What should be taught in a 1st course on smooth manifolds? - MathOverflow

What should be taught in a 1st course on smooth manifolds?

2011-03-18T20:35:07Z

I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic coursework in point-set topology and multivariable calculus, but may not know the definition of differentiable manifold. I am following the textbook Differential Topology by Guillemin and Pollack, supplemented by Milnor's book.

My question is: What are good topics to cover that are not in assigned textbooks?

Answer by YBL for What should be taught in a 1st course on smooth manifolds?

2011-03-18T21:16:38Z

Thierry Aubin's book "A course in differential geometry" is really good for an introductory course. It covers the basic definitions of manifolds and vector bundles, orientability and integration (Stokes formula) and then focuses on Riemannian geometry defining the Levi-Civita connection, curvature tensor etc...

The only important missing topics are Lie groups and de Rham cohomology. Many courses in differential geometry don't talk about these subjects leaving them to specialised courses in Lie theory or Algebraic topology but I think it's a mistake.

Answer by Georges Elencwajg for What should be taught in a 1st course on smooth manifolds?

2011-03-18T23:29:15Z

I nominate Ehresmann's theorem according to which a proper submersion between manifolds is automatically a locally trivial bundle. It is incredibly useful, in deformation theory for example, but is sadly neglected in introductory courses and books on manifolds. It is completely elementary: witness these lecture notes by Peter Petersen, where it is proved in a few lines on page 9, the prerequisites being about two pages long.

Bjørn Ian Dundas and our friend Andrew Stacey also have online documents proving this theorem.

Answer by R. Andrew Hicks for What should be taught in a 1st course on smooth manifolds?

2011-03-19T01:13:58Z

I don't believe either of those books covers distributions and the theorem of Frobenius. Connections to partial differential equations in general I think are good topics.

Guillemin and Pollack is a book I like a lot, but chapters 2 & 3 (transversality and intersection) always seemed a bit specialized for a first course. Although, the title is, after all, "Differential Topology". My experience is that people tend to cover just chapters 1 & 4.

The definition of a manifold in G&P is as a subset of $\mathbb{R}^n$ (as in Milnor). As I recall the the definition of diffeomorphism is such that a cube and a sphere are considered not to be diffeomorphic. This is because G&P define a map at point of a manifold to be smooth if it can be extended to a map on an open set of the ambient space that is smooth in the sense that it is a map from an open set in $\mathbb{R}^n$ to $\mathbb{R}^m$. I never understood, or saw, how this approach can be used to think about different differentiable structures on manifolds. Since there is only one differential structure on $S^2$, the definition I mention above of diffeomorphism seems to at odds with the general one, given for example in Spivak volume 1. (If anyone could explain this to me I'd be grateful. As a student I found this confusing and still do.)

What I am getting at in the above paragraph is that an additional topic might be the general definition of differentiable manifold. It's nice have projective spaces and Grassmanians at least in ones collection of examples.

Answer by Hong Liu for What should be taught in a 1st course on smooth manifolds?

2011-03-19T01:42:10Z

I think fibre bundles should be introduced to give a modern viewpoint of tensor analysis.

Answer by Anton Petrunin for What should be taught in a 1st course on smooth manifolds?

2011-03-19T02:38:51Z

If this is the first course in diffgeometry, you should not go further than Gauss--Bonnet for surfaces. (I would not even consider anything with dimension >2.)

You can cover more but there is no reason to force your students. At the level of Gauss--Bonnet they have enough to play with and if they like diffgeometry, they could take an other course.

Answer by Giuseppe for What should be taught in a 1st course on smooth manifolds?

2011-03-19T08:26:27Z

In a first course aiming to introduce differentiable manifolds as the spaces on which do calculus, you could give to the students the notion of connection at least on vector bundles.

In order to reflect on the reason for this choice, I report the words of S.S.Chern closing the introduction of Global Differential Geometry, MAA Studies in Math.27, 1989:

The Editor is convinced that the notion of a connection in a vector bundle will soon find its way into a class on advanced calculus, as it is a fundamental notion and its applications are wide-spread. His chapter, "Vector Bundles with a Connection," hopefully will show that it is basically an elementary concept.

Answer by Paul Siegel for What should be taught in a 1st course on smooth manifolds?

2011-03-19T19:46:40Z

I think there are two ways to approach a first course on manifolds: one can focus on either their geometry or their topology.

If you want to focus on geometry, then I think Anton Petrunin's suggestion is the end of the story. I'm a fourth year graduate student, and practically every time I find myself confused about something in differential geometry I realize that the root cause of my confusion is that I never properly learned surfaces. And I've taken lots of geometry courses.

If you want to focus on topology, I really think it makes a lot of sense to teach some Morse theory. It's rather elementary, it's extremely powerful and virtually ubiquitous in differential topology, and most of all it really feels like topology in a way that differential forms don't.

Finally, from looking at only the two books you mentioned in your question, I would be a little worried that your students won't have a lot of examples to work with. What about introducing Lie groups?

Answer by Deane Yang for What should be taught in a 1st course on smooth manifolds?

2011-03-20T01:41:25Z

I'm not sure if the original question is about a one semester or year course.

If this is the first course the students have ever had in differential geometry, then I still agree with Anton that at least the first semester should be about only 2-dimensional manifolds embedded in $R^3$ and Gauss-Bonnet. The point here is that everything can be understood visually, but you learn how to deploy linear algebra and calculus to prove what seems obvious visually. The full power of differential geometry is displayed very nicely. Guillemin and Pollack provides a nice textbook to base the course on. I also like O'Neill's elementary differential geometry textbook.

I would not introduce the more abstract machinery until the second semester, and even then try to be selective about what is discussed because there is just too much. It seems best to focus on basic Riemannian geometry and what, say, sectional curvature means (this builds nicely on what was done in the first semester). It is of course important to introduce many different examples. Although the basic abstract definitions and properties of Lie groups and algebras could be introduced, I believe the focus should be on how to build interesting geometric spaces from standard matrix groups ($GL(n)$, $SL(n)$, $SO(n)$, $SU(n)$).

Answer by Charlie Frohman for What should be taught in a 1st course on smooth manifolds?

2011-03-20T20:36:36Z

The problem will be that the students do not have a firm grasp of multivariable calculus.

You should probably start with a rigorous review of multivariable calculus including the definition of the differentiable, C^1 implies differentiable on open sets, mixed partials are equal, inverse function theorem, local immersion theorem, local submersion theorem. That will allow you to segue into the definition of smooth manifold as a parametrized subset of R^n as in Guilleman and Pollack.

Guillemann and Pollack is a softening of Milnor's "Topology from a Differentiable Viewpoint" and as such is about the lowest level approach you can take to introducing the students to the "stuff" of topology. The exercises are good. I like to have the students divide up the long guided exercise sections to present at the board. I like to supplement the book by proving the Morse Lemma, having a discussion of linking number, and proving the the Hopf fibration is not homotopic to a constant map using linking numbers. I also like touching on complex variables by proving the argument principle. Finally, I like proving that two maps from a closed oriented n-manifold to the n-sphere are homotopic if and only if they have the same degree. I don't do all of these in any one year as there is not time. I generally key off of what seems to interest the particular group of students in the class that year.

Be careful in the section on integration, they leave out (or left out in an earlier edition) that you need to be using orientation preserving parametrizations to define the integral.

After teaching such a course for about 15 years, I changed directions and started teaching the foundations of smooth manifolds in the place of the Guilleman and Pollack course, so that students could learn a more mature definition of smooth manifold, and introduce vector bundles, tensors, and Lie Groups. I have used both the books by Jack Lee and by Boothby. Each has its strong points and weak points (at least in use with graduate students at Iowa.) This turned out to be better for the graduate program as a whole because kids who wanted to do representation theory or PDE could get exposed to the ideas they would see in their research. It also allowed the Differential Geometry sequence to run more regularly. If you decided to go that route, it would still be wise to start with multivariable calculus, as really, very few kids going to graduate school in math have a sufficient background in the calculus.

However, the students are much less happy about taking the foundations of smooth manifolds, because it does not offer the immediate gratification of studying degree and winding number. In fact, when I teach the course as foundations of smooth manifolds, there will always be a block of 3 or 4 students who resent having taken the class. When I teach out of Guilleman and Pollack, even the students who never develop a clue, still enjoy the experience.

Answer by Spencer for What should be taught in a 1st course on smooth manifolds?

2011-03-20T23:46:01Z

This is in agreement with Igor's comment on Anton's answer, but became too long.

I'd say whatever approach you ultimately take, for a first-year grad course it surely has to be done 'properly', i.e. starting from intrinsic definition of a smooth manifold and using the 'modern' language and general definitions of tensor bundles, connections etc.

Absolutely crucially (and here's what inspired this comment), the course simply has to teach people that there is more to manifolds than 2D surfaces because that's why the theory is quite so useful and so prominent in modern mathematics. The whole point is surely the sheer diversity of objects amenable to geometric thought (whatever that means). The job of the teacher would then be to maintain the intuition of "surfaces in R^3" while using general definitions. I believe this can be done. If it cannot, then what on Earth are we all doing?

By the look of the books mentioned in the question, it certainly looks like a course on what I would call "Differential Topology". Sure, there is nothing wrong with a good course on Differential Topology! However, it doesn't seem to me to be synonymous with "A First Course on Smooth Manifolds". My go to book for the latter is John Lee's Introduction to Smooth Manifolds.

Answer by John Sidles for What should be taught in a 1st course on smooth manifolds?

2011-03-21T16:38:13Z

Update: it may be Spivak's new book Physics for Mathematicians: Mechanics I covers most of the material that this answer had in mind. I've just ordered a copy, and will report on it when it arrives.

Neither Milnor's book nor Guillemin and Pollack's book contains the word "symplectic" ... which is a great pity!

Since the manifolds under study are smooth, they have a cotangent bundle; this bundle is associated to a tautological one-form whose exterior derivative is a (canonical) symplectic form.

If in addition the base manifold has a metric, then a canonical (quadratic) Hamiltonian function too is defined on the tangent bundle.

Hmmm ... what might be the integral curves of this Hamiltonian function? It is instructive for students to discover for themselves that the curves are simply the geodesics of the base manifold.

In this way, students gain an appreciation that all of dynamics (both classical and quantum) is intimately linked to the geometry and topology of smooth manifolds ... this appreciation is good preparation for many careers in math, science, and engineering.

Answer by John Klein for What should be taught in a 1st course on smooth manifolds?

2011-03-21T20:18:45Z

I do have one addition to make to the above. At our university we usually use a combination of Guillemin and Pollack and Milnor. There is another approach at a first course which some have found useful: Bott and Tu's book,

                  Differential forms in algebraic topology

This text covers an alternative set of topics that overlap both manifold theory and algebraic topology.

Disclaimers: (1) I have never used the text myself, but several colleagues have said in the past that it is a good book to use---and I am personally a big fan of Bott's approach to mathematical writing.

(2) If one uses Bott and Tu, then one has to sacrifice

                          transversality.

Andrew Ranicki once told me that transversality counts as one of the most important gems of 20th century mathematics.

Answer by David MJC for What should be taught in a 1st course on smooth manifolds?

2011-03-22T01:16:24Z

Differential forms.

Books by Darling (Differential forms and connections) and Madsen-Tornehave (From calculus to cohomology: de Rham cohomology and characteristic classes) may help.