3

I'm looking for good books in English about differentiable manifolds in finite dimensional spaces, as well as Banach manifolds (optionally, containing some information about calculus in Banach spaces). As for my general math knowledge, I'm familiar with some advanced topics like vector calculus, measure theory and functional analysis, so I'm not looking for an elementary level book (I even wonder if there exists any). By «good» I mean the following:

  • providing a large amount of illustrations while describing finite dimensional theory;
  • having unrigorous explanations before diving into formalizing statements;
  • containing exercises to do.

As a reference, I can come up with «A Geometric Approach to Differential Forms» by David Bachman. Even though it is extremely unrigorous in some places, I found it to be a good introduction to differential forms, which helped to start reading more formal books on this topic.

Smiley1000
  • 4,219
Yalikesi
  • 2,865
  • Banach manifolds? Such things exists? There is a difference with Riemannian manifolds? EDIT: the last book about "Banach manifolds" that I've found is of 1978... it seems that this topics is slightly discontinued –  Sep 05 '21 at 20:11
  • 1
    @Masacroso Banach manifolds are kind of generalization of finite-dim manifolds, in sense that they are required to be "locally Banach", while ordinary manifolds are "locally $\mathbb{R}^n$". However, I think that some things may generalize the same way finite-dim calculus generalizes to calculus in Banach spaces (i.e. defining limits using norm, derivatives as linear operators and so on). So, I will be glad to get at least some finite-dim recommendations :) – Yalikesi Sep 05 '21 at 20:33
  • 2
    I don't know of any "good" book by your definition, but the book on differentiable manifolds by Serge Lang does treat Banach manifolds. There are some minor changes here and there, for example, one requires the kernel of the derivatives of a submersion to be complemented before getting special "coordinates" (whatever this means now), etc. – Ivo Terek Sep 05 '21 at 20:38
  • 2
    For basic dufferential calculus in Banach spaces, there are lots of sources. Here are a few: Differential Calculus by Cartan, Volume 2 of Mathematical Analysis by Zorich, Foundations of Mathematical Analysis by Dieudonné, Advanced Calculus by Loomis and Sternberg. For manifolds, there aren't as many, but I know Lang's Fundamentals of Differential Geometry does things in this level of generality. I haven't looked at Bourbaki's Fascicule de résultats on manifolds (which omits proofs) in a long time, so I'm not sure what the level of generality is. – Anonymous Sep 05 '21 at 20:41
  • @IvoTerek I've looked through Lang's book, but found it inconvenient to start reading with some intro to category theory. Can this book be read further without diving into cat. th.? – Yalikesi Sep 05 '21 at 20:47
  • Lang uses Category Theory language, but not Category Theory per se (if you know what a category, functor, and natural transformation are, that's really enough). Out of the books Anonymous just pointed out, apart from Lang's I had only heard of the one by Cartan (but never explored it). Anything by Bourbaki should be extremely general. They seem like good suggestions. – Ivo Terek Sep 05 '21 at 20:54
  • 1
    @IvoTerek I’ve got it, thanks. As Lang references to Dieudonné, I think it’s better to start with his “Foundations”. – Yalikesi Sep 05 '21 at 21:00
  • 1
    @Masacroso: Mathoverflow has a Banach manifold tag (maybe not always used as here, however). See also Reference on Infinite Dimensional Manifold AND Reference request: infinite-dimensional manifolds AND How are infinite-dimensional manifolds most commonly treated? AND Refrence Request - Book for Manifolds over Banach Spaces. – Dave L. Renfro Sep 05 '21 at 21:25
  • @Dave thanks, I will try to read about infinite-dimensional manifolds, seems interesting –  Sep 06 '21 at 14:46
  • Related: https://math.stackexchange.com/q/5029969/491450 , https://math.stackexchange.com/q/295358/491450 , https://math.stackexchange.com/q/2227945/491450 , https://math.stackexchange.com/q/1272537/491450 , https://math.stackexchange.com/q/705175/491450 , https://math.stackexchange.com/q/2080992/491450 , How are infinite-dimensional manifolds most commonly treated? (MathOverflow) ... – Smiley1000 Jan 31 '25 at 16:25
  • ... Infinite dimensional Riemannian geometry (MathOverflow) – Smiley1000 Jan 31 '25 at 16:25

0 Answers0