In the winter of , I decided to write up complete solutions to the starred exercises in. Differential Topology by Guillemin and Pollack. Victor William Guillemin · Alan Stuart Pollack Guillemin and Polack – Differential Topology – Translated by Nadjafikhah – Persian – pdf. MB. Sorry. 1 Smooth manifolds and Topological manifolds. 3. Smooth . Gardiner and closely follow Guillemin and Pollack’s Differential Topology. 2.
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For a start, for differential topology, I think I must read Stokes’ theorem and de Rham theorem with complete proofs. Differential geometry is a bit more difficult. What is a connection? Which notion should I use? I want to know about parallel transport and holonomy. What are the most important and basic theorems here? Are there concise books which can teach me the stuff faster than the voluminous Spivak books?
Suggestions about important theorems and concepts to learn, and book references, will be most helpful. I have compiled what I think is a definitive collection of listmanias at Amazon for a best selection of books an references, mostly in increasing order of difficulty, in almost any branch of geometry and topology. In particular the books I recommend below for differential topology and differential geometry ; I hope to fill in commentaries for each title as I have the time in the future.
If you want to have an overall knowledge Physics-flavored the best books are Nakahara ‘s “Geometry, Topology and Physics” and above all: Frankel ‘s “The Geometry of Physics” great book, but sometimes his notation can bug you a lot compared to standards.
If you want to learn Differential Topology study these in this order: To start Algebraic Topology these two are of great help: For modern differential geometry I cannot stress enough to study carefully the books of Jeffrey M. Both are deep, readable, thorough and cover a lot of topics with a very modern style and notation. In particular, Nicolaescu’s is my favorite. From manifolds to riemannian geometry and bundles, along with amazing summary appendices for theory review and tables of useful formulas.
It is also filled with LOTS of figures and classic drawings of every construction giving a very visual and geometric motivation. It even develops Riemannian geometry, de Rham cohomology and variational calculus on manifolds very easily and their explanations are very down to Earth.
If you can get a copy of this title for a cheap price the link above sends you to Amazon marketplace and there are cheap “like new” copies I think it is worth it. Nevertheless, since its treatment is a bit dated, the kind of algebraic formulation is not used forget about pullbacks and functors, like Tu or Lee mentionthat is why an old fashion geometrical treatment may be very helpful to complement modern titles.
In the end, we must not forget that the old masters were much more visual an intuitive than the modern abstract approaches to geometry. If you are interested in learning Algebraic Geometry I recommend the books of my Amazon list. They are in recommended order to learn from the beginning by yourself. In particular, from that list, a quick path to understand basic Algebraic Geometry would be to read Bertrametti et al.
But then you are entering the world of abstract algebra. For differential topology, I would add Poincare duality to something you may want to know. For differential geometry, I don’t really know any good texts. Besides the standard Spivak, the other canonical choice would be Kobayashi-Nomizu’s Foundations of Differential Geometrywhich is by no means easy going.
I have not looked at it personally in depth, but it has some decent reviews. It covers a large swath of the differential topology, and also the basic theory of connections.
A book I’ve enjoyed and found useful though not so much as a textbook is Morita’s Geometry of differential forms. I’m doing exactly the same thing as you right now. I’m self-learning differential topology and differential geometry. An Introduction to Curvature” highly enough. The attention to detail that Lee writes with is so fantastic.
When reading his texts that you know you’re learning things the standard way with no omissions. And of course, the same goes for his proofs.
Plus, the two books are the second and third in a triology the first being his “Introduction to Topological Manifolds”so they were really meant to be read in this order.
Of course, I also agree that Guillemin and Pollack, Hirsch, and Milnor are great supplements, and will probably emphasize some of the topological aspects that Lee doesn’t go into. Like the other posters, I think Lee’s books are fantastic. I’d start with his Introduction to Smooth Manifolds. For differential geometry, I’d go on to guilpemin Riemannian Manifolds and then follow up with do Carmo’s Riemannian Geometry.
That’s what I did. Guillemin and Pollack’s “Differential Topology” is about the friendliest introduction to the subject you could hope for.
It’s an excellent non-course book. Good supplementary books would be Milnor’s “Topology from a differentiable viewpoint” much more terseand Hirsch’s “Differential Topology” much more elaborate, focusing on the key analytical theorems. For topoloyy geometry it’s much more of a mixed bag as it really depends on where you want to go.
I’ve always viewed Ehresmann connections as the fundamental notion of connection. But it suits my tastes. But I don’t know much in the way of great self-learning differential geometry texts, they’re all rather quirky special-interest textbooks or undergraduate-level grab-bags of light topics.
I haven’t spent any serious amount of time with the Spivak books so I don’t feel comfortable giving any advice on them. About 50 of these books are 20th or 21st century books which would ane useful as introductions to differential geometry. I give some brief indications of the contents and suitability of most of the books in this list. It’s currently a free and legal download.
Guillemin & Pollack, Differential Topology | Pearson
Guilllemin an entry level text and the prior responders have put a lot of effort into giving outstanding suggestions. But I thought it might be of interest.
You can look at it on Google books to decide if it fits your style. If you are a Mathematica user, I think this is a wonderful avenue for self-study, for you can see and manipulate all the central constructions yourself. I use Gray’s code frequently; I was a fan. Here is how he died: I would recommend Jost’s book “Riemannian geometry and pollac analysis” as well as Sharpe’s “Differential geometry”. The first book is pragmatically written and guides the reader to a lot of interesting stuff, like Hodge’s theorem, Morse guilleemin and harmonic maps.
The second book is mainly concerned with Cartan diffeeential, but before that it has an excellent chapter on differential topology. Furthermore it treats Ehresmann connections in appendix A. I highly recommend the following Differential Geometry book. This is a very informative book which covers many important topics including nature and purpose of differential geometry, a concept of mapping, coordinates in Fifferential space, vectors in Euclidean space, basic rules of vector calculus in Euclidean space, tangent and normal plane, osculating plane, involutes, and evolutes, Bertrand curves etc.
Home Questions Tags Users Unanswered. Teaching myself differential topology and differential geometry Ask Question. Harddaysknight 3 6 3. I enjoyed do Carmo’s “Riemannian Geometry”, which I found very readable. Of course there’s much more to differential geometry than Riemannian geometry, but it’s a start This book is probably way too easy for you, but I learned differential geometry from Stoker and I really love this book even though most people seem to not know about it.
I personally found de Carmo to be a nice text, but I found Stoker to be far easier to read.
I think a lot of the important results are in this book, but you will have to look elsewhere for the most technical things. Again, possibly at too low a level, but everything I know about algebraic geometry I learned from working through Cox, Little, and O’Shea.
This book is great for self study, in my opinion. I have tried to read the major algebraic geometry texts, but they are way over my head; this book on the other hand always makes complete sense to me. A word of advice: It’s about pages of not-so-easy complex analysis review.
Or, do get caught up in it, if that’s your thing.
You can see their table of contents at Amazon. That’s certainly a nice list! But your amazon link doesn’t work.
Would the list you recommend help me or should I start reading guilemin basics books? I am going to take abstract algebra, complex analysis, and analysis 1 next semester. Then, books like Runde’s and Munkres’ on topology will be at your level and you should by all means try them.
Keep studying and everything will be at your reach! I can’t help you with algebraic geometry. As you seem to know a bit a lot about this, could you suggest what would be a nice book to start with if someone is interested in harmonic analysis fopology PDEs and wants to know how to do this kind of stuff on non-Euclidean spaces I guess that is what Diff Geom is about?
Also, do you have a reference where diffrrential things are applicable in PDE or harmonic analysis?