What kind of class is differential geometry?

[Eclair_de_XII]

This is my college's description for it:

Differential Geometry (3) Properties and fundamental geometric invariants of curves and surfaces in space; applications to the physical sciences. Pre: Calculus IV, and Introduction to Linear Algebra; or consent.

I was doing pretty well in all my Calculus classes, but I'm very certain that I would need to review a bit, since it's been over a year since I took Calculus IV. It's also implied that I need to know my Linear Algebra. I'm pretty good at solving systems of linear equations, but kind of had trouble with the orthonormalization and orthogonalization portion of the Linear Algebra class I took last summer.

So what kind of concepts would I need to review before going into this course? It talks about spaces and curves, so I assume I would need to review some Calculus III, as well? I'm a bit skeptical about taking this course over Intermediate Microeconomics, which I need for my VEE credit. On one hand, I might face a shortage of math classes that I can take within the constraints of my work schedule in the next year and a half. On the other, I kind of want to get Microeconomics out of the way so I don't have to worry about getting validation from that actuarial society anymore. It's a weak excuse, I know.

I'm also taking these two courses:

Introduction to Real Analysis (3) A rigorous axiomatic development of one variable calculus. Completeness, topology of the plane, limits, continuity, differentiation, integration. Pre: Calculus II, and Proof-writing.

Mathematical Modeling: Probabilistic Models (3) Probabilistic mathematical modeling emphasizing models and tools used in the biological sciences. Topics include stochastic and Poisson processes, Markov models, estimation, Monte Carlo simulation and Ising models and Ising models. A computer lab may be taken concurrently. Pre: Calculus II.

Anyway, I'm kind of rambling; what kind of course is Differential Geometry?

 

[fresh_42]

Anyway, I'm kind of rambling; what kind of course is Differential Geometry?

This book here is a good introduction:
http://www.springer.com/gp/book/9780387903576

Have a look at the table of contents. It pretty much sums up any answer which could be given.
Linear algebra and all kinds of analysis (real, complex, vector) would be a good preparation.

 

[Eclair_de_XII]

all kinds of analysis (real, complex, vector) would be a good preparation.

Question... Do you think it would still be doable if I didn't take any of those analysis courses?

I'm taking only one of them in the spring, and have not taken the other two yet.

 

[fresh_42]

I think differentiation and integration are most important. Of course you can always learn things in addition if you need them on a case by case basis. It always depends on how the course is held, whether applications and calculations or the more theoretical principles are emphasized. As a very short description it is about analysis on curved surfaces. Therefore tangent spaces (linear algebra) play a role as well as the analytic part (differentiation and integration), and the better you know the Euclidean cases, the easier you see the similarities. The point is, that those curved surfaces don't need to be embedded anywhere. They are all you have to do calculations on. In order to do that, the usual Euclidean cases serve as template and local approximation, and not as the global surrounding space anymore.

 

[Eclair_de_XII]

Well, if analysis courses aren't completely mandatory, then I suppose I can do decently in the course.

Also, when you say "Euclidean cases", do you refer to differentiation and integration of curves in Euclidean space? Such as parabolas and hyperbolas bounded by some line and such?

 

[fresh_42]

Well, if analysis courses aren't completely mandatory, then I suppose I can do decently in the course.

Also, when you say "Euclidean cases", do you refer to differentiation and integration of curves in Euclidean space? Such as parabolas and hyperbolas bounded by some line and such?

Yes, with Euclidean cases I mean everything which happens in $\mathbb{R}^n$ or $\mathbb{C}^n$, i.e. functions defined there. In differential geometry we cancel this condition. The spaces are now e.g. spheres, groups or any other curved space. They don't need to be inside a Euclidean space anymore. Partly it began with measuring distances on earth. The planet isn't something in a three dimensional box, it is all we have. So the question came up, if analysis on such an object can be done. Can we differentiate curved function graphs which are themselves on a curved surface? Paths on a sphere, and not paths in an angular box. Differential geometry was the positive answer to these questions. It is a bit tricky, for you need to have some abstraction skills, as it leaves known territory, but calculation skills as well, as in the end it is about computations. Black holes are extreme examples for something differential geometry is needed for in order to understand their geometry. The many questions we get about them are often because people think of them as an especially heavy ball. But the entire space around them is curvy - no Euclidean geometry near.

 

[Eclair_de_XII]

In differential geometry we cancel this condition. The spaces are now e.g. spheres, groups or any other curved space. They don't need to be inside a Euclidean space anymore.

So what you're saying is that differential geometry now considers functions on subspaces that are not subsets of Euclidean space? I can kind of see where the linear algebra comes in... I mean, I've covered material on subspaces and its eight main rules in the two linear algebra classes I've taken. So I'm a bit glad that I can apply what I learned before here.

But I am surprised that the upper division math classes I've taken so far don't make much use of the material I learned from the later parts of the Calculus series, namely that of III and IV. A little bit disappointed to be honest.

Black holes are extreme examples for something differential geometry is needed for in order to understand their geometry.

I've also heard that Einstein formulated some theory about how matter produces distortions (gravity) in space-time. I ask out of curiosity: Would differential geometry also apply here?

 

[Orodruin]

I've also heard that Einstein formulated some theory about how matter produces distortions (gravity) in space-time. I ask out of curiosity: Would differential geometry also apply here?

Yes. Differential geometry is the language in which general relativity (and many other physics models) is naturally written. You cannot learn GR without knowing differential geometry or learning it in parallel.

 

[Eclair_de_XII]

Oh, I see. I was just wondering. Thanks for answering.