Thread: Undergraduate Differential Geometry

So registration has begun for our winter quarter here at UCLA, and ever since completing the standard multivariate calculus course, I have been interested in this particular subject. How this came about, is that besides the computational methods of calculus everyone should know, this course was really (from my point of view) a topics "survey" of elementary differential geometry.

For example, we discuss curves and their parametrization, arc length, curvature, local TNB frames, local osculating circles, etc. However, a deeper investigation into this subject leads one to concepts such as torsion and the Frenet formulas, which evidently characterize curves completely (in addition to more sophisticated theory and computational techniques involving, for example, material from linear algebra). Then these ideas are extended to surfaces, where new concepts are introduced such as intrinsic and extrinsic curvature, fundamental forms, and perhaps a culmination to the Gauss-Bonet theorem.

As another example, when studying elementary vector analysis we encounter curves, surfaces and volumes embedded in ambient space (R^n), and of course the calculus of vector and scalar fields "on" these geometric objects. Again, a deeper investigation leads one to realize that all of these geometric objects which we do calculus over are really in some sense one in the same, at least with respect to the fact that they are all simply smooth continuous maps from an "external" parameter domain to the "ambient" space of R^n, with their "dimension" characterized by the number of free parameters in the domain. This of course, leads one to the more general characterization of a differentiable manifold. We also come to see vector fields as really a special case of differential forms, where vector operations such as curl, div and grad correspond to basic manipulations of differential forms using the exterior algebra/wedge product, and the theorems of Stokes, Gauss, etc. as special cases of a much more general statement about manifolds and differential forms (*THE* Stoke's Theorem).

While I find both of these subjects very interesting, I was especially captivated by the generality of differential forms and Stoke's Theorem. Of course, I find the theory of curves/surfaces interesting as well, and perhaps this is prerequisite to the generalization of surfaces/vector theorems to manifolds/Stoke's Theorem.

In any case, I expected our two quarter differential geometry sequence to cover these topics, but it appears that it really is limited to the theory of curves and surfaces.

Here is the catalog description, which accounts for both courses:

Curves in 3-space, Frenet formulas, surfaces in 3-space, normal curvature, Gaussian curvature, congruence of curves and surfaces, intrinsic geometry of surfaces, isometries, geodesics, Gauss/Bonnet theorem. P/NP or letter grading.

What is your guys' experience in undergraduate differential geometry? Is the generalization of Stoke's Theorem and the (introductory) study of differential forms and manifolds really a graduate level only topic?

Can these topics be studied independently (at least at the basic undergraduate level) or is one area of study (beyond the basics of multivariate calculus/linear algebra) requisite to the other?

What is your guys' experience in undergraduate differential geometry?

I am currently taking an undergraduate course in diff geo (see below for a description). I don't know if I'll take the second course (see below for a description of this as well)--I might just take the grad course on Riemannian geometry, differential topology, or Lie groups. I like diff geo, but I like generality (to an obscene degree) and so I find learning things in such specificity to be annoying--although this can give you good intuition for the more general notions.

Is the generalization of Stoke's Theorem and the (introductory) study of differential forms and manifolds really a graduate level only topic?

Not really, depending on how you do it. I know here (at UMD) we offer a two-sequence course on undergraduate differential geometry. The first covers what you have quoted (a lot more actually, but presumably this is just because its a course description) and the second is a course on the generalization of this to real manifolds of finite type, not thought of as immersed in --differential forms--Lie groups over such manifolds. So, I think if you want to you can do it as an undergraduate course. If you're interested I've attached an example of a midterm and final in this course from a past year.

Can these topics be studied independently (at least at the basic undergraduate level) or is one area of study (beyond the basics of multivariate calculus/linear algebra) requisite to the other?

Anything can be done independently (that's the only way I like it!). But I think just basic multivariate calculus and linear algebra might be a little sparse. If you instead meant mutlivariable analysis (and what you call "vector analysis") and more advanced linear algebra (things like tensor algebras) you should be fine.