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Motivation of question ahead: I have been accepted to study a Master's degree in pure mathematics at an overseas university starting in August. There are quite a few courses that I am interested in taking there that require prerequisites that I do not yet have. To rectify that I am taking a course in ring and module theory and another in abstract analysis. Unfortunately, however, no one at my institution is offering any course on differential geometry/differential topology, which I also need to learn before I go.

To rectify this I've initiated a study group at the university, where we basically work through Tu's book. Unfortunately I am very busy with the other courses mentioned, and with teaching duties, so the pace is only a meagre 10-15 pages a week. This is a problem, because when I arrive at the new institution I'm expected to know the following:

"The notion of manifold, smooth maps, immersions and submersions, tangent vectors, Lie derivatives along vector fields, the flow of a vector field, the tangent space (and bundle), the definition of differential forms, de Rham operator (and hopefully the definition of de Rham cohomology),"

But by my calculations, I will not be able to linearly study the book until I reach the de Rham cohomology. This means that I am going to have to, unfortunately, skip some sections for now.

Actual Question: The content page for the book is very detailed (available on the linked Amazon page), and I am hoping that someone can tell me what sections I can safely leave out without excessively hampering my ability to understand the concepts specifically mentioned above, and without interrupting the flow of ideas terribly. Thanks in advance.

P.S. I realise that this question does not really comply with the guidelines set out for asking questions, but I am not sure where else to go to find the information I am seeking, and I do require this information for non-trivial reasons. So if you are going to vote to close, I'd very much appreciate guidance as to where I should go to find an answer for this question instead (besides at my university, as I am already trying that route concurrently, but our department has very few people knowledgeable enough in the area of differential geometry to be able to answer this question it seems.)

J W
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K.Power
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  • Follow up: For forms, integration, de Rham, etc, I suggest Sections 3, 4 (forms on open subsets of $\mathbb R^n$, 13 (support), 17-19 (forms on open subsets of manifolds), 20.4 (interior multiplication), 21-23 (integration), 24-27, 28.1 (de rham cohomology). I think Section 7 (quotients) is a prerequisite to Section 28.2-3 and that Section 7 is used as a prerequisite only for a few examples or exercises in Sections 8-27. –  May 29 '19 at 03:34

2 Answers2

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I've read this book and it is a very beautiful one. If you understand the derivative as a linear map already then you can skip the section on Euclidean Spaces and Foliations. In the Euclidean spaces section he does motivate very well where the notion of forms come from. He shows the construction for general finite dimensional vector spaces and dual vector spaces. The correspondence will be that the vector you are interested in is the tangent space and your differential forms will be coming from the co-tangent space. Also, if you are well equipped with the concept of orientation, you can skip a few sections there, however I don't think most people understand orientation on general manifolds.

Faraad Armwood
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    Thank you for answer! I have already almost finished the Euclidean Spaces portion of the book, so too late to skip that. I will skim through orientations to see whether I get the idea or not without going into too much detail. – K.Power Feb 17 '17 at 22:31
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    Would you perhaps know whether the accepted answer on this question is accurate in stating that I will be able to understand the section on De Rham theory if I only manage to get through the section on manifolds and tangent spaces beforehand? Thank you very much for your time. – K.Power Feb 17 '17 at 22:34
  • @K.Power I think you need to know forms. See my comment. –  May 29 '19 at 03:35
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I'm currently taking a course that covers those topics and I have been using that book. My recommendation would be to do chapters 5, 6, 8, 9, 11, 12, 14, 16 in that order. For chapter 12, I would skip the last three sections. That will take care of most of the stuff you are supposed to know. For the differential forms you can probably get away with reading Chapter 4 (differential forms on $\mathbb{R}^n$.) I just started learning about the De Rham Cohomology, so I don't have a very well-informed opinion yet, but there isn't very much to the definition. So if you only need to know the definition, then I wouldn't worry about it.

Clyde
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    Thank you very much for the detailed answer. All the best for your studies further – K.Power Feb 18 '17 at 16:13
  • Follow up: For forms, integration, de Rham, etc, I suggest Sections 3, 4 (forms on open subsets of $\mathbb R^n$, 13 (support), 17-19 (forms on open subsets of manifolds), 20.4 (interior multiplication), 21-23 (integration), 24-27, 28.1 (de rham cohomology). I think Section 7 (quotients) is a prerequisite to Section 28.2-3 and that Section 7 is used as a prerequisite only for a few examples or exercises in Sections 8-27. –  May 29 '19 at 03:34