# Quick Diff Geometry question

1. Jun 3, 2005

### sparkster

I'm taking a class in differential geometry in the fall and I wanted to be sure to be well prepared. Are there any basics that I might need to bone up on before showing up?

2. Jun 3, 2005

### quetzalcoatl9

You might find it useful to learn a few fundamental concepts to give you a headstart in the course.

For example, viewing a vector as a differential operator:

$$\vec{v} (f) = v^i \frac{\partial{f}}{\partial{x^i}} = D_v f$$

When first learning diff.geom., i found this a tough idea to get my arms around, but it really does make sense.

This is different than how you are probably used to viewing vectors. Normally you think of: given a function, what is the directional derivative with respect to a vector? Here, we are saying that given a fixed vector, we can vary the function to give the directional derivative. Notice that the directional derivative is a real number, so $\vec{v} : F \rightarrow R$

Since it really doesn't matter what function it is, we can say $\vec{v} = v^i \frac{\partial}{\partial{x^i}}$, so that the vector basis is actually just a function of our coordinates.

So reading up on tangent vectors and coordinate systems might be good. Also learning the definition of a differentiable manifold is probably the best place to start before any of this.

3. Jun 3, 2005

### mathwonk

review derivatives: i.e. the definition of the derivative of f at a point p, as a linear transformation L such that for every direction vector v at p, the directional derivative of f in the direction v at p, equals L(v).

thats the main thing.

they usually review stuff like greens theorem and the inverse function theorem, but that wouldn't hurt either.

thats pretty much all there is to know actually: the concept of a derivative as a linear map, stokes theorem, and the inverse function theorem.

4. Jun 3, 2005

### sparkster

Thanks guys. It sounds like it's pretty much those sections of linear algebra that you talked about and then some vector calculus.

Do you know if Schaum's Outlines in Diff Geo is any good? And have you ever seen Millman's book on Diff Geo?

5. Jun 3, 2005

### mathwonk

no, but i liked millmans book on non eucliden geometry. i.e. he is smart and writes well.

6. Jun 4, 2005

### Haelfix

You will want to know what a tangent space is (there are three good definitions that are equivalent, see for instance Spivak vol 1), and the cotangent space (ie the dual of the tangent space). Thats really the basics, everything else is generalization from there.

Really the study of differential geometry is a case study in triviality at this point in time, the only hard part is getting the notation down and unlearning some bad habits (which is surprisingly hard the first time around).

7. Jun 4, 2005

### sparkster

He's teaching the class and using his own book. Some texts I've had in the past have been next to useless so sometimes I'll buy a backup. If you liked his other book, then the diff geo one will probably be pretty good too.

8. Jun 4, 2005

### mathwonk

If this class is being taught by Richard S Millman, I think it should be excellent. In my view the world's best differential geometry book is by spivak, especially volume 2. vol 1 is a big load of abstract background on manifolds, and makes it hard, or at least long, to get to vol 2.

it is probably important to also have a short version, like maybe millman and parker, or manfred docarmo, or the notes by Ted Shifrin from University of Georgia.

9. Jun 4, 2005

### sparkster

Yeah, he's the guy; he's new to the university. I think the class text is "Elements of Differential Geometry" but I'll ask him about the short version. I usually like to have a couple of texts for classes like this.

10. Jun 4, 2005

### mathwonk

actually i meant millman and parker IS the short version, compared to spivaks 5 volumes.

check out your new prof's bio on the web. hes a big time university professor, teacher, author and administrator, even former univ president. and he has participated in a lot of high level instructional activity, as well as written intelligent books which treat a subject thoughtfully, not ones designed to make the most money possible by talking down to the audience.

so if this diff geom book is like his geometry book, it will have a good pedagogical point of view, as well as significant mathematical underpinnings. You never know what someone's classroom style will be like, but I would welcome the chance to learn the subject from him myself.

Last edited: Jun 4, 2005
11. Jun 4, 2005

### sparkster

Oh, I see. I've not gotten the book yet, so I didn't know how long or short it was.

I had heard that was a former university president, but I didn't know much else about him. I do know that's really excited about the class; I went to his office to ask a few questions about it and got an impromtu lesson on lie algebra. After your posts I'm looking forward to the class even more. Thanks!

12. Jun 8, 2005

### pmb_phy

That depends on how quickly you can pick up ideas. You should learn what you need to know in class if the teacher and material is good. Pick up a text on differential geometry and learn it over the summer. That's what I used to do and it worked out well for me. Differential Geometry, by Erwin Kreyszig is pretty good.

Pete

13. Jun 8, 2005

### pmb_phy

Pete

14. Jun 8, 2005

### sparkster

The math library had a copy of schaum's outlines for differential geometry. I'm working my way through it.

I'll let you know when I pick it up.

15. Jun 8, 2005

### mathwonk

those schaum's notes are going to be extremely old fashioned and old style in both concept and notation. you do not want to be stuck in the bind of only knowing indicial notation for differential geometry as so many physicts are even today. i recommend you look at ted shifrin's free notes on his website at the university of georgia. shifrin studied with chern and is a master both of the subject and of exposition.

see:

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

16. Jun 9, 2005

### sambo

Having been put through the workout that was Spivak, as well as Bishop and Goldberg, Warner, de Rham (actually quite good, albeit difficult), and more, I can honestly say that all of these make a poor choice for a first course on differential geometry. While Spivac keeps things... simple(?), I found that the results are all hidden under mounds of text, which in turn made it difficult to figure out what was actually important. On the other hand, the others seemed to be intended for an audience that already had a basic working knowledge of the subject, hence making them inaccessable to a beginner.

Then I found "Lectures on Differential Geometry", by S. S. Chern et. al. which had been, in my opinion, the single best introductory text on the subject I hae come across to date. The pace is quick, yet nothing is skipped, the examples are excellent, and the pedagodgy is very well thought out. So, if you want to go into this course smelling like roses, I would recommend this text in conjunction with the exercises in Bishop and Goldberg.

Just my opinion though...

17. Jun 9, 2005

Millman and Parker's "Elements of Differential Geometry" was published in 1977 (I think) by Prentice Hall. I didn't know it's been republished. It's a decent introductory text. I think the authors use the first and second fundmanental forms. O'Neill's "Elementary Differential Geometry" is pitched at a similar level, but is more modern in outlook (despite being being published in 1967): the book could be subtitled, "The method of moving frames a la Elie Cartan, for children." Another more recent book which I'd unequivocally recommend is John Oprea's "Differential Geometry and It's Applications." The approach is similar to O'Neill, but a bit more elementary (e.g. he doesn't use exterior calculus, connection forms and Cartan's structural equations but does use shape operator and covariant derivatives). You should work through chapters 1,2,3, 5, and 6. You can get used copies (first editions) for maybe \$10-15. I don't like do Carmo's undergrad text, though I wax lyrical about his grad text (Riemannian Geometry). Steer clear of both the Schaum series book and Kreyszig: they're both way too old-fashioned. Once you've made your way through one of the undergrad texts above, you'll be in a position to appreciate Spivak's magnum opus.

Postscript: There's also a relatively recent book out by Andrew Pressley, titled "Elementary Differential Geometry", published by Springer. It's a bit old-fashioned, but he gets to the meat of the subject rapidly. It's more elementary than the other texts. I'd still stick with Oprea or O'Neill.

Last edited: Jun 9, 2005
18. Jun 9, 2005

### mathwonk

another excellent book, unfortunately out of print for years, is by noel j. hicks, something like "notes on differential geometry".

but please check out shifrin, at his UGA webpage. these are excellent modern notes and FREE, for the time being.

19. Jun 10, 2005