1. ## What is this?

I am interesting in learning about the math which was greatly developed by Riemann. How is it called? And what does one need to know to follow it?

2. I think you are referring to differential geometry - right?

Know the story on how Gauss chose (wickedly!) this subject for Riemann to present as his doctoral thesis?

3. Originally Posted by Rebesques
I think you are referring to differential geometry - right?
That sounds right. Can you give me different topics in differential geometry?

Originally Posted by Rebesques
Know the story on how Gauss chose (wickedly!) this subject for Riemann to present as his doctoral thesis?
No. I would like to.
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Do you think I can follow it? I respect Riemann very much, but I do not know much about his work because unlike Gauss, Euler he primarily focused in only one area.

4. Well, you can begin with the differential geometry of curves and surfaces. But note that it requires some calculus and differential equations, especially when you get into details. Believe me, it is rewarding. Plus, the formulas are really exquisite (and so really beautiful), check the Gauss-Bonnet formula.

To get into manifolds is a different matter. It requires some topology and some heavy calculus. But still, the results are striking!

Good luck and -above all- maintain your persistance.

5. Is it supposed to be some generalization of regions and surfaces?
(That formula was too advanced for me )

Originally Posted by Rebesques
To get into manifolds is a different matter. It requires some topology and some heavy calculus. But still, the results are striking!
How well do you know it? For some reason this topic and set-theory are two topics that people never seem to know. I never met a person (at least I do not think so) that know these topics on a very advanced level. My guess it that in college such a course was optional and most poeple omitted it.

This is mine 21th post!!!

6. Is it supposed to be some generalization of regions and surfaces?
Actually, it is a unification of these. They are just called manifolds!

That formula was too advanced for me
Maybe I chose a tough link It is a really pretty relation about the jumps of the derivative of a piecewise smooth "length-minimizing" curve (geodesic) and the way the manifold "bends in itself" (gaussian curvature). It is a real beauty! Plus, there is a strong algebra-geometric context to it, as the "number of holes" (genus) of the manifold appears out of the blue!

Let me see if i can find some link to a better treatment of the subject...

My guess it that in college such a course was optional and most poeple omitted it.

Well, in the special case they even exist, most people just avoid them - nothing wrong in that, it's a matter of personal gust.

http://people.hofstra.edu/faculty/St...f_geom/tc.html

8. Originally Posted by galactus