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I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology.

I have decided to fix this lacuna once for all. Unfortunately I cannot attend a course right now. I must teach myself all the stuff by reading books.

Towards this purpose I want to know what are the most important basic theorems in differential geometry and differential topology. 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?

Also finally I want to read into some algebraic geometry and Hodge/Kähler stuff.

Suggestions about important theorems and concepts to learn, and book references, will be most helpful.

  • 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...2010-12-09
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    This book is probably way too easy for you, but I learned differential geometry from [Stoker](http://books.google.com/books?id=5Qk7WwXePuQC&printsec=frontcover&dq=stoker+differential+geometry&hl=en&ei=wSsATdnEFsb_lgeC_ajxCA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCsQ6AEwAA#v=onepage&q&f=false) 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.2010-12-09
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    Again, possibly at too low a level, but everything I know about algebraic geometry I learned from working through [Cox, Little, and O'Shea](http://books.google.com/books?id=yCsDO425PC0C&printsec=frontcover&dq=computational+algebraic+geometry&hl=en&ei=YS4ATb-wLYOKlwf-2PW3Dg&sa=X&oi=book_result&ct=result&resnum=7&ved=0CEkQ6AEwBg#v=onepage&q&f=false). 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.2010-12-09
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    Also, Griffiths & Harris is a pretty standard "classical algebraic geometry" book. A word of advice: don't get caught up in chapter 0. It's about 100 pages of not-so-easy complex analysis review. (Or, do get caught up in it, if that's your thing.)2010-12-09
  • 0
    Are there any good courses videos of MIT/standford etc.?2013-02-22
  • 1
    I don't see why the edit of this question was approved. @rlgordonma?2013-02-22

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