# Possibility of an Elementary Differential Geometry Course

I have to admit I'm not sure if this is an appropriate question. It's related to research in math education, but not directly to math.

I've found that in talking to professional physicists and engineers, most of them find some use for differential geometry nowadays. One theoretical physicist went as far as to say you could "do nothing serious without it." Yet at most schools (at least the few I've looked at) differential geometry is reserved for graduate students in math and advanced math undergraduates. No schools I looked at had an elementary differential geometry class in, say, a similar style as the calculus sequence. Some of the people I talked to also expressed a lot of difficulty in learning it for the first time on their own. I myself am taking an advanced graduate course in General Relativity, and a good portion of the difficulty of the students is in misuderstanding the fundamental concepts of differential geometry.

To cover differential geometry rigorously, of course one needs quite a bit of advanced mathematics, including topology and analysis. But universities teach elementary calculus classes, most of which are not terribly rigorous, but are sufficient for the purposes of non-mathematicians. Linear algebra, multivariate calculus, and a bit of differential equations would (in my mind) be sufficient to teach a course for engineers. You might argue that one needs to know the theory of manifolds first, but I see this as analagous to studying calculus without really knowing the structure of $\mathbb{R}$.

From my viewpoint, differential geometry is the logical extension of calculus. Based on it's huge (and growing) impact on applied disciplines, It seems logical to have a course in it for engineers and physicists, which I would put immediately after the final semester of calculus (assuming the students have also had linear algebra).

So my question is this: Are there specific instances, either textbooks or courses at a university, of differential geometry classes taught with the intent of being useful for engineers and scientists, which assume only basic calculus knowledge and linear algebra? (Obviously, there are books like "Differential Geometry for Physicists," but I really mean something that would be used by mathematicians teaching such a course). If so, how successful have these courses/books been? If not, or if the attempts have been unsuccessful, is there any particular reason as to why it is not feasable/common?

• We created a 4th year differential geometry course here at U.Vic with the idea that the course would be of interest to mechanical engineering students (think Lagrangian and Hamiltonian classical mechanics), physics students (relativity), and math students interested in manifold theory. I don't know to what extent other places are doing this but I imagine it's becoming a fairly typical story nowadays. – Ryan Budney Mar 11 '11 at 23:49
• It seems to me that a course on using differential geometry (instead of proving theorems about differential geometry) is a good idea. It seems to me that the only prerequisites needed are linear algebra and vector calculus. You don't even need analysis and topology (which are needed mainly for proofs). But I don't know of any textbook that takes this approach. – Deane Yang Mar 11 '11 at 23:57
• (Cont:) (2) Will students take it? After all, the people you talked to benefit from hindsight, but would they have known to take the class? Science students in particular strike me on average as having very definite opinions about what courses they ought to take (because they're the kind of folks who read ahead in the textbook and look stuff up online). Nevertheless, I know for a fact that they're not always right. – Thierry Zell Mar 12 '11 at 0:00
• To a certain degree, that's what Arnold's Mathematical Methods book is about. One may argue, though, it is not sufficiently elementary. Maybe we can convince Dick Palais to write another book? :) – Willie Wong Mar 12 '11 at 0:23
• I'm not a differential geometer. But it seems like a lot of the mathematics in do Carmo's Differential geometry of curves and surfaces would be doable by someone who has had the calculus sequence. I wouldn't recommend that book for such a course -- you'd have to tell the students to skip too many things, and as far as I remember it has no applications -- but this seems like one possible jumping-off point. (I can't say anything more specific; my copy of do Carmo is on the other side of North America from me.) – Michael Lugo Mar 12 '11 at 1:11

## 10 Answers

I think, one of the big problem is that aside from theoretical physics (string theory, general relativity), most mathematicians aren't terribly aware of what engineers and scientists use differential geometry for. This certainly makes it difficult to write/plan a course in that regard.

It was only recently when I heard a talk by Alain Goriely did I find out that biologists care about differential geometry too! But during the talk there were quite a few theorems about curves in three dimensional space that I've never heard of, and I do geometry PDEs and general relativity for a living. This at least provides an isolated data point to illustrate the above, on how mathematicians typically don't know what is or is not important for applications to other fields.

Ideally such a course/textbook should be prepared by someone with great interdisciplinary familiarity.

In terms of differential geometry "as a natural extension of calculus", I think you may have better luck going to older textbooks, where instead of calling it differential geometry, the subject is just called "advanced calculus". Quite a few books are written back then with an eye toward the applied mathematician (but of course, I am incapable of giving recommendations).

Let me add that I am currently supervising a third-year undergraduate course in University of Cambridge on differential geometry. It fits half of your bill: it does not assume more than basic calculus and linear algebra (partly due to the funny way the Cambridge maths curriculum is rather scant on analysis); the current set of lecture notes is written by Gabriel Paternain (if you are interested you can try asking him for a copy). Unfortunately the way the degree program works, the course won't attract much non-pure-mathematicians other than the future-theoretical-physicists. So I can't really comment on how well it works for engineers and other scientists.

The course is divided in essentially four parts:

1. Definition of manifolds as submanifolds in Euclidean space, diffeomorphisms and smooth maps, Sard's theorem and degree mod 2.
2. Curves and surfaces in space. Frenet frame, curvature, torsion of curves; isoperimetric inequality. First and second fundamental form, mean and Gaussian curvature.
3. Calculus of variations, geodesics, minimal surfaces.
4. More about curvature, leading up to Gauss-Bonnet.

One more note: I just remembered that Gary Gibbons is teaching a course titled "Applications to Differential Geometry to Physics". It is not necessarily elementary, but certainly has a lot of applications. Being taught from the point of view of a polymath, the examples given in the notes do cover some more ground than is typical.

• Willie, this sounds good. The only thing I would add to this are explicit examples starting with but going beyond the standard constant curvature spaces. Real, complex, and quaternionic projective spaces. Compact Lie groups. Homogeneous spaces. Things like that. – Deane Yang Mar 12 '11 at 1:23
• Willie, is this the "3rd year course for people who want to escape", or "3rd year course for those who we want to train up for Part III"? – Yemon Choi Mar 12 '11 at 6:30
• (Off-topic, but I'd be interested to hear your thoughts on the amount and nature of analysis in the Cambridge maths syllabus) – Yemon Choi Mar 12 '11 at 6:31
• @Yemon: It's a D course rather than a C course, if that's what you're asking. I wouldn't say that it's preparation for Part III because we don't do abstract manifolds or even abstract Riemannian metrics (everything is done by pullback from the embedding); there is a Part III Differential Geometry course which starts again from scratch. It's a pity, because I'm not sure how one is supposed to do, say, Lie theory without already knowing some differential geometry... – Zhen Lin Mar 12 '11 at 15:45
• There's also a Part IB course called "Geometry" which covers very elementary differential geometry (including, oddly enough, abstract Riemannian metrics on open subsets of $\mathbb{R}^2$). There's quite a bit of overlap between the two courses, in my opinion, and I feel quite strongly that Part II students should be given the definition of an abstract manifold (if only because the Riemann Surfaces course gives such a definition) and a Riemannian metric (if students can understand it in the context of General Relativity, they can understand it here!). – Zhen Lin Mar 12 '11 at 15:49

I see that someone has mentioned this in the comments already, but I think it deserves to be left as an answer.

Here at UGA we do have a regular undergraduate course fitting your approximate description. It is an undergraduate course in differential geometry. The prerequisites are multivariable calculus and linear algebra (it is hard to see how one could get away with any less than that!). Especially, real analysis is not a prerequisite for the course: in fact, to get an undergraduate math major at UGA one needs to take only one of: (a) real analysis (b) complex analysis (c) this differential geometry course. (To be honest, I am not thrilled that real analysis is not required, but I most certainly digress.)

This course is often taught by Ted Shifrin, the most distinguished and veteran teacher on our faculty and someone with more years of experience than he'd probably like me to quantify on the subject of differential geometry (his thesis advisor was Chern). In particular the course is now being taught by Ted, out of a preliminary draft of a book written by Ted. The course webpage is here, from which you can find links to the course text, the syllabus, the problem sets, and so forth.

• Yes, Ted was a legendary teacher already when he was a Moore instructor at MIT. – Deane Yang Mar 12 '11 at 14:18
• In hindsight, the problems in Ted's differential geometry course are still quite difficult. But that's how one builds a strong geometric intuition. – John Jiang Mar 13 '11 at 5:22

Although not with physicists or engineers in mind, in Edinburgh we do have a third year (in a four-year degree) course on differential geometry focussing on surfaces embedded in $\mathbb{R}^3$ which uses nothing more than calculus (including several variable calculus and ordinary differential equations) and linear algebra. The course is modern in that it uses the language of differential forms (in $\mathbb{R}^n$, so no need of manifolds).

A list of the lectures in the course (at least the last time I taught it, which was in 2007-8) is the following:

1. Surfaces
2. Vector fields
3. One-forms and line integrals
4. Differential forms
5. Moving frames and connection forms
6. The fundamental forms
7. Curvature
8. The meaning of curvature
9. Isometry and Gauss’s Theorema Egregium
10. Geodesics
11. Integration
12. Minimal surfaces
13. Stokes’s Theorem
14. The Gauss–Bonnet Theorem

This is delivered in 16 50-minute slots. The course was designed by my colleague Toby Bailey and it is taken by 40-60 students every year, so it seems to be quite popular. I think it is a very good introduction to differential geometry and ending with Gauss–Bonnet gives a nice way to complete the course.

• Nice! Do you use lecture notes or a book? – Thierry Zell Mar 12 '11 at 0:53
• Taking a cue from Willie Wong, may I ask [without critical intent]: Does the course touch on the applications of the topics covered to other fields? – Joseph O'Rourke Mar 12 '11 at 0:57
• One of the problems with applications is there's such a zoo of them. Do you focus on giving people the tools to rapidly pick-up applications, or do you stroll through the zoo picking up what you need as you go along? Something like what Jose mentions looks good -- you could turn it into a full-year course by adding a little "application zoo" to the end of those lecture notes. – Ryan Budney Mar 12 '11 at 1:14
• @Ryan: that's a really mixed metaphor you got going there. But considering you want it to be little and able to "pick up" stuff, I guess what you really meant is an "application petting zoo"? – Willie Wong Mar 12 '11 at 2:10
• @Willie: Or maybe an application orchard. If you pick up too many you get the runs. So be careful to fortify your meal with some meaty theory. – Ryan Budney Mar 12 '11 at 2:24

Such a course was standard in the 19th century. Picard's Traité d'analyse or Goursat's Cours d'analyse mathématique are textbooks for such a course (including Calculus and Differential Equations).

I like the book 'Geometrical methods of mathematical physics' by Bernard Schutz:

http://books.google.co.uk/books?id=HAPMB2e643kC

I'd say that the approach is fairly close to what the OP asked for. I'm also a big fan of 'A first course in general relativity' by the same author:

http://books.google.co.uk/books?id=qhDFuWbLlgQC

I once used Callahan's "Spacetime geometry" as the background for a course on differential geometry. If you concentrate on what you need for deriving Newton's laws of motion from the curvature of time you can actually make it work, at least in a course for physicists.

• Hmmm, starting from Newton-Carton theory can actually lead to an interesting introduction of the Erlangen program. I wonder if there are any texts taking this approach. – Willie Wong Mar 12 '11 at 23:25

From my experience (Italian university) the situation regarding physics and engeneering is quite different.

In physics elementary differential geometry is partly taught inside what can be called Calculus II (mainly computations of length of curves and area of surfaces) and partly inside more advanced math courses (typical denomination: Math Methods for physics) which are, however, usually centered towards manifold thoery and the like, no need to explain why. So it is not exceptional to meet someone that knows the definition of Riemannian curvature tensor but does not know what the torsion of a space curve is. Textbooks are usualy those already mentioned in the discussion.

In engeneering there was almost nothing of this kind until few years ago. Recently some course denominated "elementary differential geometry for..." appeared (dots can be: industrial design, mechanical eng. etc.). Non standard textbooks here, however. Gray "Modern Differential geometry of curves and surfaces" was quite considered for a while due to its extensive use of Mathematica, I personally got lot of inspiration from Galliers' "Geometrich methods and applications for computer science and engeneering" Springer, whih contains a full chapter on Elementary differential geometry.

My personal opinion is that many problems in differential geometry could be avoided if one starts with the classical geometry of curves and surfaces, however, strictly sticks to differential forms and tensors. This has the following advantages:

1. No prior knowledge of differential geometry is assumed.
2. Not much background with something beyond calculus and basic linear algebra is assumed. Interestingly, there are books that do not even introduce topological spaces in detail to treat surfaces.
3. Curves and Surfaces are very intuitive as to some extent one can still imagine all that. When one deals with Jacobi-fields on Riemannina manifolds, it is very ckever to try to imagine some surface in order to visualize this abstract concept.

Especially point three should clarify that it advisable to design an elemantary course which sticks to the easier/or: classical aspects of differential geometry. In Munich, it is possible to take such a course in semester number 4. But students motivated enough are already taking it in the first half of their second year. Sometimes with astonishing results...

Classical differential geometry has wide applications, simply think of Gauss who used his theory of surfaces to measure Hannover (the country Hannover in that days). I think after having understood the differential geometry of surfaces it will be more easy to follow a course on differential geometry which is designed to meet the mathematical prerequisites of for instance general relativity.

I support the previous suggestions, especially Ted Shifrin's fine notes.

I am quite ignorant of differential geometry myself, hence am sympathetic to this proposal. I have recently realized, through my love of Euclidean geometry, what others probably all know, that there is a natural sequence of topics: euclidean geometry, spherical geometry, hyperbolic geometry (these all being surfaces of constant curvature), and then other surfaces of constant curvature, namely quotients of the previous ones.

Next one naturally progresses to surfaces of varying curvature, i.e. the realm of differential geometry proper, Riemannian surfaces. Finally one raises the dimension.

Thus the natural elementary course to teach seems to me to be spherical geometry, then....

The point is to teach curvature, and emphasize that euclidean geometry is the unique geometry of constant zero curvature, (with another hypothesis that lines are infinitely long).

In particular we seem to miss an opportunity when we teach non euclidean, i.e. hyperbolic geometry, without a link to differential geometry via curvature, but this is normally done in non euclidean geometry courses.

E.g. curvature is easily presented in the elementary way that Riemann (and Gauss) described it, as the defect of the angle sum of an infinitesimal triangle. Gauss Bonnet easily follows for the basic surfaces. In this regard, I suggest a first step in learning differential geometry is to discuss whether a cylinder is or is not curved, and why or why not.

Along this line of reasoning, John Stillwell has a very nice book on surfaces of constant curvature that would impart a lot of useful concepts at the advanced undergraduate level..

http://www.amazon.com/Geometry-of-Surfaces-Universitext-ebook/dp/B000WDQJQY

I even sketched out a plan form such a course to present to brilliant 10 year olds, assuming neither calculus, topology, linear algebra or even trig, which could be taught in the course

For more general differential geometry, there is a book by David Henderson, which attempts to teach an intuitive understanding of the ideas, curvature, parallel transport, holonomy...

http://www.amazon.com/Differential-Geometry-A-Geometric-Introduction/dp/0135699630

But the basis of this suggestion is the hypothesis that concepts are more fundamental than techniques for computing them. Since it seems that the most important concept in differential geometry is curvature, the first job is to convey an appreciation for curvature and its role in geometry. This can be done naturally in a very elementary setting. Only afterwards does it seem important to train someone in the means of computing it, i.e. tensor calculus and differential forms, chern classes, etc....

It seems to me most courses smash many topics as a differential geometry course; it can be confusing for a beginner. As far as i am concerned, differential geometry should only contain the following so that it can be seen as advanced calculus :

1) definition of smooth manifolds(embedding, differential structure and so on). 2) tangent spaces(which can be equated as learning how to differentiate functions on a nonlinear space), bundles and cotangent space... 3) vector fields, integral curves, flow... 4) tensor calculus (learning how to integrate functions on
a nonlinear space)

The rest of the stuff like curvature, geodesic, connections, orientation and so on should be given as a second course (Riemannian geometry). I am sure things will be much simpler for us guys oriented to applying it to some practical problem. From my experience however one has to push as far as riemannian geometry and some group theory(lie group, specifically) to see much of the practical benefits of it. But who is to say what can be useful ....