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Greene and Yau say: "It is our hope that the three volumes of these proceedings, taken as a whole, will provide a broad overview of geometry and its relationship to mathematics in toto, with one obvious exception; the geometry of complex manifolds Thus the reader seeking a complete view of geometry would do well to add the second volume on complex geometry from the Proceedings to the present three volumes".

Concerning advanced differential geometry textbooks in general: There's a kind of a contradiction between "advanced" and "textbook". By definition, a textbook is what you read to reach an advanced level.

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A really advanced DG book is typically a monograph because advanced books are at the research level, which is very specialized. Anyway, these are my suggestions for DG books which are on the boundary between "textbook" and "advanced". These are in chronological order of first editions. Honestly, no one needs ONE book which cover all the topics on your list.

Alan Kennington's very extensive list of textbook recommendations in differential geometry offers several suggestions, notably. Serge Lang, Fundamentals of differential geometry. Walter Poor, Differential geometric structures , with contents:. Let me mention Peter Michor 's great books. There are more lecture notes and books on his publications page.

Over time, I looked up various advanced topics in those books above, and found the explanations quite readable, even so I'm not an expert in differential geometry.

Fundamentals of Differential Geometry - Serge Lang.pdf

Many of the topics you mention are treated, so I would still say that those books are advanced enough. Besse's Einstein Manifolds.

Differential Geometry - Math History - NJ Wildberger

Despite the name, it is about a lot more than Einstein manifolds. It covers the state of the art circa , so bear that in mind, but it has a wealth of material and behind Besse lies a collective of some of the foremost differential geometers of the time. It covers quite a bit of territory:. For Riemannian geometry you want the comparison theorems and discussion of non-smooth spaces e. Burago-Burago-Ivanov is great. For complex manifolds you want a discussion of sheaf cohomology and Hodge theory probably Griffiths and Harris is best, but I like Wells' book as well.

For symplectic manifolds you want some discussion of symplectic capacities and the non-squeezing theorem I think McDuff and Salamon is still the best here, but I'm not sure. This book comes the closest to covering the wide range of topics in which you are interested. At least, it comes the closest of all the books of which I am aware. I have been studying all the topics you mentioned reasonably intensively for the last 50 years, so, that's a lot of books. At least 1, The title is a little misleading, this book is more about differential geometry than it is about algebraic geometry.

However, it does cover what one should know about differential geometry before studying algebraic geometry.

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Also before studying a book like Husemoller's Fiber Bundles. If you know a little algebraic topology - like the definition of the homology and cohomology groups - and if you have a basic understanding of holomorphic i. It would be good and natural, but not absolutely necessary, to know Differential Geometry to the level of Noel Hicks "Notes on Differential Geometry," or, equivalently, to the level of Do Carmo's two books, one on Gauss and the other on Riemannian geometry.

Of course, you need the prerequisites for do Carmo's books before you are ready for Griffith and Harris. Regarding algebraic topology, very little is required because this book explains a lot of the basic material, such as the Kunneth formula. Regarding several complex variables, this book picks it up from the beginning, even explaining Oka's lemma.

No background is needed. On the other hand, the book is not too advanced either. K-theory is not touched on.

Fundamentals of Differential Geometry (Graduate Texts in Mathematics) - PDF Free Download

Homotopy theory is barely mentioned. There is no p-adic analysis or finite fields. Everything is over the real or complex numbers.

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BTW, Voisin has by far the best explanation of what sheaves are all about that I have ever seen. I recommend reading that before reading Griffith and Harris's explanation.

Fundamentals of Differential Geometry

There is one requirement you mentioned that you mentioned that this book does not exactly qualify for. Namely that if give light treatments. However, if the book is so excellently written that you can skip the proofs and probably get the kind of "surface" understanding that you are looking for. Don't let the size - pages - discourage you. The page count if you omit the proofs is a lot less. Also there are some topics, like "the quadratic line complex," 70 pages, that you might not be interested in.

A drawback of the discussion of Chern classes omits the intuitive Fiber bundle explanation. Algebraic geometry: a first course Graduate Texts in Mathematics Riemannian geometry Graduate Texts in Mathematics , 2nd Edition.

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