10
$\begingroup$

My current research has brought me into an area the requires me to learn some infinite dimensional Riemannian and Kähler geometry. Can someone recommend some good books or survey articles to help me get started with the basics? I've seen Lang's book but that's it.

$\endgroup$
8
  • 2
    $\begingroup$ For infinite dimensional smooth and Riemannian manifolds, Serge Lang's books are popular as a start. $\endgroup$ Mar 30, 2014 at 2:30
  • 4
    $\begingroup$ Why don't you list (in the question) the books you've already "seen", then? $\endgroup$ Mar 30, 2014 at 8:35
  • 3
    $\begingroup$ Klingenberg's Riemannian geometry was recommended reading when I was an undergrad, and uses manifolds modeled on Banach spaces. It might help to say a bit more about what you need to learn. $\endgroup$
    – S. Carnahan
    Mar 30, 2014 at 21:53
  • 2
    $\begingroup$ Andrew Stacey's 2007 talk "Variations on a Theme: Riemannian Geometry in Infinite Dimensions" is eminently digestible: only 7 pages. PDF download $\endgroup$ Mar 30, 2014 at 23:18
  • 1
    $\begingroup$ I've been alerted to this question. The right thing to read depends on the context. What infinite dimensional spaces are you interested in? Specifically, what model spaces for your spaces? (Please edit your question with that information.) $\endgroup$ Apr 2, 2014 at 16:56

2 Answers 2

10
$\begingroup$

Lempert, László The Dolbeault complex in infinite dimensions. III. Sheaf cohomology in Banach spaces. Invent. Math. 142 (2000), no. 3, 579-603.

Lempert, László The Dolbeault complex in infinite dimensions. II. J. Amer. Math. Soc. 12 (1999), no. 3, 775-793.

Lempert, László The Dolbeault complex in infinite dimensions. I. J. Amer. Math. Soc. 11 (1998), no. 3, 485-520.

Hamilton, Richard S. The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65–222.

$\endgroup$
1
  • $\begingroup$ Thanks! A colleague just turned me on to the Hamilton paper, it seems to be a good entry point for the subject. $\endgroup$
    – Wintermute
    Apr 3, 2014 at 11:23
4
$\begingroup$

I would also suggest the beautiful paper by Arnold

http://www.ams.org/mathscinet-getitem?mr=202082

and the book by

Kriegl-Michor: The convenient setting of global analysis,

see also the references therein.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.