What are some good sources to learn about real analytic manifolds?

Question

Asking this question as someone with a graduate student level understanding of smooth differential/Riemannian geometry (May be a bit more that Riemannian Geometry by Do Carmo). I am trying to upgrade my knowledge to the real-analytic set up.

Also my target is not to go in the direction of complex algebraic geometry (like Griffiths-Harris). I am primarily interested in building "analytic diffeomorphisms" and "analytic flows" on abstract analytic manifolds. I am interested in some basic materials which will eventually lead me in that direction.

Another question I asked, that may be more precise: What are some general strategies to build measure preserving real-analytic diffeomorphisms?

Answer 1

Till someone who actually knows something gets here, I will try to summarize some haphazard information I was collected:

Some articles (non survey styles) which might be useful:

(1) From a differentiable to a real analytic perturbation theory, applications to the Kupka Smale theorems - H.W. Broer and F.M. Tangerman.

(2) A non-stabilizable jet of a singularity of a vector field; the analytic case. In Algebraic aand Differential Topology - Global Differential Geometry - F. Takens (There is some kind of analytic partition of unity described here, where do you find this paper?)

(3) New Banach space properties of the disc algebra and $H^\infty$ - J. Bourgain (Also some kind of partition of unity discussed here, too hard to read)

I will keep searching and will add as I find for information.

Answer 2

This is a group of references, so to say, for basic foundations on the field. Whenever one deals with real analytic functions, one gets involved with complex analytic functions and analytic sheaves. Even for differentiable matters this is true. Good examples are Malgrange's and Tougeron's classical books on Ideals of differentiable functions, or Bierstone-Milman solution of Glaeser's Problem (I mention this last as a top research example). Also, in my own experience, to do anything with real analytic functions one needs Complex Analytic Geometry. For this I would recommend Naramsimhan's Theory of Analytic Spaces, Gunning-Rossi's Several Complex Variables, or a book by Whitney, I'm afraid a bit forgotten: Complex Analytic Varieties. To put a context to this comment, which may be far form your specific interests (and then I apologize), I refer to Is there a field of 'real analytic geometry'? Another important matter is approximation: to what extent one can make analytic anything differential. The basic tool is Whitney's Approximaton Thm, and a very good reference for this is Narasimhan's Real and Complex Analysis.