Advanced Differential Geometry Textbook

Question

I tried this post on StackExchange with no luck. Hopefully the experts at MathOverflow can help.

In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual graduate courses.

They are Switzer Algebraic Topology: Homology and Homotopy and Whitehead Elements of Homotopy Theory. These are both excellent books that (theoretically) give you overviews and introduction to most of the main topics that you need for becoming a modern researcher in algebraic topology.

Differential Geometry seems replete with excellent introductory textbooks. From Lee to do Carmo to so many others.

Now you might be thinking that Kobayashi/Nomizu seems natural. But the age of those books is showing in terms of what people are really doing today compared to what you learn from using those books. They just aren't the most efficient way to learn modern differential geometry (or so I've heard).

I am looking for a book that covers topics like Characteristic Classes, Index Theory, the analytic side of manifold theory, Lie groups, Hodge theory, Kahler manifolds and complex geometry, symplectic and Poisson geometry, Riemmanian Geometry and geometric analysis, and perhaps some relations to algebraic geometry and mathematical physics. But none of these topics completely, just as Switzer does with a unifying perspective and proofs of legitimate results done at an advanced level, but really as an introduction to each of the topics (Switzer does this with K-theory, spectral sequences, cohomology operations, Spectra...).

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.

Any recommendations for great textbooks/monographs would be much appreciated!

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 1989 Proceedings to the present three volumes". However most of the articles are research level articles and lack the coherence and unified vision of a textbook/monograph.

Answer 1

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.)

• Bishop/Crittenden, "Geometry of manifolds" (1964). Quite advanced, although not too difficult, despite the 1964 date.
• Cheeger/Ebin, "Comparison theorems in Riemannian geometry" (1975). This is on the boundary between textbook and monograph. Definitely advanced, despite the 1975 date.
• Greene/Wu, "Function theory on manifolds which possess a pole" (1979). Monograph/textbook about function theory on Cartan-Hadamard manifolds, including extensive coverage of Kähler manifolds.
• Schoen/Yau, "Lectures on Differential Geometry" (1994). This is about as advanced as it gets. You need to read at least 5 other DG books before starting this one.
• Theodore Frankel, "The geometry of physics: An introduction" (1997, 1999, 2001, 2011). This has lots of advanced DG, but in the physics applications, not so much on topological DG questions.
• Peter Petersen, "Riemannian geometry" (1998, 2006). Very definitely advanced. You need to read at least 3 other DG books before this one.
• Serge Lang, "Fundamentals of differential geometry" (1999). This is definitely advanced, although it nominally starts at the beginning. It's what I call a "higher viewpoint" on DG. Very thorough and demanding.
• Morgan/Tián, "Ricci flow and the Poincaré conjecture" (2007). Advanced monograph on the Poincaré conjecture solution, but written almost like a textbook.
• Shlomo Sternberg, "Curvature in mathematics and physics" (2012). Definitely advanced. On the boundary between DG and physics.

I would say that all of these books are beyond the John M. Lee and Do Carmo textbook level.

Answer 2

Honestly, no one needs ONE book which cover all the topics on your list. Say for Riemmanian Geometry and Geometric Analysis I would suggest

• Cheeger--Ebin "Comparison theorems in Riemannian geometry"
• Burago--Burago--Ivanov "Metric Geometry"
• Gromov "Sign and geometric meaning of curvature"
• Berger "A Panoramic View of Riemannian Geometry"

Answer 3

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:

• Chapter 1: An Introduction to fiber bundles (principal and associated bundles, vector bundles and section)
• Chapter 2: Connection theory for vector bundles
• Chapter 3: Riemannian vector bundles (Levi-Civita connection, Gauss-Bonnet theorem)
• Chapter 4: Harmonic Theory (Laplace-Beltrami operator, Chern's formula for the Laplacian)
• Chapter 5: Geometric vector fields on Riemannian manifolds (harmonic fields, Killing fields, conformal fields, affine fields, projective fields)
• Chapter 6: Lie groups (Lie algebras, homegeneous spaces)
• Chapter 7: Symmetric spaces
• Chapter 8: Symplectic and Hermitian vector bundles (complex manifolds, curvature on Kähler manifolds)
• Chapter 9: Other differential geometric structures (parallelsm on principal fibre bundles, holonomy and curvature, Cartan connections, spin structures)

Answer 4

Let me mention Peter Michor's great books

• Peter W. Michor: Topics in Differential Geometry. Graduate Studies in Mathematics, Vol. 93 American Mathematical Society, Providence, 2008.
• Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997. 618 pages. Zbl 889.58001, MR 98i:58015
• Ivan Kolár, Jan Slovák, Peter W. Michor: Natural operations in differential geometry. Springer-Verlag, Berlin, Heidelberg, New York, (1993), vi+434 pp., MR 94a:58004, ZM 782:53013.

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.

Answer 5

Unless I missed it, nobody has mentioned my favourite book in Differential Geometry: Arthur L. Besse's Einstein Manifolds. Despite the name, it is about a lot more than Einstein manifolds. It covers the state of the art circa 1987, 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.

Answer 6

I would check out "Heat Kernels and Dirac Operators" by Berline, Getzler, and Verne. It covers quite a bit of territory:

-Characteristic classes: Much stronger than most books; develops Chern-Weil theory in the setting of principal bundles, includes the equivariant case

-Index theory: This is one of the standard textbooks for the heat kernel proof of the index theorem and local index theory in general

-Lie groups: Proves the Weyl and Kirillov character formulas

-Kahler manifolds & complex geometry: Proves the Riemann-Roch formula as a special case of the index theorem, but otherwise not much

-Not much symplectic/poisson geometry (though maybe a little in the discussion of coadjoint orbits)

-Riemannian geometry: Proves Gauss-Bonnet-Chern and does some serious computations with curvature, but no comparison theorems

-Geometric analysis: heat kernels and Dirac operators are after all the theme of the book, but there's not really much discussion of standard elliptic operator theory or pseudo-differential operator theory, and there are no nonlinear operators

For the areas where the coverage is poorer - Riemannian geometry, complex manifolds / algebraic geometry, symplectic / Poisson geometry, non-linear geometric analysis - a more focused book is probably required because the techniques are much more specialized. For Riemannian geometry you want the comparison theorems and discussion of non-smooth spaces (e.g. 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).

Answer 7

I strongly recommend "Principles of Algebraic Geometry," by Griffiths and Harris.

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 1,000.

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.e., analytic) functions of a (single) complex variable, then this might be the book for you.

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. In fact, Griffith and Harris could be viewed as a nice introduction to Hodge Theory and Complex Analytic Geometry by Claire Voisin. 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 - 800 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.

Kahler manifolds are introduced and discussed in the first chapter, "Foundational Material."

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.g. Also are spectral sequences and things like that. Also "resolutions." There are probably "folk theorems" lying about that would make a good dissertation. 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.

Deacon John Aiken, PhD in mathematical physics, 1972, LSU.

Louis Nirenberg was my advisor at the Courant Institute, NYU where I obtained my masters.

Answer 8

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.

Answer 9

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.

Answer 10

A nice recent (2017) book that covers some of items listed in the question is Differential Geometry: Connections, Curvature, and Characteristic Classes by Loring W. Tu.

Answer 11

Concerning Kähler manifolds:
Looking through the indexes of the DG books on my bookshelf, I found Kähler manifolds mentioned in only two:

• Walter Poor (1981), pages 262–273 (of which pages 270–273 is a section titled "the curvature of Kähler manifolds"),
• Greene and Wu, "Function theory on manifolds which possess a pole", Springer-Verlag 1979, which has substantial coverage of Kähler manifolds, although this is more of a monograph than a textbook.

Most of your other topics are fairly widely covered.

Answer 12

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.