11

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.

Martin Sleziak
  • 53,687
Saal Hardali
  • 4,759
  • 2
    begginer=novice that implore to learn? – Taladris Oct 21 '15 at 14:28
  • Helgason's book "Differential Geometry, Lie Groups, and Symmetric Spaces" hits many of those bullets. – Matthew Leingang Oct 21 '15 at 14:33
  • spelling of beginner double n, not double g. – Narasimham Oct 21 '15 at 14:42
  • 1
    @MatthewLeingang I tried Helgason. After overcoming the initial phase of being daunted by the level of detail I realized it's overly sofisticated for what I need. It would really better serve me after I have a good grounding in the basic theory and know exactly what to look for in there. – Saal Hardali Oct 21 '15 at 23:26
  • 1
    @SaalHardali I've had a similar problem. I wanted the exact same thing described in the title of your question. But I'm sorry to tell you I did not find anything suitable. – Aloizio Macedo Oct 21 '15 at 23:46
  • I'm not certain one needs much Lie theory to understand a whole plethora of symplectic geometry. If I recall correctly, there's quite a lot of basic Lie group/homogeneous space theory in standard differential geometry texts like Lee's "Introduction to Smooth Manifolds". Do you wish to go beyond that? For the record, I don't think many people have spent a lot of time thinking about more than one thing on your proposed list of applications. – PVAL-inactive Oct 21 '15 at 23:55
  • @AloizioMacedo Did you find anything suitable after all these years? I'm having the same trouble you both had. I don't really like the very algebraic path which most of the books seems to follow. I'm not sure at all if such book exists, but I'd be willing to find it (or wait for someone to write it someday) – EternalBlood Jul 04 '18 at 04:52
  • @EternalBlood I think I realized over the years 2 slightly conflicting things 1) After the fundmentals which are in most good diff. geometry textbooks one should study the compact case first (as it is much simpler) 2) The compact case is almost entirely controlled by algebra/combinatorics. For this reason I think its a difficult subject from a pedagogy POV – Saal Hardali Jul 04 '18 at 04:58

3 Answers3

4

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.

Brian Klatt
  • 2,212
  • While this book looks very promising I added a bounty since i feel this question (regarding the how and what of studying lie theory) could use a variety of expert opinions. – Saal Hardali Oct 23 '15 at 16:22
4

Relates issues:

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".

enter image description here

Han de Bruijn
  • 17,070
2

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.

Lucas Seco
  • 291