Question

Lee vs Tu

Answer 1

Note i took Lee's text as an undergrad and read Tu's post the course.

Tu's is better where i was as a math student but Lee's is a more indepth well rounded source. So as i am now, i think lees is better. I think the level of rigour in both texts are easily upper undergrad lower grad level. But Tu's tends to focus more on being easily accessible while Lee's focuses on giving a complete picture.

Answer 2

I self studied from both Tu and Lee. My analysis background wasn't that strong, so I initially found Lee overwhelming. Also, Lee is more comprehensive, so will will take quite a while to work through.

What worked for me was to go through Tu first to get a general idea of the subject, and then dive into the relevant sections of Lee when I needed more background on specific areas. I think that might give you the best of both worlds.

Answer 3

My favorite book about Riemannian geometry is Jost. He explains connections and curvature in the best way I've ever seen.

Probably the best thing to do is read Tu and then after that, read Lee and/or other more specialized books about topics like vector bundles or Riemannian metrics.

I don't think I've ever actually read Lee straight through. I mostly referenced it as needed while reading other books and eventually covered most of it that way.

Answer 4

I recommend Tu. It is a more barebones treatment that gives you a lot of necessary skills quickly. Do all the exercises (they are not very hard).

Answer 5

Jeffrey Lee instead of John Lee. Less verbose and more content than Tu. Tu also has a sencond book that covers bundles and riemannian metrics.
Sadly Tu doesn't cover lie groups and homogeneous manifolds properly

Answer 6

Lee is very popular and its coverage is quite nice, but I found it very hard to learn from as a student. And even now that I know all of the material, I find it very (unnecessarily) difficult to read. Tu's book is definitely less encyclopedic, but I think it is nicer to learn from. You could also consider the first volume of Spivak's "comprehensive course on differential geometry".

I would advise that it is wrong to be too concerned about what is included and what is excluded, especially for big topics like Riemannian geometry and vector bundles. There's no need to get everything from a single book, it's better to focus on what you find you can learn from the easiest.

Answer 7

>I don't know what exactly to skip.

That really depends on what you want to do later. But the 'core' of the book is definitely chapters 1-6 and 11-14. You are not wrong that vector bundles are a huge deal, so I would not skip 5. I think it's would be rare for that to be bad advice. (But, of course, it's *possible*.)

7 and 8 will contain some results that you need for later stuff. 9 is entirely about Lie groups, which you might or might not care about. 10 is going to be the least important chapter for most people.

15-16 are an introduction to cohomology but I recall being confused there, until I'd first learned singular cohomology properly in an algebraic topology class.

17-20 are kind of technical. Lots of appeals to results from the theory of ODEs, for example.

What I did is read most of the 'core' stuff, and then go back to Lee when I felt like I needed to understand something else to actually do or learn something specific. Over the course of about a year and a half of starting to work with my advisor, that wound up meaning that I've read almost all of it. But definitely not all in one go and not all in the right order.

Answer 8

>I am currently using Lee's "Introduction to Smooth Manifolds" and I find it well written, though it seems overly verbose at times

in the preface, he writes that this is intentional, because students need to get over the hump of the abstract definitions of manifolds, vector/tensor fields, partitions of unity, etc. before a more informal & terse style should be used. I agree with him on that front.

Answer 9

I used Bott and Tu as an undergrad and used Lee's book as a graduate student.  Both books are equally rigorous and are more than sufficient for covering a semesters worth of differential topology and (if this is important to you) enough to prepare anyone for a qualifying exam in the subjects.  


I personally prefer Lee's book, but if the comments are any evidence, this is not necessarily everyone's opinion. I'll add that I'm an algebraist (see: flair) and came into this course with an algebraic geometry background. Lee is chock full of great illustrations that clearly describe what's going on. The formulas that articulate what the photo shows are often extremely long and difficult to parse, but IMHO that is a symptom of differential manifolds and not a symptom of Lee. If you don't allow people to encapsulate information in functors/sheaves, then you're necessarily going to drown in notation, as functors/sheaves encode all the complex notation into themselves so you can "ignore" it until you actually do a computation, where the complexity will show itself all the same.

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Bott and Tu's book lacks exposition and the pictures aren't that great, which makes it hard to really see what's going on. In particular, it is a really good book for people who already know and are comfortable with some amount of differential topology/geometry and have an extremely strong grasp on linear algebra. This makes it a book beloved by professors, but as a student (especially a sophomore in undergrad) I found it impregnable.

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Some of this may be due to the fact that I worked through Tu first (and was mildly underprepared) then took a course with Lee where I was slightly overprepared. That being said, I do like how Lee works and find the exposition long but pretty important to my understanding.

Answer 10

Read
Introduction to Topological Manifolds too