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.
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Many of the topics you mention are treated, so I would still say that those books are advanced enough. Besse's Einstein Manifolds. 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.
Differential Geometric Structures by Poor Walter a
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. Homotopy theory is barely mentioned. There is no p-adic analysis or finite fields. Everything is over the real or complex numbers. 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. 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.
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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.
G-H only gives the well-known method of computing them from differential geometry. You mentioned that you are interested in becoming a researcher in algebraic topology.
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It seems to me that the most fruitful field for a researcher in algebraic topology these days is algebraic geometry. For example, category theory is involved in essential ways in a.
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Also are spectral sequences and things like that. Also "resolutions. The key here is to find an advisor in algebraic geometry who publishes a lot. For example, a friend of mine who is a recent graduate in algebraic geometry tells me that there is no Kunneth formula in the theory of motives. To me, that looks like an interesting research area for an algebraic topologist right there. Special submanifolds e.
The Atiyah-Singer Index theorem provides a deep link between invariants associated to a linear differential operator on a manifold and algebraic topological invariants. This has led to a general study of index theories, which includes the development of new cohomology theories and analysis on path and loop spaces. Skip to main content Skip to navigation Differential Geometry is the study of geometric structures on manifolds.
Geometric structures Riemannian structures in which an inner product is specified on the tangent space at each point of the manifold. Symmetric structures, including hyperbolic structures.
Geometric structures on complex manifolds
Examples of manifolds include surfaces in 3-space, complex projective space, and matrix Lie groups e. The way in which coordinate charts are pieced together give the manifold a differentiable structure or a complex structure. Riemannian structures in which an inner product is specified on the tangent space at each point of the manifold. This allows us to measure length of curves, angles and most importantly, to define curvature.
Symplectic structures in which a closed skew-symmetric bilinear form is specified on the tangent space at each point of the manifold. This structure was inspired by considerations from classical mechanics Poisson bracket, etc.