Advanced beginners textbook on Lie theory from a geometric viewpoint

Question

There are several questions resembling this one but none of them are quite the same I believe.

I have a background in differential geometry and topology, as well as analysis (locally convex spaces). I have a very basic familiarity with lie groups since they tend to pop out quite a lot in differential geometry.

Lately I found out I have a tendency to think about a lot of problems in differential geometry in terms of lie groups. Unfortunately whenever I do that kind of thing i get stuck due to a lack of solid background.

I've decided to pick up a serious text on lie theory and fill this gap. I prefer a book that would help me understand the geometry better rather than the algebraic framework. Here is a list of the applications I have in mind for the theory.

• Recognizing homogeneous spaces, describing them in terms of coset spaces and proving stuff about them using that information. (For example the answer to this MSE question)

• Spin geometry and symplectic geometry.

• G-bundles and gauge theory.

• Holonomy groups and Riemannian symmetric spaces.

Prefarably the book would contain some exercise problems besides the theory.

Answer 1

Walter A. Poor's text, "Differential Geometric Structures" hits all the points you mentioned above in various amounts of detail. Lie groups and homogeneous spaces are discussed in Chapter 6, symplectic geometry in Chapter 8, principal bundles and spin geometry in Chapter 9, symmetric spaces in Chapter 7, and holonomy in a variety of places throughout. The point of view of the whole book is to think of "geometric structures" broadly as a notion of parallel transport of information along curves. It's also a Dover book, so you can get it on Amazon for less than $20, likely including shipping.

However, it's not a book on Lie theory per se. For a geometric introduction to Lie theory, maybe try Wulf Rossmann's, "Lie Groups: An Introduction Through Linear Groups" or John Stillwell's, "Naive Lie Theory." Stillwell's book in particular takes a hands-on geometric approach, including pictures and explicit calculations. You should be able to see the contents and read the introductions to both books on Amazon if you want a feel for whether these would be good starting points for you.

Answer 2

Relates issues:

• Exponential of a function times derivative
• How to properly apply the Lie Series
• How to derive these Lie Series formulas

Don't know how to combine "advanced" and "beginners" textbook in the first place; because I think that's more or less a contradictio in terminis. But anyway, my absolute favorite is the following one, with a lot of geometry contained in it indeed. But, what's more important, written by someone who has been close to the founder of Lie group theory himself:

• Sophus Lie, Vorlesungen über Differentialgleichungen mit bekannten Infinitesimalen Transformationen, bearbeitet und herausgegeben von Dr. Georg Wilhelm Scheffers,
  Leipzig (1891). Availability: Amazon.

It's written in German. Don't know if that may be called "unfortunately".

Answer 3

You may enjoy the following reference as I do:

"Structure and Geometry of Lie Groups" by Joachim Hilgert and Karl-Hermann Neeb, Springer Monographs in Mathematics, 2012.