[Math] 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.