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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?

Martin Sleziak
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Aleksei Averchenko
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    http://mathoverflow.net/questions/14877/how-much-of-differential-geometry-can-be-developed-entirely-without-atlases-clo – Ryan Budney Sep 30 '11 at 14:56
  • @RyanBudney, IMHO getting rid of atlases is not imperative, both Serre and Lang kept them in their respective texts and it was fine. – Aleksei Averchenko Sep 30 '11 at 15:00
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    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. – Matt Sep 30 '11 at 15:38
  • Oops. I should have clicked that link before commenting. I guess that is basically what the old post was about. – Matt Sep 30 '11 at 15:41

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