The only book I have found that is sort of along these lines is Nicolaescu's Lectures on the Geometry of Manifolds , but this book misses many topics. This was inspired by page viii of Lee's excellent book: link where he lists some of these other topics and almost implies that they would take another volume. I'm wondering whether that advanced volume exists. Edit : there are many excellent recommendations I particularly like the Index theory text mentioned by Gordon Craig in the comments as it doesn't shy away from analysis, and does so many things in geometry plus has extensive references below.
One other reference that I found which people may find interesting is the following: link and link2 where Prof. 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. 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.
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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. 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. 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. 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. 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. 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. If one looks for such a wide variety of arguments in a single text he will have, of course, to miss something from the point of view of how deep the text is going into details. I find that a very intriguing balance between variety, deepness and details is obtained by the three-volumes text by Dubrovin, Novikov, Fomenko: Modern Geometry.
Other interesting texts in this perspective are those aimed at physicists like Nakahara: Geometry, Topology and Physics and Schutz: Geometrical Methods of Mathematical Physics , together with the text by Frankel already mentioned in other comments. I'm not sure either how advanced you'd consider this or how much of your interests it covers, but I recently spent some time referring to Greub, Halperin, and Vanstone's Connections, Curvature, and Cohomology.
I'll also put in a second for Wells's Differential Analysis on Complex Manifolds, which is very readable. Parabolic Geometries by Cap and Slovak is a good introduction to Cartan geometry, which includes Riemannian geometry and more specialized parabolic geometries such as projective and conformal geometry. It gives you a good general picture of many of the geometries people study today from the point of natural differential operators, Lie groups, Lie algebras, and representation theory. A nice recent book that covers some of items listed in the question is Differential Geometry: Connections, Curvature, and Characteristic Classes by Loring W.
Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. This work was done, for the first time, by Buchin Su , myself , and myself and R. Mishra [19, 20]. Symmetric and recurrent Finsler manifolds The concept of Kappa spaces introduced by H. Ruse , and later extended by A.
Walker  in the name of Recurrent Riemannian spaces has been further extended to Finsler geometry and to the generalized metric and non-metric spaces. A class of Finsler manifolds, in which the covariant derivative of a non-null curvature tensor is expressible as an inner product of the curvature tensor and an arbitrary non-null covariant vector field, is said to be recurrent. Such manifolds are studied by A. Mishra and H.
Pande , R. Sen , myself , myself and F. Meher  and others.go to link
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Infinitesimal transformations preserving parallelism of a pair of vectors are called affine motions and have been studied for above kind of Finsler manifolds by me ; myself and F. Meher [13, 15, 18] and myself, R. Mishra and Nawal-Kishore . Infinitesimal transformations preserving the geodesic character of curves define projective motions.
Such transformations are studied, in different types of Finsler manifolds discussed above, by me , myself and F. Meher [14, 16] and myself, Nawal-Kishore and P. Pandey . Study of motions, affine motions, projective motions, conformal and homothetic motions generated by infinitesimal transformations of different types in Finsler manifolds is very vast and fruitful for its exploration. The last one is the latest amongst these. Following abstract methods and employing the newest technique much has been done in the geometry of Riemannian manifolds, mainly in the European and American schools, but the study of Finsler manifolds in context with this logical approach has been done by Japanese geometers: Makoto Matsumoto [3, 4, 5]; K.
Yano ; K. Yano and Y. Muto ; K. Yano and T.
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Paris A , France 54 , Study Group of Geometry, Japan 5 Definitions, infinitesimal transformations and isometries. Kyoto Univ.
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