I am blown away by exposition of analytic manifolds in Serre's Lie algebras and Lie groups, I want more! Is there a text that treats classic topics of differential geometry like connections, Riemannian metrics, holonomy, de Rham cohomology, (almost) complex manifolds, homogeneous spaces, symmetric spaces in such a manner? Or maybe some more analytic manifolds, towards the modern research topics?

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    $\begingroup$ mathoverflow.net/questions/14877/… $\endgroup$ – Ryan Budney Sep 30 '11 at 14:56
  • $\begingroup$ @RyanBudney, IMHO getting rid of atlases is not imperative, both Serre and Lang kept them in their respective texts and it was fine. $\endgroup$ – Alexei Averchenko Sep 30 '11 at 15:00
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    $\begingroup$ Warner's Foundations of Differentiable Manifolds and Lie Groups emphasizes the structure sheaf approach. I found this to be quite amazing to read after I had the basics down (from John Lee's books). It isn't quite what you're looking for and doesn't do most things that differently, but worth a look in my opinion. $\endgroup$ – Matt Sep 30 '11 at 15:38
  • $\begingroup$ Oops. I should have clicked that link before commenting. I guess that is basically what the old post was about. $\endgroup$ – Matt Sep 30 '11 at 15:41

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