. Some of the topics we studied was Hilbert spaces, transition functions, fibre bundles, tangent bundles, cotangent bundles, tensors, and one forms. It is a really rewarding course and it introduces you to the language used in higher level physics.

7. Dec 8, 2014

### ShayanJ

So its a good idea to take GR and then DG. But in the DG course, you should be learning about manifold geometry so that things are more general than the thins you learn in the GR course. Its good for both physics and mathematics. The book that sara mentioned and also the topics are a good example of what you should be expecting.(Although I don't think Hilbert Spaces is a standard topic in a DG course!) But as sara mentioned, schutz's book is for physicists. But I think in a DG course, the professor is a mathematician and uses textbooks written by mathematicians.

8. Dec 9, 2014

### Arsenic&Lace

DG is like grammar; there's far more to GR than DG, in the same way that writing a novel isn't really, at the end of the day, about English grammar.

You'll pick up the language to some extent studying GR so DG might make more sense; in the same manner, learning the grammar really well might make GR a bit more accessible, but you can easily take one without the other.

The real question to ask is what do you think you want to do, and which seems more interesting?

9. Dec 9, 2014

### pierce15

Thanks, everyone. I've decided to pursue GR.