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

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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?

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You mean to cover more topics than are covered in those books combined?! How about Morse Thoery, the h-cobordism theorem, and the Smale-Hirsch theory of immersions? – Mark Grant Mar 18 2011 at 20:50

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@Mark: you must be kidding. The students jlk mentions are only 1st graduate students & undergraduates, and you are suggesting Smale-Hirsch theory as a topic! – John Klein Mar 18 2011 at 22:05

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Personally I would be quite happy with just a course that covers the material in Guillemin and Pollack thoroughly... – Qiaochu Yuan Mar 18 2011 at 22:53

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Based on Mark's punctuation, I think he is making more of a rhetorical point than an actual suggestion (though I didn't read the original question the way he seems to have had) – Yemon Choi Mar 18 2011 at 23:56

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@John, Yemon: You're quite right, I forgot to hit the sarcasm button. – Mark Grant Mar 19 2011 at 10:56
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## 13 Answers

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.

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IMO it makes a fine homework problem in an introductory manifolds course, right around when one learns the proof of the tubular neighbourhood theorem. – Ryan Budney Mar 19 2011 at 6:14

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I agree; for students who are going on to other fields (e.g. algebraic geometry) where differential topology plays a role (both as techincal background in some situations, and as more general motivational background), this is one of the most useful results to take away from a manifolds course. Also, although it is not particularly a differential topology course, just teaching the definition of proper map would be helpful (and greatly appreciated by students going on to algebraic geometry!). – Emerton Mar 19 2011 at 20:14

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Dear jlk, I think that Ehresmann's theorem is the natural point at which properness appears. I think that the "morphisms as families" point of view is not usually emphasized in courses in differential topology the way it is in algebraic geometry, but I don't see why you couldn't discuss it. (And thus explain that it is important to have a notion --- i.e. proper submersion --- which captures the idea of a smooth family of compact manifolds without requiring that the base or total space themselves be proper.) Note also that the case of Ehresmann's theorem with equidimensional source and ... – Emerton Mar 20 2011 at 20:42

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... target (i.e. zero dimensional fibres) ties in with covering space theory, which the students already know (presumably --- if not, then that might be a better topic than Ehresmann). I'm sorry that I can't give a more interesting answer. Best wishes, Matthew – Emerton Mar 20 2011 at 20:44

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Dear jlk, You're welcome. Note though that my comment about differential topology courses not emphasizing "morphisms as families" may have been a bit hasty, since this is one of the focuses of Morse theory. So from an algebraic geometer's perspective (and for other reasons too), Morse theory is another excellent possibility to consider. Regards, Matthew – Emerton Mar 21 2011 at 13:39
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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.

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I completely agree with this! – Deane Yang Mar 19 2011 at 2:57

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I could not agree more. In particular, I would avoid all of the big bureaucracy sometimes confused with differential geometry (there is absolutely no need to inflict upon students the general definition of tensors, vector bundles and what not!) – Mariano Suárez-Alvarez Mar 19 2011 at 6:19
Agree with Mariano too. – Deane Yang Mar 19 2011 at 12:49

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For most graduate students who won't be specializing in geometry/topology their first course on manifolds will also be their last. Focusing on dimension 2 is a great approach for an upper level undergraduate course, but not for the first year graduate one. – Igor Belegradek Mar 19 2011 at 13:07
The undergraduate topology/geometry requirement that we have at my school has a term on intro to algebraic topology, a second term on intro to diff.geometry which is similar to what this post describes, and a last term on diff. topology which incidentally uses the same textbooks as in the OP, and I think that's similar to what is being asked here. – Gjergji Zaimi Mar 19 2011 at 20:43
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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.

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Thanks! These are very good suggestions. – jlk Mar 21 2011 at 5:01
I've heard rumours of a 2nd edition to Guillemin and Pollack being near completion. – Ryan Budney Mar 21 2011 at 17:13

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Great answer, especially since it is based on real experience. – Deane Yang Mar 21 2011 at 18:01
I want to second the motion that Jack Lee's book <i>Introduction to Smooth Manifolds</i> is beautifully written. Our UW QSE Group has translated substantial portions of Nielsen and Chuang's <i>Quantum Computation and Quantum Information</i> into the geometric language of Lee's <i>Introduction to Smooth Manifolds</i> ... the high quality of both texts made this quite pleasurable. As a reading experience, Lee's text is like rafting the Mississippi river ... a river that is large and somewhat slow ... and yet, with patience, its flow carries the reader smoothly for an immense distance. – John Sidles Mar 21 2011 at 21:06

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?

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The suggestion of Morse theory is a good one. I wrote in a comment on another answer that in differential topology courses the idea of "morphisms as families" is often not emphasized. However, that comment was probably a bit hasty: this is the focus of Morse theory, and thinking about how the topology of the level sets change as the parameter varies is not only very interesting in itself, but provides good preparation for later arguments in lots of different contexts. – Emerton Mar 21 2011 at 13:38

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That's a really good point - I have never made that connection until now. Yet another good reason to introduce Morse theory! – Paul Siegel Mar 21 2011 at 15:13

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 ($GL(n)$, $SL(n)$, $SO(n)$, $SU(n)$).

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I am with you. A first course in Differential Geometry should be very concrete. Generally, what I am up against is that the students have very little experience with multivariable calculus, and they need to be forced to work lots of concrete examples, so that they can get that into their heads. – Charlie Frohman Mar 20 2011 at 9:59

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.

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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.

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I taught a course like this last year, and I used Lee's book. It is a great book, because of the fact that it is so wordy. (Others may disagree, but I am a firm believer that wordy is better than terse for textbooks that students are expected to read and learn from.) – Spiro Karigiannis Mar 21 2011 at 14:03

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.

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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.

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I'm not sure Bott-Tu is right for the first course, but it has my vote as a great introduction to many important and fundamental topics in differential topology. – Deane Yang Mar 21 2011 at 20:35

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.

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If you take the point of view, as in these books, that smooth manifolds are certain subsets of $\mathbb R^n$ and inherit their smooth structure from $\mathbb R^n$, then there is no such thing as two smooth structures on the same set. But there is such a thing, obviously, as two smooth manifolds related by a homeomorphism that is not a diffeomorphism. And there is also such a thing, not obviously, as two smooth manifolds related by a homeomorphism but not by any diffeomorphism. – Tom Goodwillie Mar 19 2011 at 12:24
Thanks for that clarification. So it really is a different definition of diffeomorphism, and so perhaps is deserving of a different name, like 'ambient diffeomorphism'. – R. Andrew Hicks Mar 19 2011 at 16:16
No, I don't believe it is a different notion. Perhaps the confusion is that a cube in the sense of G&P is not a smooth manifold--thinking of a sphere as homeomorphically embedded in R^n as a cube does not put a smooth manifold structure on the sphere. – Jack Huizenga Mar 19 2011 at 20:24
Another way of putting it is this: every smooth manifold has an embedding in some R^n, in such a way that the smooth functions on the manifold are (pullbacks of) those functions on the image which can be locally extended to smooth functions on open neighborhoods. – Jack Huizenga Mar 19 2011 at 22:21

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Every smooth manifold in $\mathbb R^n$ in the sense of the Milnor book ("concrete manifold") is canonically a smooth manifold in the abstract sense. If $M$ and $N$ are concrete manifolds, then the smooth maps between them in the concrete sense are precisely the smooth maps between them in the abstract sense. That is, we have a full and faithful functor from the one category to the other. In fact, it is an equivalence of categories -- that is, every abstract manifold is diffeomorphic to some concrete manifold -- that is, every abstract manifold can be smoothly embedded in some $\mathbb R^n$. – Tom Goodwillie Mar 20 2011 at 3:54
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I think fibre bundles should be introduced to give a modern viewpoint of tensor analysis.

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Differential forms.

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

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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.

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I don't have a problem with a course introducing symplectic geometry via physics. It seems to me that a first course should choose one focused topic and goal and do just enough to achieve the goal. Guillemin and Pollack chose the Gauss-Bonnet theorem and do a beautiful job of staying focused on that. Another possibility is Hamiltonian mechanics. But what would be the goal (analogous to Gauss-Bonnet)? – Deane Yang Mar 21 2011 at 16:55
What would be the goal? Hmmmm ... that would depend upon the class. For a class of engineers and/or scientists, I would suggest ... hmmm ... thermostatic flow s(Liouville's Theorem), with applications in synthetic biology (because that's where the jobs are). For mathematicians, maybe ... hmmm .... de Rham cohomology? Definitely, the pedagogic challenge here is not too few good options for continued study, but rather, far too many of them. – John Sidles Mar 21 2011 at 17:06
On further consideration of Deane Yang's (excellent) question "What would be the goal", a very useful final two weeks of of lectures might survey the topic "Some origins of metric and symplectic structures in mathematics, science, and engineering." And yet, an entire course surely could be devoted to this topic alone. – John Sidles Mar 21 2011 at 17:23
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