Differential Geometry of Curves and Surfaces

Question

I'm self-studying differential geometry using Lee's Intro to Smooth Manifold and Do Carmo's Riemannian Geometry. However, I've never studied the subject so-called "differential geometry of curves and surfaces" (such as the one dealt with by Do Carmo's Differential Geometry of Curves and Surfaces). Since this topic exclusively deals with 3-dimensional space, it doesn't attract me as much as other DG topics so far.

What's the point of studying this topic? How important is it for those who will later study more advanced DG, especially Riemannian Geometry? If you think it's a kind of an optional subject, what would you learn instead? If you think it's an integral part of DG sequence, could you give me the reason why it is so important as well as some of its interesting applications and theorems?

Answer 1

I'm not an expert and would be interested in an answer myself but some remarks.

I believe that studying the DG of curves and surfaces can give you some intuition to the more general approach and after you studyied Riemannian geometry, some topics in the "curves and surfaces" are just special cases.

One of the main differences, however, is that DG of surfaces and curves focuses more on their embeddings to $\mathbb{R}^3$ and the general Riemannian theory studies more the intrinsic properties (properties of embeddings is just one of many topics people are studying in Riemannian geometry). In DG of curves and surfaces, the only intrinsic property is the Gaussian curvature and the global topology of the surface. The intrinsic curvature of any curve is zero, but in courses on "DG of curves and surfaces" you will learn that a curve has curvature and torsion, which are in fact properties of the embedding. This might be confusing at the first encounter.

Also, there are some interesting questions on surfaces, such as the classification of surfaces with constant Gaussian curvature, or Gauss-Bonet theorem (the most beautiful Theorem in this course!) -- that I'm not sure what are the theorems generalizing this to higher dimensions.

To answer one of your questions, I don't think that DG of curves and surfaces is a prerequisity to the general Riemannian geometry.

Answer 2

If you are just starting out in differential geometry, I would go as far as to say that the study of curves and surfaces is a necessary supplement.

For a novice geometer, curves and surfaces are essential because they are easy to visualize, so developing specific techniques for working with them will help give intuition for higher dimensions and make difficult concepts easier to grasp.

As an example, the Gauss-Bonnet theorem for a closed Riemannian $2n$-manifold $(M,\mathrm{g})$, where $n>1$, deals with integrating the Pfaffian of the curvature form. This can get messy. On the other hand, for $n=1$, we get $$\chi(M)=\frac{1}{2\pi}\int_M K \omega,$$ where $\chi(M)$ is the Euler characteristic of $M$, $K$ is the Gaussian curvature, and $\omega$ is the area form. This is beautifully clean, and it motivates the case for arbitrary $n$.

The study of curves and surfaces will also introduce some useful tricks with frames. They aren't used very often, in my experience, but they're useful to know nonetheless.

Just to make sure I'm clear though, do NOT detach yourself from modern tools when studying curves and surfaces. Always develop the theory in terms of the modern tools. Otherwise, the methods you develop from studying curves and surfaces will be painful to implement when you try to use them in the modern setting.

Answer 3

Differential geometry of curves and surfaces is very important. It is a main mathematical component of a branch of mechanical engineering called: the theory shells. Shell constructions are everywhere: airplanes, ships, rockets, cars, pressure vessels, etc. If you open any book devoted the shells theory (or, say, the Finite Element Analysis of shells), you would see that those books are mostly mathematics (with some examples from mechanical or civil engineering). As an example, you may consider the following book, named "Computational Geometry of Surfaces and its Application to the Finite Element Analysis of Shells": http://www.amazon.com/Computational-Geometry-Surfaces-Application-Analysis/dp/0646594044

The authors also have a webpage devoted to the book:

http://members.ozemail.com.au/~comecau/quad_shell.htm