What math is necessary to learn manifolds?

Question

I want to learn about manifolds, but I'm only a senior in high school and obviously have a while to go. I'm in AP Calc BC. What should I study to eventually learn manifolds? Linear Algebra? What else?

You need to learn linear algebra independently of what ulterior interest you might have, really. — Mariano Suárez-Álvarez, Jan 05 '12 at 02:41
I agree with Mariano's answer. In fact, manifolds are quite far away from freshman calculus -- at least two solid years of study away, in my opinion -- and in between lies a bunch of things you will have to take anyway, especially multi-variable calculus, linear algebra and basic real analysis. Really I think it's too soon to be worrying about this, and I don't mean this in the discouraging sense, I literally mean that you needn't worry about it yet... — Pete L. Clark, Jan 05 '12 at 03:39
...If you want advice, here it is: first, enroll in a university (or very good liberal arts college -- i.e., in the Amherst-Swarthmore-Williams class) with a very strong math department. Second, concentrate now on fully mastering the material up to and including BC calculus. Really having this down will serve you well, even unto your study of manifolds. Good luck! — Pete L. Clark, Jan 05 '12 at 03:40
@Mariano: I'm not completely sure if you're seriously asking the question (I also don't know why someone unfamiliar with the American high school system would possibly know this, so...) there are two varieties of "Advanced Placement Calculus", one called "AB" and the other called "BC". If there is some actual meaning to these letters I have never been able to divine it, but the essence is that the "BC" course is more rigorous and covers more material.... — Pete L. Clark, Jan 05 '12 at 05:35
... Roughly speaking "AB" would place you out of the first semester of "freshman calculus" (which I am now worldly enough to understand is itself a peculiarly American institution), whereas the highest possible score on "BC" really should be good for two semesters. Or something. I am actually more describing my own memories of my student experiences than any recent information. You could scarcely pay me enough money to look at a contemporary AP calculus exam: I fear that I would be deeply traumatized by what I saw. — Pete L. Clark, Jan 05 '12 at 05:37
(@Mariano: if in fact you are unfamiliar with the American high school system, that is. You are such a knowledgeable guy that I probably shouldn't assume that just because you live in South America you wouldn't know these things...) — Pete L. Clark, Jan 05 '12 at 05:41
@Pete, I knew about the AB variety only! Thanks for the info :) — Mariano Suárez-Álvarez, Jan 05 '12 at 06:38
@Pete L. Clark: During 1956-1968 there was just one exam. Then, beginning in 1969, both the AB and BC exams were given. Originally, A referred to certain precalculus topics, B referred to certain topics that mostly involved derivatives, and C referred to certain topics that mostly involved integration techniques and applications of definite integrals. The AB exam tested A and B, the BC exam tested B and C. Most of the A material was omitted by the 1990s, but the names were too entrenched by then to change. — Dave L. Renfro, Jan 05 '12 at 16:10
@Dave: thanks for this explanation. As it happens I took the "BC" exam in 1993, so this explains my relative fuzziness on the nomenclature. — Pete L. Clark, Jan 06 '12 at 05:51
A side note: I picked up Tu's Introduction to Manifolds well before I was really ready to get much out of it. At first, it was terse and foreign to the point of being traumatizing. As I keep revisiting it, it broadened my horizons tremendously. And it tied together ideas had been loose threads for years. (eg. the chapter on functors that answered a question I'd had since AP calculus: "Is something -- the space we're in, maybe -- changing when we take derivatives?" Answer from my calc. teacher: "Not to my knowledge." Tu's answer: "Yep, you're changing categories." I like insights like that.) — steve_0804, Apr 29 '17 at 00:05

score 22 · Answer 1 · edited Apr 13 '17 at 12:21

22

I hate to add yet another answer to the list, but I would like to be thorough. The pre-requisites for learning about manifolds are as follows:

Multivariable calculus
Linear algebra
Real analysis
Point-set topology (a.k.a. General topology)

For multivariable calculus and linear algebra, most of the standard texts will do. Note that multivariable calculus and linear algebra can be learned independently of one another (so it doesn't matter which one you learn first).

There are many good texts for learning real analysis, some of which are mentioned in the answers to this question. It will be important that your real analysis education cover not only single-variable differentiation and integration, but also multivariable differentiation and integration as well. In particular, it's important that you learn about the Inverse Function Theorem and Implicit Function Theorem.

Where point-set topology is concerned, some of it will hopefully be covered when you learn real analysis. (Indeed, many analysis texts consider point-set topology to be a part of analysis.) You will need to learn about metric spaces and topological spaces. An excellent book for this is "Topology" (Munkres).

Finally, once you've gotten through all of this, I would say the text to use for manifold theory is "Introduction to Smooth Manifolds" (John Lee). In fact, the book has an appendix at the end which gives a rapid treatment of each of these four subjects. So, if you absolutely cannot contain your curiosity, this might be worth looking into.

Optional: One more helpful (but not necessary) pre-requisite is elementary differential geometry (or classical differential geometry), which is a beautiful topic to learn once you've finished with multivariable calculus. My recommended textbook is "Elementary Differential Geometry" (Pressley).

edited Apr 13 '17 at 12:21

Community

1

answered Jan 05 '12 at 03:25

Jesse Madnick

31,524

Is there a way to streamline this? Certainly each of these areas merits its own attention, and there are lovely little nooks, crannies and whole villages to explore, see and appreciate, but if I just want to start to Lee's book quickly what would I need? – Erik G. Jan 19 '13 at 01:31
@ErikG: If you don't know any multivariable calculus or linear algebra, I'd say it'd be pretty difficult to get started on Lee's "Intro to Smooth Manifolds" book. But like I said, if you're in a rush, take a look at the appendix to Lee's book: it covers (in a rapid but useful fashion) all of the pre-requisite knowledge. – Jesse Madnick Jan 19 '13 at 01:39
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Lee also wrote a prequel called "Introduction to Topological Manifolds" for the topology background. The pre-requisites to that book are fairly light, so maybe that's the way to go. Ultimately, it's similar to Munkres' "Topology" book, but with an emphasis on topological manifolds. (The multivariable calculus and real analysis mainly comes into play when studying smooth manifolds. Note that smooth manifolds have found many more applications in mathematics, so the term "manifold" generally refers to smooth ones.) – Jesse Madnick Jan 19 '13 at 01:43
Very useful to know thanks. My sense is that there is all the difference between studying math you like, even if it is advanced; and, going about it in the typical curricular fashion of first take these courses first. – Erik G. Jan 19 '13 at 02:06
Lee's appendix to TM is also superb. I think you could in theory start with the appendix of TM, read TM ch1-4, read appendices B and C of SM, and then start SM proper. The biggest difficulties of this route would be becoming comfortable with $\delta$'s and $\epsilon$'s, the notion of differential, and vector spaces, so you could insert something like Rudin ch2 (resp ch4 and the first bit of ch9) before starting TM (resp SM). – juan arroyo Dec 21 '16 at 02:18

score 6 · Answer 2 · answered Jan 05 '12 at 01:21

6

You should most likely have a background in linear algebra as you said, multivariable calculus, and real and complex analysis. For instance, Rudin begins a discussion of manifolds after discussing standard multivariable analysis in his Principles of Mathematical Analysis.

answered Jan 05 '12 at 01:21

analysisj

2,700

I like this answer, but don't think complex analysis is really necessary. – Jesse Madnick Jan 05 '12 at 03:09
3

Speaking from experience, I also don't think Rudin's discussion of manifolds is particularly friendly to beginners. – Jesse Madnick Jan 05 '12 at 03:28
Well, if you want to study complex manifolds, then complex analysis certainly seems to be in order. Anyway, complex analysis sure can't hurt (I am struggling to think of what could...). And I agree, the material on manifolds in Rudin's Principles is about as bad as the preceding part of the book is good. Just about anywhere else would be better. – Pete L. Clark Jan 05 '12 at 03:45
@Pete I completely agree,it may be the single worst presentation of multivariable calculus in the history of textbooks.I personally have always thought the last 2 chapters of Rudin are forced and tacked on. – Mathemagician1234 Jan 05 '12 at 11:05

score 5 · Answer 3 · answered Jan 05 '12 at 02:01

I think the prerequisites you will need to study manifold theory depend on which aspect of manifold theory that you wish to study. For example, a good knowledge of algebraic topology is more essential if you wish to study differential topology than if you wish to study differential geometry (although you should eventually learn algebraic topology in some depth no matter which aspect of manifold theory you pursue). However, the prerequisites to study the standard theory of differentiable manifolds are (generally speaking) point-set topology, linear algebra and advanced (multivariable) calculus. A good knowledge of point-set topology and linear algebra implies that you have the mathematical maturity necessary to study manifold theory as well as the necessary knowledge; thus, it is important to carefully study these two subjects.

For example, you might wish to look at Topology: A First Course by James Munkres and Linear Algebra Done Right by Sheldon Axler which will provide you with more knowledge in these subjects than is strictly necessary in manifold theory (but this knowledge will be essential in your study of other branches of mathematics). Finally, I think Principles of Mathematical Analysis by Walter Rudin furnishes a solid knowledge of the elements of advanced calculus (both single-variable and multivariable) that will be necessary for manifold theory.

In short, it would be a good idea to use manifold theory as a means to advance your knowledge of other (essential) branches of mathematics because the prerequisites for manifold theory are more fundamental in mathematics (as a whole) than manifold theory itself.

I hope this helps!

score 3 · Answer 4 · answered Jan 05 '12 at 01:29

I agree with @analysisj 's response (and up-voted it). I just wanted to add that it might be useful to review some topology depending on how your real and complex analysis courses were taught. Some analysis courses use books that are light on the point set topology theory.

score 3 · Answer 5 · answered Jan 05 '12 at 02:35

In addition to linear algebra, analysis and topology as others have suggested, learning some classical differential geometry probably wouldn't be a bad idea either. The very abstract definitions one encounters in the theory of manifolds are inspired by the principles of differential geometry much as point-set topology was inspired by analysis. Seeing these concepts made tangible by concrete calculations will give more meaning to the more elaborate machinery of manifolds and differential forms.

score 3 · Answer 6 · answered Jan 05 '12 at 02:54

Here's a nice basic introduction -- it gives an idea of what manifolds are, and you don't need to know linear algebra or multivariable calculus to read it: http://www.math.washington.edu/~lee/Books/ITM/c01.pdf

A good resource for people who haven't even had calculus but who want to learn some basic ideas in topology is http://www.learner.org/courses/mathilluminated/units/4/textbook/01.php

Before you can even learn the precise definition of manifolds and theorems about manifolds you should be familiar with topological notions that students typically learn in analysis. It's difficult to have any intuition behind manifolds if you don't have a good idea of $\mathbb{R}^n$. But you should take a look at Chapter 2 of Munkres's Topology. Try to understand what you can. If you get too lost, consult an introductory analysis book like Rosenlicht's Introduction to Real Analysis and hop back and forth between the two. If you want to do calculus on manifolds, you should certainly learn some linear algebra and multivariable calculus first.

What math is necessary to learn manifolds?

6 Answers6